Polyadic systems, representations and quantum groups

POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS STEVEN DUPLIJ arXiv:1308.4060v3 [math.RT] 21 Jan 2018 A BSTRACT. Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomor- phisms, a ’heteromorphism’ which connects polyadic systems having unequal arities, is introduced via an explicit formula, together with related definitions for multiplace representations and multiac- tions. Concrete examples of matrix representations for some ternary groups are then reviewed. Ternary algebras and Hopf algebras are defined, and their properties are studied. At the end some ternary gen- eralizations of quantum groups and the Yang-Baxter equation are presented. C ONTENTS 1. I NTRODUCTION 2 2. P RELIMINARIES 3 3. S PECIAL ELEMENTS AND PROPERTIES OF POLYADIC SYSTEMS 6 4. H OMOMORPHISMS OF POLYADIC SYSTEMS 11 5. M ULTIPLACE MAPPINGS OF POLYADIC SYSTEMS AND HETEROMORPHISMS 13 6. A SSOCIATIVITY QUIVERS AND HETEROMORPHISMS 18 7. M ULTIPLACE REPRESENTATIONS OF POLYADIC SYSTEMS 24 8. M ULTIACTIONS AND G- SPACES 29 9. R EGULAR MULTIACTIONS 31 10. M ULTIPLACE REPRESENTATIONS OF TERNARY GROUPS 33 11. M ATRIX REPRESENTATIONS OF TERNARY GROUPS 37 12. T ERNARY ALGEBRAS AND H OPF ALGEBRAS 40 13. T ERNARY QUANTUM GROUPS 44 14. C ONCLUSIONS 47 R EFERENCES 47 Date: 20 May 2012. 2010 Mathematics Subject Classification. 16T05, 16T25, 17A42, 20N15, 20F29, 20G05, 20G42, 57T05. Published: Journal of Kharkov National University, (ser. Nuclei, Particles and Fields), Vol. 1017, 3(55) (2012) 28-59. 1 2 STEVEN DUPLIJ 1. I NTRODUCTION One of the most promising directions in generalizing physical theories is the consideration of higher arity algebras K ERNER [2000], in other words ternary and n-ary algebras, in which the binary com- position law is substituted by a ternary or n-ary one DE A ZCARRAGA AND I ZQUIERDO [2010]. Firstly, ternary algebraic operations (with the arity n = 3) were introduced already in the XIX-th century by A. Cayley in 1845 and later by J. J. Sylvester in 1883. The notion of an n-ary group was introduced in 1928 by D ÖRNTE [1929] (inspired by E. Nöther) and is a natural generalization of the notion of a group. Even before this, in 1924, a particular case, that is, the ternary group of idempotents, was used in P R ÜFER [1924] to study infinite abelian groups. The important coset theorem of Post explained the connection between n-ary groups and their covering binary groups P OST [1940]. The next step in study of n-ary groups was the Gluskin-Hosszú theorem H OSSZ Ú [1963], G LUSKIN [1965]. Another definition of n-ary groups can be given as a universal algebra with additional laws D UDEK ET AL . [1977] or identities containing special elements RUSAKOV [1979]. The representation theory of (binary) groups W EYL [1946], F ULTON AND H ARRIS [1991] plays an important role in their physical applications C ORNWELL [1997]. It is initially based on a matrix realization of the group elements with the abstract group action realized as the usual matrix multi- plication C URTIS AND R EINER [1962], C OLLINS [1990]. The cubic and n-ary generalizations of matrices and determinants were made in K APRANOV ET AL . [1994], S OKOLOV [1972], and their physical application appeared in K AWAMURA [2003], R AUSCH DE T RAUBENBERG [2008]. In gen- eral, particular questions of n-ary group representations were considered, and matrix representations derived, by the author B OROWIEC ET AL . [2006], and some general theorems connecting representa- tions of binary and n-ary groups were presented in D UDEK AND S HAHRYARI [2012]. The intention here is to generalize the above constructions of n-ary group representations to more complicated and nontrivial cases. In physics, the most applicable structures are the nonassociative Grassmann, Clifford and Lie al- gebras L ÕHMUS ET AL . [1994], L OUNESTO AND A BLAMOWICZ [2004], G EORGI [1999], and so their higher arity generalizations play the key role in further applications. Indeed, the ternary ana- log of Clifford algebra was considered in A BRAMOV [1995], and the ternary analog of Grassmann algebra A BRAMOV [1996] was exploited to construct various ternary extensions of supersymmetry A BRAMOV ET AL . [1997]. The construction of realistic physical models is based on Lie algebras, such that the fields take their values in a concrete binary Lie algebra G EORGI [1999]. In the higher arity studies, the standard Lie bracket is replaced by a linear n-ary bracket, and the algebraic structure of the corresponding model is defined by the additional characteristic identity for this generalized bracket, corresponding to the Jacobi identity DE A ZCARRAGA AND I ZQUIERDO [2010]. There are two possibilities to construct the generalized Jacobi identity: 1) The Lie bracket is a derivation by itself; 2) A double Lie bracket vanishes, when antisymmetrized with respect to its entries. The first case leads to the so called Filippov algebras F ILIPPOV [1985] (or n-Lie algebra) and second case corresponds to generalized Lie algebras M ICHOR AND V INOGRADOV [1996] (or higher order Lie algebras). The infinite-dimensional version of n-Lie algebras are the Nambu algebras NAMBU [1973], TAKHTAJAN [1994], and their n-bracket is given by the Jacobian determinant of n functions, the Nambu bracket, which in fact satisfies the Filippov identity F ILIPPOV [1985]. Recently, the ternary Filippov algebras were successfully applied to a three-dimensional superconformal gauge theory describing the effective worldvolume theory of coincident M2-branes of M-theory BAGGER AND L AMBERT [2008a,b], G USTAVSSON [2009]. The infinite-dimensional Nambu  POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 3 bracket realization H O ET AL . [2008] gave the possibility to describe a condensate of nearly co- incident M2-branes L OW [2010]. From another side, Hopf algebras A BE [1980], S WEEDLER [1969], M ONTGOMERY [1993] play a fundamental role in quantum group theory K ASSEL [1995], S HNIDER AND S TERNBERG [1993]. Previously, their Von Neumann generalization was introduced in D UPLIJ AND L I [2001], D UPLIJ AND S INEL’ SHCHIKOV [2009], L I AND D UPLIJ [2002], their actions on the quantum plane were classified in D UPLIJ AND S INEL’ SHCHIKOV [2010], and ternary Hopf algebras were defined and studied in D UPLIJ [2001], B OROWIEC ET AL . [2001]. The goal of this paper is to give a comprehensive review of polyadic systems and their representa- tions. First, we classify general polyadic systems and introduce n-ary semigroups and groups. Then we consider their homomorphisms and multiplace generalizations, paying attention to their associa- tivity. We define multiplace representations and multiactions, and give examples of matrix represen- tations for some ternary groups. We define and investigate ternary algebras and Hopf algebras, study their properties and give some examples. At the end we consider some ternary generalizations of quantum groups and the Yang-Baxter equation. 2. P RELIMINARIES Let G be a non-empty set (underlying set, universe, carrier), its elements we denote by lower-case Latin letters gi ∈ G. The n-tuple (or polyad) g1 , . . . , gn of elements from G is denoted by (g1 , . . . , gn ). n z }| { 1 The Cartesian product G × . . . × G = G×n consists of all n-tuples (g1 , . . . , gn ), such that gi ∈ G, i = 1, . . . , n. For all equal elements g ∈ G, we denote n-tuple (polyad) by power (g n ). If the number of elements in the n-tuple is clear from the context or is not
important, we denote it with one bold letter (g), in other cases we use the power in brackets g (n) . We now introduce two important constructions on sets. (n) Definition 2-1. The i-projection of the Cartesian product G×n on its i-th “axis” is the map Pri : G×n → G such that (g1 , . . . gi , . . . , gn ) 7−→ gi . Definition 2-2. The i-diagonal Diagn : G → G×n sends one element to the equal element n-tuple g 7−→ (g n ). The one-point set {•} can be treated as a unit for the Cartesian product, since there are bijections between G and G×{•}×n , where G can be on any place. On the Cartesian product G×n one can define a polyadic (n-ary, n-adic, if it is necessary to specify n, its arity or rank) operation µn : G×n → G. For operations we use small Greek letters and place arguments in square brackets µn [g]. The operations with n = 1, 2, 3 are called unary, binary and ternary. The case n = 0 is special and corresponds to (c) fixing a distinguished element of G, a “constant” c ∈ G, and it is called a 0-ary operation µ0 , which (c) (c) maps the one-point set {•} to G, such that µ0 : {•} → G, and formally has the value µ0 [{•}] = c ∈ G. The 0-ary operation “kills” arity, which can be seen from the following B ERGMAN [1995]: the composition of n-ary and m-ary operations µn ◦ µm gives (n + m − 1)-ary operation by µn+m−1 [g, h] = µn [g, µm [h]] . (2.1) (c) Then, if to compose µn with the 0-ary operation µ0 , we obtain (c) µn−1 [g] = µn [g, c] , (2.2) 1We place the sign for the Cartesian product (×) into the power, because the same abbreviation will also be used below for other types of product. 4 STEVEN DUPLIJ because g is a polyad of length (n − 1). So, it is necessary to make a clear distinction between the (c) 0-ary operation µ0 and its value c in G, as will be seen and will become important below. Definition 2-3. A polyadic system G is a set G which is closed under polyadic operations. We will write G = hset|operationsi or G = hset|operations|relationsi, where “relations” are some additional properties of operations (e.g., associativity conditions for semigroups or cancellation prop- erties). In such a definition it is not necessary to list the images of 0-ary operations (e.g. the unit or zero in groups), as is done in various other definitions. Here, we mostly consider concrete polyadic systems with one “chief” (fundamental) n-ary operation µn , which is called polyadic multiplication (or n-ary multiplication). Definition 2-4. A n-ary system G n = hG | µn i is a set G closed under one n-ary operation µn (without any other additional structure). Note that a set with one closed binary operation without any other relations was called a groupoid by Hausmann and Ore H AUSMANN AND O RE [1937] (see, also C LIFFORD AND P RESTON [1961]). However, nowadays the term “groupoid” is widely used in category theory and homotopy theory for a different construction with binary multiplication, the so-called Brandt groupoid B RANDT [1927] (see, also, B RUCK [1966]). Alternatively, and much later on, Bourbaki B OURBAKI [1998] intro- duced the term “magma” for binary systems. Then, the above terms were extended to the case of one fundamental n-ary operation as well. Nevertheless, we will use some neutral notations “polyadic system” and “n-ary system” (when arity n is fixed/known/important), which adequately indicates all of their main properties. Let us consider the changing arity problem: Definition 2-5. For a given n-ary system hG | µn i to construct another polyadic system hG | µ′n′ i over the same set G, which has multiplication with a different arity n′ . The formulas (2.1) and (2.2) give us the simplest examples of how to change the arity of a polyadic system. In general, there are 3 ways: (1) Iterating. Using composition of the operation µn with itself, one can increase the arity from n to n′iter (as in (2.1)) without changing the signature of the system. We denote the number of ℓ iterating multiplications by ℓµ , and use the bold Greek letters µnµ for the resulting composition of n-ary multiplications, such that ℓµ z   }|  { def ×(n−1) ×(n−1) µ′n′ = µℓnµ = µn ◦ µn ◦ . . . µn × id . . . × id , (2.3) where n′ = niter = ℓµ (n − 1) + 1, (2.4) ℓ which gives the length of a polyad (g) in the notation µnµ [g]. Without assuming associativity ℓ there many variants for placing µn ’s among id’s in the r.h.s. of (2.3). The operation µnµ is named a long product D ÖRNTE [1929] or derived D UDEK [2007].  POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 5 (2) Reducing (Collapsing). Using nc distinguished elements or constants (or nc additional 0-ary (c ) operations µ0 i , i = 1, . . . nc ), one can decrease arity from n to n′red (as in (2.2)), such that2  n  z }|c { (c ...c ) def (c ) (c ) µ′n′ = µn′1 nc = µn ◦ µ0 1 × . . . × µ0 nc × id×(n−nc )  , (2.5) where n′ = nred = n − nc , (2.6) (c ) and the 0-ary operations µ0 i can be on any places. (3) Mixing. Changing (increasing or decreasing) arity may be done by combining iterating and reducing (maybe with additional operations of different arity). If we do not use additional operations, the final arity can be presented in a general form using (2.4) and (2.6). It will depend on the order of iterating and reducing, and so we have two subcases: (a) Iterating →Reducing. We have n′ = niter→red = ℓµ (n − 1) − nc + 1. (2.7) The maximal number of constants (when n′iter→red = 2) is equal to nmax c = ℓµ (n − 1) − 1 (2.8) and can be increased by increasing the number of multiplications ℓµ . (b) Reducing →Iterating. We obtain n′ = nred→iter = ℓµ (n − 1 − nc ) + 1. (2.9) Now the maximal number of constants is nmax c =n−2 (2.10) and this is achieved only when ℓµ = 1. To give examples of the third (mixed) case we put n = 4, ℓµ = 3, nc = 2 for both subcases of opposite ordering: (1) Iterating →Reducing. We can put (c ,c )′   µ8 1 2 g (8) = µ4 [g1 , g2 , g3, µ4 [g4 , g5 , g6 , µ4 [g7 , g8 , c1 , c2 ]]] , (2.11) which corresponds to the following commutative diagram ǫ id×6 ×µ4 id×3 ×µ4 G×8 P✲ G×8 × {•}2 ✲ G×4 ✲ G×7 PP PP PP PP µ4 PP (2.12) (c ,c )′ PP µ8 1 2 PP PP ❄ PqP G (2) Reducing →Iterating. We can have (c ,c )′   µ4 1 2 g (4) = µ4 [g1 , c1 , c2 , µ4 [g2 , c1 , c2 , µ4 [g3 , c1 , c2 , g4 ]]] , (2.13) 2In (c ...cnc ) D UDEK AND M ICHALSKI [1984] µn 1 is named a retract (which term is already busy and widely used in category theory for another construction). 6 STEVEN DUPLIJ such that the diagram ǫ
×3 id×6 ×µ4 id×3 ×µ4 G×4 ✲ G × {•}2 ×G ✲ G×4✲ G×7 ❳❳ ❳❳❳ ❳❳❳ ❳❳ ❳❳❳ µ4 ❳❳❳ (2.14) (c ,c )′ µ4 1 2 ❳❳❳ ❳❳ ❳❳❳ ❄ ❳❳ ③ G is commutative. It is important to find conditions where iterating and reducing compensate each other, i.e. they do not change arity overall. Indeed, let the number of the iterating multiplications ℓµ be fixed, then (0) we can find such a number of reducing constants nc , such that the final arity will coincide with the initial arity n. The result will depend on the order of operations. There are two cases: (0) (1) Iterating →Reducing. For the number of reducing constants nc we obtain from (2.4) and (2.6) n(0) c = (n − 1) (ℓµ − 1) , (2.15) such that there is no restriction on ℓµ . (0) (2) Reducing →Iterating. For nc we get (n − 1) (ℓµ − 1) n(0) c = , (2.16) ℓµ (0) and now ℓµ ≤ n − 1. The requirement that nc should be an integer gives two further possibilities ( n−1 , ℓµ = 2, n(0) c = 2 (2.17) n − 2, ℓµ = n − 1. The above relations can be useful in the study of various n-ary multiplication structures and their presentation in special form is needed in concrete problems. 3. S PECIAL ELEMENTS AND PROPERTIES OF POLYADIC SYSTEMS Let us recall the definitions of some standard algebraic systems and their special elements, which will be considered in this paper, using our notation. Definition 3-1. A zero of a polyadic system is a distinguished element z (and the corresponding 0-ary (z) operation µ0 ) such that for any (n − 1)-tuple (polyad) g ∈ G×(n−1) we have µn [g, z] = z, (3.1) where z can be on any place in the l.h.s. of (3.1). There is only one zero (if its place is not fixed) which can be possible in a polyadic system. As in the binary case, an analog of positive powers of an element P OST [1940] should coincide with the number of multiplications ℓµ in the iterating (2.3). Definition 3-2. A (positive) polyadic power of an element is   g hℓµ i = µℓnµ g ℓµ(n−1)+1 . (3.2)  POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 7 Definition 3-3. An element of a polyadic system g is called ℓµ -nilpotent (or simply nilpotent for ℓµ = 1), if there exist such ℓµ that g hℓµ i = z. (3.3) Definition 3-4. A polyadic system with zero z is called ℓµ -nilpotent, if there exists ℓµ such that for any (ℓµ (n − 1) + 1)-tuple (polyad) g we have µℓnµ [g] = z. (3.4) Therefore, the index of nilpotency (number of elements whose product is zero) of an ℓµ -nilpotent n-ary system is (ℓµ (n − 1) + 1), while its polyadic power is ℓµ . Definition 3-5. A polyadic (n-ary) identity (or neutral element) of a polyadic system is a distinguished (e) element e (and the corresponding 0-ary operation µ0 ) such that for any element g ∈ G we have   µn g, en−1 = g, (3.5) where g can be on any place in the l.h.s. of (3.5). In binary groups the identity is the only neutral element, while in polyadic systems, there exist neutral polyads n consisting of elements of G satisfying µn [g, n] = g, (3.6) where g can be also on any place. The neutral polyads are not determined uniquely. It follows from (3.5) that the sequence of polyadic identities en−1 is a neutral polyad. Definition 3-6. An element of a polyadic system g is called ℓµ -idempotent (or simply idempotent for ℓµ = 1), if there exist such ℓµ that g hℓµ i = g. (3.7) Both zero and the identity are ℓµ -idempotents with arbitrary ℓµ . We define (total) associativity as the invariance of the composition of two n-ary multiplications µ2n [g, h, u] = µn [g, µn [h] , u] = invariant (3.8) under placement of the internal multiplication in r.h.s. with a fixed order of elements in the whole polyad of (2n − 1) elements t(2n−1) = (g, h, u). Informally, “internal brackets/multiplication can be moved on any place”, which gives n relations     µn ◦ µn × id×(n−1) = . . . = µn ◦ id×(n−1) ×µn . (3.9) There are many other particular kinds of associativity which were introduced in T HURSTON [1949] and studied in B ELOUSOV [1972], S OKHATSKY [1997]. Here we will confine ourselves the most general, total associativity (3.8). In this case, the iteration does not depend on the placement of internal multiplications in the r.h.s. of (2.3). Definition 3-7. A polyadic semigroup (n-ary semigroup) is a n-ary system in which the operation is associative, or G semigrp n = hG | µn | associativityi. In a polyadic system with zero (3.1) one can have trivial associativity, when all n terms are (3.8) are equal to zero, i.e. µ2n [g] = z (3.10) for any (2n − 1)-tuple g. Therefore, we state that 8 STEVEN DUPLIJ Assertion 3-8. Any 2-nilpotent n-ary system (having index of nilpotency (2n − 1)) is a polyadic semigroup. In the case of changing arity one should use in (3.10) not the changed final arity n′ , but the “real” arity which is n for the reducing case and ℓµ (n − 1)+1 for all other cases. Let us give some examples. D case withEn = 2, ℓµ = 3, nc = 1, we have a ternary Example 3-9. In the mixed (interacting-reducing) (c) system hG | µ3 i iterated from a binary system G | µ2 , µ0 with one distinguished element c (or an 3 additional 0-ary operation) (c) µ3 [g, h, u] = (g · (h · (u · c))) , (3.11) E D (c) where for binary multiplication we denote g · h = µ2 [g, h]. Thus, if the ternary system G | µ3 (c) is nilpotent of index 7 (see 3-4), then it is a ternary semigroup (because µ3 is trivially associative) independently of the associativity of µ2 (see, e.g. B OROWIEC ET AL . [2006]). It is very important to find the associativity preserving conditions (constructions), where an asso- ciative initial operation µn leads to an associative final operation µ′n′ during the change of arity. Example 3-10. An associativity preserving reduction can be given by the construction of a binary associative operation using (n − 2)-tuple c consisting of nc = n − 2 different constants (c) µ2 [g, h] = µn [g, c, h] . (3.12) Associativity preserving mixing constructions with different arities and places were considered in D UDEK AND M ICHALSKI [1984], M ICHALSKI [1981], S OKHATSKY [1997]. Definition 3-11. An associative polyadic system with identity (3.5) is called a polyadic monoid. The structure of any polyadic monoid is fixed P OP AND P OP [2004]: it can be obtained by iterat- ing a binary operation Č UPONA AND T RPENOVSKI [1961] (for polyadic groups this was shown in D ÖRNTE [1929]). In polyadic systems, there are several analogs of binary commutativity. The most straightforward one comes from commutation of the multiplication with permutations. Definition 3-12. A polyadic system is σ-commutative, if µn = µn ◦ σ, or µn [g] = µn [σ ◦ g] , (3.13)
where σ◦g = gσ(1) , . . . , gσ(n) is a permutated polyad and σ is a fixed element of Sn , the permutation group on n elements. If (3.13) holds for all σ ∈ Sn , then a polyadic system is commutative. A special type of the σ-commutativity µn [g, t, h] = µn [h, t, g] , (3.14) where t is any fixed (n − 2)-polyad, is called semicommutativity. So for a n-ary semicommutative system we have     µn g, hn−1 = µn hn−1 , g . (3.15) If a n-ary semigroup Gsemigrp is iterated from a commutative binary semigroup with identity, then G semigrp is semicommutative. 3 This construction is named the b-derived groupoid in D UDEK AND M ICHALSKI [1984].  POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 9 Example 3-13. Let G be the set of natural numbers N, and the 5-ary multiplication is defined by µ5 [g] = g1 − g2 + g3 − g4 + g5 , (3.16) then G N5 = hN, µ5 i is a semicommutative 5-ary monoid having the identity eg = µ5 [g 5 ] = g for each g ∈ N. Therefore, G N5 is the idempotent monoid. Another possibility is to generalize the binary mediality in semigroups (g11 · g12 ) · (g21 · g22 ) = (g11 · g21 ) · (g12 · g22 ) , (3.17) which, obviously, follows from binary commutativity. But for n-ary systems they are different. It is seen that the mediality should contain (n + 1) multiplications, it is a relation between n × n elements, and therefore can be presented in a matrix from. The latter can be achieved by placing the arguments of the external multiplication in a column. Definition 3-14. A polyadic system is medial (or entropic), if E VANS [1963], B ELOUSOV [1972]     µn [g11 , . . . , g1n ] µn [g11 , . . . , gn1] µn  ..  = µn  .. . (3.18) . . µn [gn1 , . . . , gnn ] µn [g1n , . . . , gnn ] For polyadic semigroups we use the notation (2.3) and can present the mediality as follows   µnn [G] = µnn GT , (3.19) where G = kgij k is the n × n matrix of elements and GT is its transpose. The semicommuta- tive polyadic semigroups are medial, as in the binary case, but, in general (except n = 3) not vice versa G ŁAZEK AND G LEICHGEWICHT [1982a]. A more general concept is σ-permutability S TOJAKOVI Ć AND D UDEK [1986], such that the mediality is its particular case with σ = (1, n). Definition 3-15. A polyadic system is cancellative, if µn [g, t] = µn [h, t] =⇒ g = h, (3.20) where g, h can be on any place. This means that the mapping µn is one-to-one in each variable. If g, h are on the same i-th place on both sides, the polyadic system is called i-cancellative. The left and right cancellativity are 1-cancellativity and n-cancellativity respectively. A right and left cancellative n-ary semigroup is cancellative (with respect to the same subset). Definition 3-16. A polyadic system is called (uniquely) i-solvable, if for all polyads t, u and element h, one can (uniquely) resolve the equation (with respect to h) for the fundamental operation µn [u, h, t] = g (3.21) where h can be on any i-th place. Definition 3-17. A polyadic system which is uniquely i-solvable for all places i is called a n-ary (or polyadic) quasigroup. It follows, that, if (3.21) uniquely i-solvable for all places, than µℓnµ [u, h, t] = g (3.22) can be (uniquely) resolved with respect to h on any place. Definition 3-18. An associative polyadic quasigroup is called a n-ary (or polyadic) group. 10 STEVEN DUPLIJ The above definition is the most general one, but it is overdetermined. Much work on polyadic groups was done RUSAKOV [1998] to minimize the set of axioms (solvability not in all places P OST [1940], C ELAKOSKI [1977], decreasing or increasing the number of unknowns in determining equa- tions G AL’ MAK [2003]) or construction in terms of additionally defined objects (various analogs of the identity and sequences Ǔ SAN [2003]), as well as using not total associativity, but instead various partial ones S OKOLOV [1976], S OKHATSKY [1997], Y UREVYCH [2001]. In a polyadic group the only solution of (3.21) is called a querelement of g and denoted by ḡ D ÖRNTE [1929], such that µn [h, ḡ] = g, (3.23) where ḡ can be on any place. So, any idempotent g coincides with its querelement ḡ = g. It follows from (3.23) and (3.6), that the polyad
ng = g n−2ḡ (3.24) is neutral for any element of a polyadic group, where ḡ can be on any place. If this i-th place is important, then we write ng;i . The number of relations in (3.23) can be reduced from n (the number of possible places) to only 2 (when g is on the first and last places D ÖRNTE [1929], T IMM [1972], or on some other 2 places). In a polyadic group the Dörnte relations µn [g, nh;i] = µn [nh;j , g] = g (3.25) hold true for any allowable i, j. In the case of a binary group the relations (3.25) become g · h · h−1 = h · h−1 · g = g. The relation (3.23) can be treated as a definition of the unary queroperation µ̄1 [g] = ḡ, (3.26) such that the diagram µn G×n ✲ G ✻ ✑ ✸ ✑ id×(n−1) ×µ̄1 ✑ (3.27) ✑ Prn ✑ G×n commutes. Then, using the queroperation (3.26) one can give a diagrammatic definition of a polyadic group (cf. G LEICHGEWICHT AND G ŁAZEK [1967]). Definition 3-19. A polyadic group is a universal algebra G grp n = hG | µn , µ̄1 | associativity, Dörnte relationsi , (3.28) where µn is a n-ary associative operation and µ̄1 is the queroperation, such that the following diagram id×(n−1) ×µ̄1 µ̄1 ×id×(n−1) G×(n) ✲ G×n ✛ G×n ✻ ✻ id ×Diag(n−1) µn Diag(n−1) ×id (3.29) Pr1 ❄ Pr2 G×G ✲G✛ G×G commutes, where µ̄1 can be only on first and second places from the right (resp. left) on the left (resp. right) part of the diagram. A straightforward generalization of the queroperation concept and corresponding definitions can be made by substituting in the above formulas (3.23)–(3.26) the n-ary multiplication µn by iterating ℓ the multiplication µnµ (2.3) (cf. D UDEK [1980] for ℓµ = 2).  POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 11 Definition 3-20. Let us define the querpower k of g recursively ḡ hhkii = (ḡ hhk−1ii ), (3.30) k hh0ii hh1ii z }| { where ḡ = g, ḡ = ḡ, or as the k composition µ̄◦k 1 = µ̄1 ◦ µ̄1 ◦ . . . ◦ µ̄1 of the queroperation (3.26). For instance G AL’ MAK [2003], µ̄◦2 n−3 ◦2 1 = µn , such that for any ternary group µ̄1 = id, i.e. one has ḡ = g. Using the queroperation in polyadic groups we can define the negative polyadic power of an element g by the following recursive relation   µn g hℓµ −1i , g n−2, g h−ℓµ i = g, (3.31) or (after use of (3.2)) as a solution of the equation   µℓnµ g ℓµ(n−1) , g h−ℓµ i = g. (3.32) It is known that the querpower and the polyadic power are mutually connected D UDEK [1993]. Here, we reformulate this connection using the so called Heine numbers H EINE [1878] or q-deformed numbers K AC AND C HEUNG [2002] qk − 1 , [[k]]q = (3.33) q−1 which have the “nondeformed” limit q → 1 as [k]q → k. Then ḡ hhkii = g h−[[k]]2−n i , (3.34) which can be treated as follows: the querpower coincides with the negative polyadic deformed power with a “deformation” parameter q which is equal to the “deviation” (2 − n) from the binary group. 4. H OMOMORPHISMS OF POLYADIC SYSTEMS Let G n = hG; µn i and G ′n′ = hG′ ; µ′n′ i be two polyadic systems of any kind (quasigroup, semi- group, group, etc.). If they have the multiplications of the same arity n = n′ , then one can define the mappings from G n to G ′n . Usually such polyadic systems are similar, and we call mappings between them the equiary mappings. ′  Let us take n + 1 mappings ϕGG i : G → G′ , i = 1, . . . , n + 1. An ordered system of mappings ′ GG ϕi is called a homotopy from G n to G ′n , if B ELOUSOV [1972] ′ h i ′ GG′ GG′ ϕGG n+1 (µ [g n 1 , . . . , g n ]) = µ n ϕ 1 (g 1 ) , . . . , ϕ n (g n ) , gi ∈ G. (4.1) In general, one should add to this definition the “mapping” of the multiplications (µµ′ ) ψnn′ µn 7→ µ′n′ . (4.2) n ′ o ′ (µµ ) In such a way, homotopy can be defined as the extended system of mappings ϕGG i ; ψnn . The corresponding commutative (equiary) diagram is ′ ϕGG G n+1 ✲ G′ ✻ ✻ µn ............. ψnn ✲ (µ) .............. µ′n (4.3) ′ ′ ϕGG ×...×ϕGG G×n 1 n ✲ (G′ ) ×n 12 STEVEN DUPLIJ (µµ′ ) The existence of the additional “mapping” ψnn acting on the second component of hG; µn i is (µµ′ ) tacitly implied. We will write/mention the “mappings” ψnn′ manifestly, e.g.,   ′ (µµ′ ) ϕGG i ;ψnn Gn ⇒ G ′n′ , (4.4) GG′ only as needed. If all the components ϕi of a homotopy are bijections, it is called an isotopy. In ′ case of polyadic quasigroups B ELOUSOV [1972] all mappings ϕGG i are usually taken as permutations  of the same underlying set G = G . If the multiplications are also coincide µn = µ′n , then ϕGG ′ i ; id is called an autotopy of the polyadic system G n . Various properties of homotopy in universal algebras were studied, e.g. in P ETRESCU [1977], H ALA Š [1994]. The homotopy, isotopy and autotopy are widely used equiary mappings in the study of polyadic ′ quasigroups and loops, while their diagonal equiary counterparts (all ϕGG i coincide), the homomor- phism, isomorphism and automorphism, are more suitable in investigation of polyadic semigroups, groups and rings and their wide applications in physics. Usually, it is written about the latter between similar (equiary) polyadic systems: they “...are so well known that we shall not bother to define them carefully” H OBBY AND M C K ENZIE [1988]. Nevertheless, we give a diagrammatic definition of the standard homomorphism between similar polyadic systems in our notation, which will be convenient to explain the clear way of its generalization. ′ A homomorphism from Gn to G ′n is given, if there exists a mapping ϕGG : G → G′ satisfying ′ h ′ ′ i ϕGG (µn [g1 , . . . , gn ]) = µ′n ϕGG (g1 ) , . . . , ϕGG (gn ) , gi ∈ G, (4.5) which means that the corresponding (equiary) diagram is commutative ′ ϕGG G ✲ G′ ✻ ✻ µn ....... ψnn ✲ (µµ′ ) ......... µ′n (4.6) ′ ×n   ϕGG G×n ✲ (G′ ) ×n ′ Usually the homomorphism is denoted by the same onenletter ϕGG , while o it would be more consis- GG′ (µµ′ ) tent to use for its notation the extended pair of mappings ϕ ; ψnn . We will use both notations on a par. We first mention a small subset of known generalizations of the homomorphism (for bibliography till 1982 see, e.g., G ŁAZEK AND G LEICHGEWICHT [1982b]) and then introduce a concrete con- struction for an analogous mapping which can change the arity of the multiplication (fundamental operation) without introducing additional (term) operations. A general approach to mappings be- tween free algebraic systems was initiated in F UJIWARA [1959], where the so-called basic mapping formulas for generators were introduced, and its generalization to many-sorted algebras was given in V IDAL AND T UR [2010]. In N OVOTN Ý [2002] it was shown that the construction of all homomor- phisms between similar polyadic systems can be reduced to some homomorphisms between corre- sponding mono-unary algebras N OVOTN Ý [1990]. The notion of n-ary homomorphism is realized as a sequence of n consequent homomorphisms ϕi , i = 1, . . . , n, of n similar polyadic systems n z }| { ϕ1 ′ ϕ2 ϕn−1 ′′ ϕn ′′′ Gn → Gn → . . . → Gn → Gn (4.7) (generalizing Post’s n-adic substitutions P OST [1940]) was introduced in G AL’ MAK [1998], and studied in G AL’ MAK [2001a, 2007].  POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 13 The above constructions do not change the arity of polyadic systems, because they are based on the corresponding diagram which gives a definition of an equiary mapping. To change arity one has to: 1) add another equiary diagram with additional operations using the same formula (4.5), where both do not change arity; 2) use one modified (and not equiary) diagram and the underlying formula (4.5) by themselves, which will allow us to change arity without introducing additional operations. The first way leads to the concept of weak homomorphism which was introduced in G OETZ [1966], M ARCZEWSKI [1966], G ŁAZEK AND M ICHALSKI [1974] for non-indexed al- gebras and in G ŁAZEK [1980] for indexed algebras, then developed in T RACZYK [1965] for Boolean and Post algebras, in D ENECKE AND W ISMATH [2009] for coalgebras and F -algebras D ENECKE AND S AENGSURA [2008] (see also C HUNG AND S MITH [2008]). To define the weak homomorphism in our notation we should incorporate into the polyadic systems hG; µn i and hG′ ; µ′n′ i ′ the following additional term operations of opposite arity νn′ : G×n → G and νn′ : G′×n → G′ and consider two equiary mappings between hG; µn , νn′ i and hG′ ; µ′n′ , νn′ i. ′ A weak homomorphism from hG; µn , νn′ i to hG′ , µ′n′ , νn′ i is given, if there exists a mapping ϕGG : G → G′ satisfying two relations simultaneously ′ h ′ ′ i ϕGG (µn [g1 , . . . , gn ]) = νn′ ϕGG (g1 ) , . . . , ϕGG (gn ) , (4.8) ′ h ′ ′ i ϕGG (νn′ [g1 , . . . , gn′ ]) = µ′n′ ϕGG (g1 ) , . . . , ϕGG (gn′ ) , (4.9) which means that two equiary diagrams commute ′ ′ ϕGG G ϕGG ✲ G′ G ✲ G′ ✻ ✻ ✻ ✻ µn ....... ψnn ✲ (µν ′ ) ......... ′ νn νn′ ......... ψ(νµ′) ✲ .......... µ′n′ (4.10) ′ ′  ′ ×n   n ′n×n′ ϕGG ϕGG G×n ✲ (G′ ) ×n G×n ′ ✲ (G′ ) ×n′ If only one of the relations (4.8) or (4.9) holds, such a mapping is called a semi-weak ho- ′ momorphism KOLIBIAR [1984]. If ϕGG is bijective, then it defines a weak isomorphism. Any weak epimorphism can be decomposed into a homomorphism and a weak isomorphism G ŁAZEK AND M ICHALSKI [1977], and therefore the study of weak homomorphisms reduces to weak isomorphisms (see also C Z ÁK ÁNY [1962], M AL’ TCEV [1957], M AL’ TSEV [1958]). 5. M ULTIPLACE MAPPINGS OF POLYADIC SYSTEMS AND HETEROMORPHISMS Let us turn to the second way of changing the arity of the multiplication and use only one relation which we then modify in some natural manner. First, recall that in any set G there always exists the additional distinguished mapping, viz. the identity idG . We use the multiplication µn with its combination of idG . We define an (ℓid -intact) id-product for the polyadic system hG; µn i as µ(ℓ n id ) = µn × (idG )×ℓid , (5.1) µ(ℓ n id ) : G×(n+ℓid ) → G×(1+ℓid ) . (5.2) (ℓ ) To indicate the exact i-th place of µn in the r.h.s. of (5.1), we write µn id (i), as needed. Here we use the id-product to generalize the homomorphism and consider mappings between polyadic systems of different arity. It follows from (5.2) that, if the image of the id-product is G alone, than ℓid = 0. 14 STEVEN DUPLIJ (n,n′ ) Let us introduce a multiplace mapping Φk acting as follows (n,n′ ) Φk : G×k → G′ . (5.3) (n,n′ ) While constructing the corresponding diagram, we are allowed to take only one upper Φk , ′ because of one G in the upper right corner. Since we already know that the lower right corner ′ is exactly G′×n (as a pre-image of one multiplication µ′n′ ), the lower horizontal arrow should be a (n,n′ ) product of n′ multiplace mappings Φk . So we can write a definition of a multiplace analog of homomorphisms which changes the arity of the multiplication using one relation. Definition 5-1. A k-place heteromorphism from G n to G′n′ is given, if there exists a k-place mapping (n,n′ ) Φk (5.3) such that the following (arity changing or unequiary) diagram is commutative Φk G×k ✲ G′ ✻ ✻ (ℓ ) µn id µ′n′ (5.4) ×n′ (Φk ) G×kn ✲ (G′ ) ′ ×n′ and the corresponding defining equation (a modification of (4.5)) depends on the place i of µn in (5.1). For i = 1 a heteromorphism is defined by the formula   µn [g1 , . . . , gn ]      g1 gk(n′ −1) gn+1  = µ′ ′ Φ(n,n′ )  ...  , . . . , Φ(n,n′ )  ..  (n,n′ )   Φk  . ..  n k k .  . (5.5) gk gkn′ gn+ℓid The notion “heteromorphism” is motivated by E LLERMAN [2006, 2007], where mappings between objects from different categories were considered and called “chimera morphisms”. See, also, P ÉCSI [2011]. In the particular case n = 3, n′ = 2, k = 2, ℓid = 1 we have        (3,2) µ3 [g1 , g2 , g3] ′ (3,2) g1 (3,2) g3 Φ2 = µ2 Φ2 , Φ2 . (5.6) g4 g2 g4 This formula was used in the construction of the bi-element representations of ternary groups B OROWIEC ET AL . [2006]. Consider the example. Example 5-2. Let G = M2adiag (K), a set of antidiagonal 2 × 2 matrices over the field K and G′ = K, where K = R, C, Q, H. The ternary multiplication µ3 is a product  of 3 matrices.  Obviously, µ3 is 0 ai not derived from a binary multiplication. For the elements gi = , i = 1, 2, we construct a bi 0 2-place mapping G × G → G′ as   (3,2) g1 Φ2 = a1 a2 b1 b2 , (5.7) g2 which is a heteromorphism, because it satisfies (5.6). Let us introduce a standard 1-place mapping by ϕ (gi ) = ai bi , i = 1, 2. It is important to note, that ϕ (gi ) is not a homomorphism, because the product g1 g2 belongs to diagonal matrices. Consider the product of mappings ϕ (g1 ) · ϕ (g2 ) = a1 b1 a2 b2 , (5.8)  POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 15 where the product (·) in l.h.s. is taken in K. We observe that (5.7) and (5.8) coincide for the commuta- tive field K only (= R, C) only, and in this case we can have the relation between the heteromorhism (3,2) Φ2 and the 1-place mapping ϕ   (3,2) g1 Φ2 = ϕ (g1 ) · ϕ (g2 ) , (5.9) g2 while for the noncommutative field K (= Q or H) there is no such relation. A heteromorphism is called derived, if it can be expressed through a 1-place mapping (not nec- essary a homomorphism). So, in the above Example 5-2 the heteromorphism is derived (by formula (5.9)) for the commutative field K and nonderived for the noncommutative K. For arbitrary n a slightly modified construction (5.6) with still binary final arity, defined by n′ = 2, k = n − 1, ℓid = n − 2,   µn [g1 , . . . , gn−1, h1 ]      g1 h1 h2  ...  , Φ(n,2)  ...  . (n,2)   Φn−1  .  = µ′2 Φ(n,2) (5.10)  ..  n−1 n−1 gn−1 hn−1 hn−1 was used in D UDEK [2007] to study representations of n-ary groups. However, no new results com- pared with B OROWIEC ET AL . [2006] (other than changing 3 to n in some formulas) were obtained. This reflects the fact that a major role is played by the final arity n′ and the number of n-ary multipli- cations in the l.h.s. of (5.6) and (5.10). In the above cases, the latter number was one, but can make it arbitrary below n. Definition 5-3. A heteromorphism is called a ℓµ -ple heteromorphism, if it contains ℓµ multiplications (n,n′ ) in the argument of Φk in its defining relation. According this definition the mapping defined by (5.5) is the 1µ -ple heteromorphism. So by analogy with (5.1)–(5.2) we define a ℓµ -ple ℓid -intact id-product for the polyadic system hG; µn i as µ(ℓ n µ ,ℓid ) = (µn )×ℓµ × (idG )×ℓid , (5.11) µ(ℓ n µ ,ℓid ) : G×(nℓµ +ℓid ) → G×(ℓµ +ℓid ) . (5.12) Definition 5-4. A ℓµ -ple k-place heteromorphism from Gn to G ′n′ is given, if there exists a k-place (n,n′ ) mapping Φk (5.3) such that the following unequiary diagram is commutative Φk G×k ✲ G′ ✻ ✻ (ℓµ ,ℓid ) µn µ′n′ (5.13) ×n′ (Φk ) G×kn ′ ✲ (G′ ) ×n′ The corresponding main heteromorphism equation is    µn [g1 , . . . , gn ] ,   ..  ℓ   .   µ        g1 gk(n′ −1) (n,n′ )  µn gn(ℓµ −1) , . . . , gnℓµ ..  , . . . , Φ(n,n′ )  ..   Φk   = µ′ ′ Φ(n,n′)  . .  . n k k   gnℓµ +1 ,     ..  gk gkn′  . ℓid   gnℓµ +ℓid (5.14) 16 STEVEN DUPLIJ F IGURE 1. Dependence of the final arity n′ through the number of heteromorphism places k for the initial arity n = 9 with the fixed number of intact elements ℓid (left) and the fixed number of multiplications ℓµ (right): =1 (solid curves), =2 (dash curves) Obviously, we can consider various permutations of the multiplications on both sides, as further additional demands (associativity, commutativity, etc.), are introduced, which will be considered be- low. The commutativity of the diagram (5.13) leads to the system of equation connecting initial and final arities kn′ = nℓµ + ℓid, (5.15) k = ℓµ + ℓid . (5.16) Excluding ℓµ or ℓid , we obtain two arity changing formulas, respectively n−1 n′ = n − ℓid , (5.17) k n−1 n′ = ℓµ + 1, (5.18) k where n−1 k ℓid ≥ 1 and n−1 k ℓµ ≥ 1 are integer. As an example, the dependences n′ (k) for the fixed ℓµ = 1, 2 and ℓid = 1, 2 with n = 9 are presented on Figure 1. The following inequalities hold valid 1 ≤ ℓµ ≤ k, (5.19) 0 ≤ ℓid ≤ k − 1, (5.20) ℓµ ≤ k ≤ (n − 1) ℓµ , (5.21) ′ 2 ≤ n ≤ n, (5.22) which are important for the further classification of heteromorphisms. The main statement follows from (5.22) (n,n′ ) Proposition 5-5. The heteromorphism Φk defined by the general relation (5.14) always decreases the arity of polyadic multiplication.  POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 17 Another important observation is the fact that only the id-product (5.11) with ℓid 6= 0 leads to a change of the arity. In the extreme case, when k approaches its minimum, k = kmin = ℓµ , the final arity approaches its maximum n′max = n, and the id-product becomes a product of ℓµ initial multiplications µn without id’s, since now ℓid = 0 in (5.14). Therefore, we call a heteromorphism defined by (5.14) with ℓid = 0 a k (= ℓµ )-place homomorphism. The ordinary homomorphism (4.1) corresponds to k = ℓµ = 1, and so it is really a 1-place homomorphism. An opposite extreme case, when the final arity approaches its minimum n′min = 2 (the final operation is binary), corresponds to the maximal value of k, that is k = kmax = (n − 1) ℓµ . The number of id’s now is ℓid = (n − 2) ℓµ ≥ 0, which vanishes, when the initial operation is binary as well. This is the case of the ordinary homomorphism (4.1) for both binary operations n′ = n = 2 and k = ℓµ = 1. We conclude that: Any polyadic system can be mapped into a binary system by means of the special k-place ℓµ -ple (n,n′ ) heteromorphism Φk , where k = (n − 1) ℓµ (we call it a binarizing heteromorphism) which is defined by (5.14) with ℓid = (n − 2) ℓµ . In relation to the Gluskin-Hosszú theorem G LUSKIN [1965] (any n-ary group can be constructed from the special binary group and its homomorphism) our statement can be treated as: Theorem 5-6. Any n-ary system can be mapped into a binary system, using a suitable binarizing (n,2) heteromorphism Φk (5.14). The case of 1-ple binarizing heteromorphism (ℓµ = 1) corresponds to the formula (5.10). Further requirements (associativity, commutativity, etc.) will give additional relations between multiplications (n,n′ ) and Φk , and fix the exact structure of (5.14). Thus, we arrive to the following Proposition 5-7. Classification of ℓµ -ple heteromorphisms: (n,n) (1) n′ = n′max = n =⇒ Φk is the ℓµ -place or multiplace homomorphism, i.e., k = kmin = ℓµ . (5.23) (n,n′ ) (2) 2 < n′ < n =⇒ Φk is the intermediate heteromorphism with n−1 k= ℓµ . (5.24) n′ − 1 In this case the number of intact elements is proportional to the number of multiplications n − n′ ℓid = ℓµ . (5.25) n′ − 1 (n,2) (3) n′ = n′min = 2 =⇒ Φk is the (n − 1) ℓµ -place (multiplace) binarizing heteromorphism, i.e., k = kmax = (n − 1) ℓµ . (5.26) In the extreme (first and third) cases there are no restrictions on the initial arity n, while in the intermediate case n is “quantized” due to the fact that fractions in (5.17) and (5.18) should be integers. Observe, that in the extreme (first and third) cases there are no restrictions on the initial arity n, while in the intermediate case n is “quantized” due to the fact that fractions in (5.17) and (5.18) should be integer. In this way, we obtain the TABLE 1 for the series of n and n′ (we list only first ones, just for 2 ≤ k ≤ 4 and include the binarizing case n′ = 2 for completeness). Thus, we have established a general structure and classification of heteromorphisms defined by (5.14). The next important issue is the preservation of special properties (associativity, commutativity, etc.), while passing from µn to µ′n′ , which will further restrict the concrete shape of the main relation 18 STEVEN DUPLIJ TABLE 1. “Quantization” of heteromorphisms k ℓµ ℓid n/n′ n= 3, 5, 7, ... 2 1 1 n′ = 2, 3, 4, ... n= 4, 7, 10, ... 3 1 2 n′ = 2, 3, 4, ... n= 4, 7, 10, ... 3 2 1 n′ = 3, 5, 7, ... n= 5, 9, 13, ... 4 1 3 n′ = 2, 3, 4, ... n= 3, 5, 7, ... 4 2 2 n′ = 2, 3, 4, ... n= 5, 9, 13, ... 4 3 1 n′ = 4, 7, 10, ... (5.14) for each choice of the heteromorphism parameters: arities n, n′ , places k, number of intacts ℓid and multiplications ℓµ . 6. A SSOCIATIVITY QUIVERS AND HETEROMORPHISMS The most important property of the heteromorphism, which is needed for its next applications to representation theory, is the associativity of the final operation µ′n′ , when the initial operation µn is associative. In other words, we consider here the concrete form of semigroup heteromorphisms. In general, this is a complicated task, because it is not clear from (5.14), which permutation in the l.h.s. should be taken to get an associative product in its r.h.s. for each set of the heteromorphism param- eters. Straightforward checking of the associativity of the final operation µ′n′ for each permutation in the l.h.s. of (5.14) is almost impossible, especially for higher n. To solve this difficulty we intro- duce the concept of the associative polyadic quiver and special rules to construct the associative final operation µ′n′ . Definition 6-1. A polyadic quiver of products is the set of elements from G n (presented as several copies of some matrix of the elements glued together) and arrows, such that the elements along arrows form n-ary products µn . For instance, for the 4-ary multiplication µ4 [g1 , h2 , g2 , u1] (elements from G n are arbitrary here) a corresponding 4-adic quiver will be denoted by {g1 → h2 → g2 → u1 }, and graphically this 4-adic quiver is g1 h1 u1 g1 h1 7 u1 ❅❅ ❅ ♥♥♥♥♥ ♥ * ♥♥♥ g2 h2 u2 g2 h2 u2 KS (6.1) corr  µ4 [g1 , h2 , g2 , u1 ] . Next we define polyadic quivers which are related to the main heteromorphism equation (5.14) in the following way:  POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 19 1) the matrix of elements is the transposed matrix from the r.h.s. of (5.14), such that different letters (n,n′ ) correspond to their place in Φk and the low index of an element is related to its position in the µ′n′ product; 2) the number of polyadic quivers is ℓµ , which corresponds to ℓµ multiplications in the l.h.s. of (5.14); 3) the heteromorphism parameters (n, n′ , k, ℓid and ℓµ ) are not arbitrary, but satisfy the arity changing formulas (5.17)-(5.18); 4) the intact elements will be placed after a semicolon. In this way, a polyadic quiver makes a clear visualization of the main heteromorphism equation (5.14), and later on it will allow us to distinguish associativity preserving heteromorphisms by precise graphical rules. For example, the polyadic quiver {g1 → h2 → g2 → u1 ; h1 , u2 } corresponds to the unequiary het- eromorphism with n = 4, n′ = 2, k = 3, ℓid = 2 and ℓµ = 1 is g1 h1 u1 g1 h1 ♥7 u1 ❅❅ ♥ ❅ ♥♥♥♥♥ * ♥♥ g2 h2 u2 g2 h2 u2 KS corr (6.2)         µ4 [g1 , h2 , g2 , u1 ] g1 g2 (4,2) ′ (4,2) (4,2) Φ3  h1  = µ2 Φ3  h1  , Φ3  h2  , u2 u1 u2 where the intact elements h1 , u2 are boxed in squares. As it is seen from (6.2), the product µ′2 is not associative, if µ4 is associative. So, not all polyadic quivers preserve associativity. Definition 6-2. An associative polyadic quiver is a polyadic quiver which ensures the final asso- ciativity of µ′n′ in the main heteromorphism equation (5.14), when the initial multiplication µn is associative. One of the associative polyadic quivers which corresponds to the same heteromorphism parameters, as the non-associative quiver (6.2), is {g1 → h2 → u1 → g2 ; h1 , u2 } which corresponds to g1 h1 u1 g1 h1 u1 ❅❅ ❅ ⑥⑥⑥> ❅❅❅ ⑥ ❅ g2 h2 u2 g2 h2 u2 KS corr (6.3)         µ4 [g1 , h2 , u1 , g2 ] g1 g2 (4,2)   = µ′2 Φ(4,2)  h1  , Φ(4,2) Φ3 h1 3 3  h2  . u2 u1 u2 Here we propose a classification of associative polyadic quivers and the rules of construction of the corresponding heteromorphism equations, and then use the heteromorphism parameters for the classification of ℓµ -ple heteromorphisms (5.24). In other words, we describe a consistent procedure for building the semigroup heteromorphisms. Let us consider the first class of heteromorphisms (without intact elements ℓid = 0 or intactless), that is ℓµ -place (multiplace) homomorphisms. In the simplest case, associativity can be achieved, when all elements in a product are taken from the same row. The number of places k is not fixed by 20 STEVEN DUPLIJ the arity relation (5.17) and can be arbitrary, while the arrows can have various directions. There are 2k such combinations which preserve associativity. If the arrows have the same direction, this kind of mapping is also called a homomorphism. As an example, for n = n′ = 3, k = 2, ℓµ = 2 we have g1 h1   g2 h2   g3 h3 (6.4) KS corr           (3,3) µ3 [g1 , g2 , g3 ] (3,3) g1 (3,3) g2 (3,3) g3 Φ2 = µ′3 Φ2 , Φ2 , Φ2 . µ3 [h1 , h2 , h3 ] h1 h2 h3 Note that the analogous quiver with opposite arrow directions is g1 hO 1  g2 hO 2  g3 h3 (6.5) KS corr           (3,3) µ3 [g1 , g2 , g3 ] (3,3) g1 (3,3) g2 (3,3) g3 Φ2 = µ′3 Φ2 , Φ2 , Φ2 , µ3 [h3 , h2 , h1 ] h1 h2 h3 The latter mapping and the corresponding vertical quiver were used in constructing the middle repre- sentations of ternary groups B OROWIEC ET AL . [2006]. For nonvertical quivers the main rule is the following: all arrows of an associative quiver should have direction from left to right or vertical, and they should not instersect. Also, we start always from the upper left corner, because of the permutation symmetry of (5.14), we can rearrange and rename variables in the necessary way. An important class of intactless heteromorphisms (with ℓid = 0) preserving associativity can be constructed using an analogy with the Post substitutions P OST [1940], and therefore we call it the Post-like associative quiver. The number of places k is now fixed by k = n − 1, while n′ = n and ℓµ = k = n − 1. An example of the Post-like associative quiver with the same heteromorphisms parameters as in (6.4)-(6.5) is g1 h1 ❅❅ ❅❅ g1 h1 ❅ ❅❅ g2 h2 ❅ g2 h ❅ ❅❅ 2 ❅❅ ❅ g3 h3 g3 h3 (6.6) KS corr           (3,3) µ3 [g1 , h2 , g3 ] (3,3) g1 (3,3) g2 (3,3) g3 Φ2 = µ′3 Φ2 , Φ2 , Φ2 . µ3 [h1 , g2 , h3 ] h1 h2 h3  POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 21 This construction appeared in the study of ternary semigroups of morphisms C HRONOWSKI [1994b,a], C HRONOWSKI AND N OVOTN Ý [1995]. Its n-ary generalization was used in the con- sideration of polyadic operations on Cartesian powers G AL’ MAK [2008], polyadic analogs of the Cayley and Birkhoff theorems G AL’ MAK [2001b, 2007] and special representations of n-groups G LEICHGEWICHT ET AL . [1983], WANKE -J AKUBOWSKA AND WANKE -J ERIE [1984] (where the n-group with the multiplication µ′2 was called the diagonal n-group). Consider the following exam- ple. Example 6-3. Let Λ be the Grassmann algebra consisting of even and odd parts Λ = Λ0̄ ⊕ Λ1̄ (see (1̄) e.g., B EREZIN [1987]). The odd part can be considered as a ternary semigroup G 3 = hΛ1̄ , µ3 i, its multiplication µ3 : Λ1̄ × Λ1̄ × Λ1̄ → Λ1̄ is defined by µ3 [α, β, γ] = α · β · γ, where (·) is (1̄) multiplication in Λ and α, β, γ ∈ Λ1̄ , so G 3 is nonderived and contains no unity. The even part can (0̄) be treated as a ternary group G 3 = hΛ0̄ , µ′3 i with the multiplication µ′3 : Λ0̄ × Λ0̄ × Λ0̄ → Λ0̄ , defined (0̄) by µ3 [a, b, c] = a · b · c, where a, b, c ∈ Λ0̄ , thus G 3 is derived and contains unity. We introduce the (1̄) (0̄) (3,3) heteromorphism G 3 → G 3 as a mapping (2-place homomorphism) Φ2 : Λ1̄ × Λ1̄ → Λ0̄ by the formula   (3,3) α Φ2 = α · β, (6.7) β (3,3) where α, β ∈ Λ1̄ . It is seen that Φ2 defined by (6.7) satisfies the Post-like heteromorphism equation (6.6), but not the “vertical” one (6.4), due to the anticommutativity of odd elements from Λ1̄ . In other (0̄) (1̄) words, G 3 can be treated as a nontrivial example of the “diagonal” semigroup of G 3 (according to the notation of G LEICHGEWICHT ET AL . [1983], WANKE -J AKUBOWSKA AND WANKE -J ERIE [1984]). Note that for the number of places k ≥ 3 there exist additional (to the above) associative quivers having the same heteromorphism parameters. For instance, when n′ = n = 4 and k = 3 we have the Post-like associative quiver g1 h1 ❆ u1 ❅ g1 h1 u1 ❅❅ ❆ ❅ ❅ ❆❆ ❅❅ g2 h2 ❆ u2 ❅ g2 h u2 ❆ ❅ ❅❅ 2 ❆❆ ❅❅ ❅ g3 h3 u3 ❅ g3 h u ❅ ❅❅ 3 ❆❆ 3 ❅❅ ❅ ❆❆ g4 h4 u4 g4 h4 u4 (6.8) KS corr             µ4 [g1 , h2 , u3 , g4 ] g1 g2 g1 g2 (4,4) ′ (4,4)  µ4 [h1 , u2 , g3 , h4 ]  = µ4 Φ3 (4,4) (4,4) (4,4) Φ3  h1  , Φ3  h2  , Φ3  h1  , Φ3  h2  . µ4 [u1 , g2 , h3 , u4 ] u1 u2 u1 u2 22 STEVEN DUPLIJ Also, we have one intermediate non-Post associative quiver g1 PP h1 PP u1 PP g1 h1 u1 g1 h1 u1 PPP PPP PPP PPP PPP PP P' P' P' g2 h2 u2 PP g2 PP h2 PP u2 g2 h2 u2 PPP PPP PPP PPP PPPP PPPP ' ' ' g3 h3 u3 g3 h3 PP u3 PP g3 PP h3 u3 PPP PPP PPP PPP P P' PPP' PPP' g4 h4 u4 g4 h4 u4 g4 h4 u4 (6.9) KS corr             µ4 [g1 , u2 , h3 , g4 ] g1 g2 g1 g2 (4,4) ′ (4,4) (4,4) (4,4) (4,4) Φ3  µ4 [h1 , g2 , u3 , h4 ]  = µ4 Φ3  h1  , Φ3  h2  , Φ3  h1  , Φ3  h2  . µ4 [u1 , h2 , g3 , u4 ] u1 u2 u1 u2 A general method of constructing associative quivers for n′ = n, ℓid = 0 and k = n − 1 can be illustrated from the following more complicated example with n = 5. First, we draw k = n − 1 (= 4) copies of element matrices. Then we go from the first element in the first column g1 to the last element in this column gn′ (= g5 ) by different k = n − 1 (= 4) ways: by the vertical quiver, by the Post-like quiver (going to the second copy of the element matrix) and by the remaining k − 2 (= 2) non-Post associative quivers, as g1 ❘ ❍❍❘❱❘❱❘❱❱h❱1❱ ❱ u1 v1 g1 h1 u1 v1 g1 h1 u1 v1 g1 h1 u1 v1 . ❍❍ ❘❘❘ ❱❱❱  ❍$ ❘❘❘ ❱❱❱❱ ) ❱* g2 h2 ❆ u2 ◗◗ v2 ❱❱❱❱g2 h2 u2 v2 g2 h2 u2 v2 g2 h2 u2 v2 ◗◗◗ ❱ ❆❆ ❆ ◗◗◗ ❱❱❱❱❱❱❱❱  ◗( ❱❱* g3 h3 u3 ❅ v3 g3 ◗◗◗ h3 u3 ❲❲❲v❲3❲ ❅❅ ◗◗◗ ❲❲❲❲❲ g3 h3 u3 v3 g3 h3 u3 v3 ❅ ◗◗◗( ❲ ❲❲ ❲❲❲❲❲  + g4 h4 u4 v4 ❉ g4 h4 u4 ❘❘❘v4 g4 h4 ❱❱❱u ❱❱4 ❱❱ v4 g4 h4 u4 v4 ❉❉ ❘❘❘ ❱❱❱❱ ❉❉ ❘❘❘ ❱❱❱❱  ! ❘❘) ❱+ g5 h5 u5 v5 g5 h5 u5 v5 g5 h5 u5 v5 g5 h5 u5 v5 vertical Post non-Post non-Post (6.10) Here we show, for short, only the quiver itself without the corresponding heteromorphism equation and only the arrows corresponding to the first product, while other arrows (starting from h1 , u1 , v1 ) are parallel to it, as in (6.9). The next type of heteromorphisms (intermediate) is described by the equations (5.15)-(5.25), it contains intact elements (ℓid ≥ 1) and changes (decreases) arity n′ < n. For each fixed k the arities are not arbitrary and presented in TABLE 1. The first general rule is: the associative quivers are non- decreasing in both,vertical (from up to down) and horizontal (from left to right), directions. Second, if there are several multiplications (ℓµ ≥ 2), the corresponding associative quivers do not intersect. Let us present some examples and start from the smallest number of heteromorphism places in (n,n′ ) Φk . For k = 2, the first (nonbinarizing n′ ≥ 3) case is n = 5, n′ = 3, ℓid = 1 (see the first row  POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 23 and second n/n′ pair of TABLE 1). The corresponding associative quivers are g1 / h1 ❅❅ g1 h1 g1 h1 g1 ❅❅ h1 g1 h1 g1 h1 ❅❅ ❅ g2 h2 g2 / h2 / g2 ❅❅ g2 h2 g2 h2 ❅❅ h2 g2 h2 ❅❅ ❅ g3 h3 g3 h3 g3 h3 g3 h3 g3 h3 / g3 h3 (6.11) KS KS corr corr       (5,3) µ5 [g1 , h1, g2 , h2, g3 ] (5,3) µ5 [g1 , h2, g2 , h3, g3 ] Φ2 Φ2 , h3 h1 where we do not write the r.h.s. of the heteromorphism equation (5.14), because it is simply related to the transposed quiver matrix of the size n′ × k. More complicated examples can be given for k = 3, which corresponds to the second and third lines of TABLE 1. That is we can obtain a ternary final product (n′ = 3) by using one or two multiplications ℓµ = 1, 2. Examples of the corresponding associative quivers are (ℓµ = 1) g1 / h1 u1 ❆❆ u1 g1 h1 u1 g1 h1 ❆❆ g2 h2 u2 / g2 h2 u2 g2 h2 u2 ❅❅ ❅ g3 h3 u3 g3 h3 / u3 / g3 h3 u3 (6.12) KS corr    µ7 [g1 , h1, u2 , g2 , h3 , u3 , g3 ] (7,3) Φ3  h2  u1 and (ℓµ = 2) g1 PP h1 u1 g1 h1 2 u1 ❅ g1 h1 u1 PPP ❅❅ PPP ❅ P' g2 h2 u2 g2 4 h2 PPP u2 g2 h u2 PPP ❅❅ 2 PPP ❅ ' g3 h3 u3 g3 h3 u3 g3 h3 u3 (6.13) KS corr    µ4 [g1 , u2 , h2 , g3 ] (4,3) Φ3  µ4 [h1, u1 , g2 , h3 ]  u3 respectively. Finally, for the case k = 4 one can construct the associative quiver corresponding to the first pair of the last line in TABLE 1. It has three multiplications and one intact element, and the 24 STEVEN DUPLIJ corresponding quiver is, e.g., g1 h1 u1 v1 g1 h1 3 u1 PP v1 g1 h1 u1 v1 ❅❅ PPP ❅ PPP - v2 ' g2 h2 ❆ u2 g2 h2 3 u2 ❅ v2 g2 h u2 v2 ❆ ❅❅ ❅❅ 2 ❆❆ ❅ ❅ g3 h3 u3 v3 4 g3 ❲❲❲h❲❲3 ❲ u3 v3 PP g3 h3 PP u3 v3 ❲❲❲❲❲ PPP PPP ❲❲❲❲❲ PPP PPP ❲❲❲+ ' P' g4 h4 u4 v4 g4 h4 u4 v4 g4 h4 u4 v4 (6.14) KS corr    µ5 [g1 , h2 , u3 , g3 , g4 ] (5,4)  µ5 [h1 , v2 , u2 , v3 , h4 ]    Φ4  µ5 [v1 , u1 , g2 , h3 , v4 ]  . u4 There are many other possibilities (using permutations and different variants of quivers) to obtain an associative final product µ′n′ corresponding the same heteromorphism parameters, and therefore we do list them all. The above examples are sufficient to understand the rules of the associative quiver construction and obtain the polyadic semigroup heteromorphisms. 7. M ULTIPLACE REPRESENTATIONS OF POLYADIC SYSTEMS Representation theory K IRILLOV [1976] deals with mappings from abstract algebraic systems into linear systems, such as, e.g. linear operators in vector spaces, or into general (semi)groups of transformations of some set. In our notation, this means that in the mapping of polyadic systems (4.4) the final multiplication µ′n′ is a linear map. This leads to some restrictions on the final polyadic structure G ′n′ , which are considered below. Let V be a vector space over a field K (usually algebraically closed) and End V be a set of linear endomorphisms of V , which is in fact a binary group. In the standard way, a linear representation of a binary semigroup G 2 = hG; µ2 i is a (1-place) map Π1 : G2 → End V , such that Π1 is a homomorphism Π1 (µ2 [g, h]) = Π1 (g) ∗ Π1 (h) , (7.1) where g, h ∈ G and (∗) is the binary multiplication in End V (usually, it is a (semi)group with multiplication as composition of operators or product of matrices, if a basis is chosen). If G 2 is a binary group with the unity e, then we have the additional condition Π1 (e) = idV . (7.2) We will generalize these known formulas to the corresponding polyadic systems along with the heteromorphism concept introduced above. Our general idea is to use the heteromorphism equation (5.14) instead of the standard homomorphism equation (7.1), such that the arity of the representation will be different from the arity of the initial polyadic system n′ 6= n. Consider the structure of the final n′ -ary multiplication µ′n′ in (5.14), taking into account that the final polyadic system G ′n′ should be constructed from End V . The most natural and physically appli- cable way is to consider the binary End V and to put G ′n′ = dern′ (End V ), as it was proposed for the ternary case in B OROWIEC ET AL . [2006]. In this way G′n′ becomes a derived n′ -ary (semi)group of ′ endomorphisms of V with the multiplication µ′n′ : (End V )×n → End V , where µ′n′ [v1 , . . . , vn′ ] = v1 ∗ . . . ∗ vn′ , vi ∈ End V. (7.3)  POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 25 Because the multiplication µ′n′ (7.3) is derived and is therefore associative by definition, we may consider the associative initial polyadic systems (semigroups and groups) and the associativity pre- serving mappings that are the special heteromorphisms constructed in the previous section. Let G n = hG; µn i be an associative n-ary polyadic system. By analogy with (5.3), we introduce the following k-place mapping (n,n′ ) Πk : G×k → End V. (7.4) A multiplace representation of an associative polyadic system G n in a vector space V is given, if there exists a k-place mapping (7.4) which satisfies the (associativity preserving) heteromorphism equation (5.14), that is    µn [g1 , . . . , gn ] ,  n′  .. ℓ  z }|  . µ     {      g1 gk(n′ −1) (n,n′ )  µn gn(ℓµ −1) , . . . , gnℓµ   Πk   = Π(n,n′)  ...  ∗ . . . ∗ Π(n,n′)  .. . , (7.5) k k   gnℓµ +1 ,     ..  gk gkn′  . ℓid   gnℓµ +ℓid and the following diagram commutes Πk G×k ✲ End V ✻ ✻ (ℓµ ,ℓid ) µn (∗)n ′ (7.6) ×n′ (Πk ) ✲ ′ (End V )×n ′ G×kn (ℓ ,ℓ ) where µn µ id is given by (5.11), ℓµ and ℓid are the numbers of multiplications and intact elements in the l.h.s. of (7.5), respectively. The exact permutation in the l.h.s. of (7.5) is given by the associative quiver presented in the previous section. The representation parameters (n, n′ , k, ℓµ and ℓid ) in (7.5) are the same as the heteromorphism parameters, and they satisfy the same arity changing formulas (5.17) and (5.18). Therefore, a general classification of multiplace representations can be done by analogy with that of the heteromorphisms (5.23)–(5.26) as follows: (1) The hom-like multiplace representation which is a multiplace homomorphism with n′ = (min) n′max = n, without intact elements lid = lid = 0, and minimal number of places k = kmin = ℓµ . (7.7) (2) The intact element multiplace representation which is the intermediate heteromorphism with 2 < n′ < n and the number of intact elements is n − n′ lid = ′ ℓµ . (7.8) n −1 (3) The binary multiplace representation which is a binarizing heteromorphism (5.26) with n′ = (max) n′min = 2, the maximal number of intact elements lid = (n − 2) ℓµ and maximal number of places k = kmax = (n − 1) ℓµ . (7.9) 26 STEVEN DUPLIJ The multiplace representations for n-ary semigroups have no additional defining relations, as com- pared with (7.5). In case of n-ary groups, we need an analog of the “normalizing” relation (7.2). If the n-ary group has the unity e, then one can put    e  (n,n′ )  . Πk .. k  = idV . (7.10)  e If there is no unity at all, one can “normalize” the multiplace representation, using analogy with (7.2) in the form
Π1 h−1 ∗ h = idV , (7.11) as follows    h̄   ... ℓµ       (n,n′ )  h̄  Πk    = idV , (7.12)  h    .   .. ℓid   h for all h ∈G n , where h̄ is the querelement of h. The latter ones can be placed on any places in the l.h.s. of (7.12) due to the Dörnte identities. Also, the multiplications in the l.h.s. of (7.5) can change their place due to the same reason. A general form of multiplace representations can be found by applying the Dörnte identities to each n-ary product in the l.h.s. of (7.5). Then, using (7.12) we have schematically   t1  ..    . g1   (n,n )  .  ′ ′  (n,n )  tℓµ   Πk .. = Πk   , (7.13)  g   gk  ..   .  ℓid  g where g is an arbitrary fixed element of the n-ary group and ta = µn [ga1 , . . . , gan−1 , ḡ] , a = 1, . . . , ℓµ . (7.14) This is the special shape of some multiplace representations, while the concrete formulas should be obtained in each case separately. Nevertheless, some conclusions can be drawn from (7.13). Firstly, (n,n′ ) the equivalence classes on which Πk is constant are determined by fixing ℓµ + 1 elements, i.e. by the surface ta = const, g = const. Secondly, some k-place representations of a n-ary group can be reduced to ℓµ -place representations of its retract. In the case ℓµ = 1, multiplace representations of a n-ary group derived from a binary group correspond to ordinary representations of the latter (see B OROWIEC ET AL . [2006], D UDEK [2007]).  POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 27 (n,2) Example 7-1. Let us consider the case of a binary multiplace representation Π2n−2 of n-ary group G n = hG; µn i with two multiplications ℓµ = 2 defined by the associativity preserving equation       µn [g1 , u1 , . . . un−2 , g2 ] g1 g2   u′1     u1     u′1     ..     ..     ..    .   .   .        (n,2)  u′n−2  (n,2)  un−2  (n,2)  u′n−2  Π2n−2   = Π 2n−2   ∗ Π 2n−2   (7.15)  µn [h1 , v1 , . . . vn−2 , h2 ]   h1   h2         v1′   v1   v1′         ..   ..   ..   .   .   .  ′ ′ vn−2 vn−2 vn−2 and normalizing condition    h    ..  n−2    .       h   h̄  (n,2)   Π2n−2    = idV , (7.16)  h       .. n−2    .      h  h̄ where h ∈G n is arbitrary. Using (7.15) and (7.16) for a general form of this (2n − 2)-place repre- sentation we have          µn [g1 , u1 , . . . un−2 , h]  g1 g1 h     h      u1   u1   ..   ..    ..     ..     . n − 2     . n − 3              .   .   h   h       h̄    (n,2) (n,2)  un−2  (n,2)  un−2  (n,2)   (n,2)  h̄  Π2n−2 = Π2n−2   = Π2n−2   ∗ Π2n−2    = Π2n−2  .  h1   h1   h    µn [h 1 , v1 , . . . vn−2 , h]            v1   v1   ..   h     ..     ..     . n − 2     ..    . n − 3   .   .   h       vn−2 vn−2 h̄  h  h̄ (7.17) The Dörnte identity applied to the first elements of the products in the r.h.s. of (7.17) together with associativity of µn gives     n−2 z }| {  µn g, . . . , g, tg , h                   h     g  h   g      .   ..   ..  n−2   ..   .. n−3   . n −2   .  . n − 2                     h     h   g     g       h̄    (n,2) (n,2)  h̄  (n,2)  tg  (n,2)   (n,2)  tg  Π2n−2 = Π2n−2   n−2   = Π 2n−2   ∗Π2n−2    = Π2n−2   ,  z }| {   g     h    g              µn g, . . . , g, th , h    .. .. .. . n −2 . n − 2     n−2               .            h     g   h   g   th th  .  .  h̄  . n − 3       h  h̄ (7.18) 28 STEVEN DUPLIJ where tg = µn [ḡ, g1 , u1 , . . . un−2] , th = µn [ḡ, h1 , v1 , . . . vn−2 ] . (7.19) (n,2) Thus, the equivalent classes of the multiplace representation Π2n−2 (7.15) are determined by the (ℓµ + 1 = 3)-element surface tg = const, th = const, g = const. (7.20) Note that the “ℓµ -place reduction” of a multiplace representation is possible not for all associativity preserving heteromorphism equations. For instance, if to exchange vi ↔ vi′ in l.h.s. of (7.15), then the associativity remains, but the “ℓµ -place reduction”, analogous to (7.18) will not be possible. In case, when it is possible, the corresponding ℓµ -place representation can be realized on the binary retract of G n (for some special n-ary groups and ℓµ = 1 see see B OROWIEC " ETn−2 AL . [2006], D UDEK # def z }| { [2007]). Indeed, let Gret 2 = hG, ⊛i = retg hG; µn i, where g1 ⊛ g2 = µn g1 , g, . . . , g, g2 , g1 , g2 , g ∈ G and we define the reduced ℓµ -place representation (now ℓµ = 2) through (7.18) as follows    g    ..    . n − 2         g    g1 def (n,2)  g1  Πret g  2 = Π2n−2  . (7.21) g2  g     ..  . n − 2         g  g2 From (7.17) and (7.16) we obtain     n−2 z }| {  µn g, . . . , g, g1 , g                g   g    g       ..   ..   .  ..    . n − 2     . n − 2     n−3                 g   g   g        g1 g1′ (n,2)  g1  (n,2)  g1 ′ (n,2)  ′ g1 Πret g ∗ Πret g     2 2 = Π2n−2   ∗ Π2n−2   = Π2n−2   g2 g2′  g     g     n−2 z }| {   ..   ..     µn g, . . . , g, g2 , g   . n − 2 2    n −        .            g   g    g   ′   g2 g2   ..   .  n − 3       g  g2′    g    .. n−2      .    g      g   ..  . n − 2        n−2     z }| {       µn g1 , g, . . . , g, g1′     g       (n,2)   (n,2)  g1 ⊛ g1′  ret g g1 ⊛ g1′ = Π2n−2   =Π 2n−2   = Π2 .  g      g    g2 ⊛ g2′  ..   ..   . n−2    . n − 2           g   g         n−2 z }| { ′  g2 ⊛ g2′  µn g2 , g, . . . , g, g2    POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 29   g1 In the framework of our classification Πret 2 g is a hom-like 2-place (binary) representation. g2 The above formulas describe various properties of multiplace representations, but they give no idea of how to build representations for concrete polyadic systems. The most common method of representation construction uses the concept of a group action on a set (see, e.g., K IRILLOV [1976]). Below we extend this concept to the multiplace case and use corresponding heteromorphisms, as it was done above for homomorphisms and representations. 8. M ULTIACTIONS AND G- SPACES Let G n = hG; µn i be a polyadic system and X be a set. A (left) 1-place action of G n on X is the (n) external binary operation ρ1 : G × X → X such that it is consistent with the multiplication µn , i.e. composition of the binary operations ρ1 {g|x} gives the n-ary product, that is, n n o o (n) (n) (n) (n) ρ1 {µn [g1 , . . . gn ] |x} = ρ1 g1 |ρ1 g2 | . . . |ρ1 {gn |x} . . . , g1 , . . . , gn ∈ G, x ∈ X. (8.1) If the polyadic system is a n-ary group, then in addition to (8.1) it is implied the there exist such (n) ex ∈ G (which may or may not coincide with the unity of G n ) that ρ1 {ex |x} = x for all x ∈ X, (n) and the mapping x 7→ ρ1 {ex |x} is a bijection of X. The right 1-place actions of G n on X are defined in a symmetric way, and therefore we will consider below only one of them. Obviously, we (n) (n′ ) cannot compose ρ1 and ρ1 with n 6= n′ . Usually X is called a G-set or G-space depending on its properties (see, e.g., H USEM ÖLLER ET AL . [2008]). The application of the 1-place action defined by (8.1) to the representation theory of n-ary groups gave mostly repetitions of the ordinary (binary) group representation results (except for trivial b- derived ternary groups) D UDEK AND S HAHRYARI [2012]. Also, it is obviously seen that the con- struction (8.1) with the binary external operation ρ1 cannot be applied for studying the most important regular representations of polyadic systems, when the X coincides with Gn itself and the action arises from translations. Here we introduce the multiplace concept of action for polyadic systems, which is consistent with heteromorphisms and multiplace representations. Then we will show how it naturally appears when X = G n and apply it to construct examples of representations including the regular ones. For a polyadic system G n = hG; µn i and a set X we introduce an external polyadic operation ρk : G×k × X → X, (8.2) which is called a (left) k-place action or multiaction. To generalize the 1-action composition (8.1), we use the analogy with multiplication laws of the heteromorphisms (5.14) and the multiplace repre- sentations (7.5) and propose (schematically)       µn [g1 , . . . , gn ] ,     n′   . .   ℓµ    z }|   .        {    µ g       g 1   g k(n ′ −1)  (n) n n(ℓµ −1) , . . . , gnℓµ    ..  . . .  ρ(n)  ..    ρk   x = ρ(n) .  .  x . . . . (8.3) k  k    g nℓµ +1 ,              ..   g k g kn ′    . ℓ id         g nℓµ +ℓid   The connection between all the parameters here is the same as in the arity changing formulas (5.17)–(5.18). Composition of mappings is associative, and therefore in concrete cases we can use 30 STEVEN DUPLIJ the associative quiver technique, as it is described in the previous sections. If G n is n-ary group, then we should add to (8.3) the “normalizing” relations analogous with (7.10) or (7.12). So, if there is a unity e ∈G n , then     e   ρk (n) ..  x = x, for all x ∈ X. (8.4)  .   e In terms of the querelement, the normalization has the form      h̄       .. ℓ        .  µ    h̄   (n)  ρk   x = x, for all x ∈ X and for all h ∈ G n . (8.5)   h      ..     . ℓ id         h  (n) (n) The multiaction ρk is transitive, if any two points x and y in X can be “connected” by ρk , i.e. there exist g1 , . . . , gk ∈Gn such that     g1   (n) ρk ..  x = y. (8.6)  .   gk (n) If g1 , . . . , gk are unique, then ρk is sharply transitive. The subset of X, in which any points are connected by (8.6) with fixed g1 , . . . , gk can be called the multiorbit of X. If there is only one multiorbit, then we call X the heterogenous G-space (by analogy with the homogeneous one). By analogy with the (ordinary) 1-place actions, we define a G-equivariant map Ψ between two G-sets X and Y by (in our notation)         g1    g1   Ψ ρk (n) ..  x  = ρ(n) ..  Ψ (x) ∈ Y, (8.7)  .   k  .   gk gk which makes G-space into a category (for details, see, e.g., H USEM ÖLLER ET AL . [2008]). In the particular case, when X is a vector space over K, the multiaction (8.2) can be called a multi-G-module which satisfies (8.4) and the additional (linearity) conditions           g1    g1    g1   ρk (n) ..  ax + by = aρ(n) ..  x + bρ(n) ..  y , (8.8)  .   k  .   k  .   gk  gk  gk where a, b ∈ K. Then, comparing (7.5) and (8.3) we can define a multiplace representation as a multi-G-module by the following formula      g1  g1   (n,n′ )  .  .. (n)  Πk .. (x) = ρk .x .   (8.9)  gk gk  In a similar way, one can generalize to polyadic systems many other notions from group action theory K IRILLOV [1976].  POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 31 9. R EGULAR MULTIACTIONS The most important role in the study of polyadic systems is played by the case, when X =G n , and the multiaction coincides with the n-ary analog of translations M AL’ TCEV [1954], so called i-translations B ELOUSOV [1972]. In the binary case, ordinary translations lead to regular represen- reg(n) tations K IRILLOV [1976], and therefore we call such an action a regular multiaction ρk . In this connection, the analog of the Cayley theorem for n-ary groups was obtained in G AL’ MAK [1986, 2001b]. Now we will show in examples, how the regular multiactions can arise from i-translations. Example 9-1. Let G 3 be a ternary semigroup, k = 2, and X =G 3 , then 2-place (left) action can be defined as    reg(3) g  def ρ2 u = µ3 [g, h, u] . (9.1) h  This gives the following composition law for two regular multiactions      reg(3) g1  reg(3) g2  ρ2 ρ u = µ3 [g1 , h1 , µ3 [g2 , h2 , u]] h1  2 h2     reg(3) µ3 [g1 , h1 , g2 ]  = µ3 [µ3 [g1 , h1 , g2] , h2 , u] = ρ2 u . h2  (9.2) Thus, using the regular 2-action (9.1) we have, in fact, derived the associative quiver corresponding to (5.6). The formula (9.1) can be simultaneously treated as a 2-translation B ELOUSOV [1972]. In this way, the following left regular multiaction     g1   reg(n) ρk ..  h def = µn [g1 , . . . , gk , h] , (9.3)  .   gk corresponds to (5.10), where in the r.h.s. there is the i-translation with i = n. The right regular multiaction corresponds to the i-translation with i = 1. The binary composition of the left regular multiactions corresponds to (5.10). In general, the value of i fixes the minimal final arity n′reg , which differs for even and odd values of the initial arity n. It follows from (9.3) that for regular multiactions the number of places is fixed kreg = n − 1, (9.4) and the arity changing formulas (5.17)–(5.18) become n′reg = n − ℓid (9.5) n′reg = ℓµ + 1. (9.6) From (9.5)–(9.6) we conclude that for any n a regular multiaction having one multiplication ℓµ = 1 is binarizing and has n − 2 intact elements. For n = 3 see (9.2). Also, it follows from (9.5) that for regular multiactions the number of intact elements gives exactly the difference between initial and final arities. If the initial arity is odd, then there exists a special middle regular multiaction generated by the i-translation with i = (n + 1) 2. For n = 3 the corresponding associative quiver is (6.5) and such 2-actions were used in B OROWIEC ET AL . [2006] to construct middle representations of ternary 32 STEVEN DUPLIJ groups, which did not change arity (n′ = n). Here we give a more complicated example of a middle regular multiaction, which can contain intact elements and can therefore change arity. Example 9-2. Let us consider 5-ary semigroup and the following middle 4-action      g    i=3      reg(5) h  ↓ ρ4 s = µ5 g, h, s , u, v  . (9.7) u        v   Using (9.6) we observe that there are two possibilities for the number of multiplications ℓµ = 2, 4. The last case ℓµ = 4 is similar to the vertical associative quiver (6.5), but with a more complicated l.h.s., that is    µ [g , h , g , h g ]    5 1 1 2 2, 3      reg(5) µ5 [h3 , g4 , h4 , g5, h5 ]  ρ4 s =   µ5 [u5 , v5 , u4, v4 , u3]     µ [v , u , v , u , v ]   5 3 2 2 1 1              g 1     g 2     g 3     g 4     g 5               reg(5) h1  reg(5)  h2  reg(5)  h3  reg(5)  h4  reg(5)  h5  ρ4 ρ ρ ρ ρ s . (9.8)   u1  4   u2  4   u3  4   u4  4   u5           v   v   v   v   v   1 2 3 4 5 Now we have an additional case with two intact elements ℓid and two multiplications ℓµ = 2 as             µ 5 [g 1 , h1 , g 2 , h2, g 3 ]       g 1     g 2     g 3            reg(5) h3  reg(5) h1  reg(5) h2  reg(5) h3  ρ4  s = ρ 4 ρ ρ s ,   µ5 [h3 , v3 , u2 , v2 , u1 ]      u1  4   u2  4   u3        v1    v   v   v   1 2 3 (9.9) with arity changing from n = 5 to n′reg = 3. In addition to (9.9) we have 3 more possible regular multiactions due to the associativity of µ5 , when the multiplication brackets in the sequences of 6 elements in the first two rows and the second two ones can be shifted independently. For n > 3, in addition to left, right and middle multiactions, there exist intermediate cases. First, observe that the i-translations with i = 2 and i = n − 1 immediately fix the final arity n′reg = n. Therefore, the composition of multiactions will be similar to (9.8), but with some permutations in the l.h.s. Now we consider some multiplace analogs of regular representations of binary groups K IRILLOV [1976]. The straightforward generalization is to consider the previously introduced regular mul- tiactions (9.3) in the r.h.s. of (8.9). Let G n be a finite polyadic associative system and KG n be a vector space spanned by G n (some properties of n-ary group rings were considered in Z EKOVI Ć AND A RTAMONOV P[1999, 2002]). This means that any element of KG n can be uniquely presented in the form w = l al · hl , al ∈ K, hl ∈ G. Then, using (9.3) and (8.9) we define the i-regular k-place representation by   g1 X reg(i)  .  Πk .. (w) = al · µk+1 [g1 . . . gi−1 hl gi+1 . . . gk ] . (9.10) gk l Comparing (9.3) and (9.10) one can conclude that all the general properties of multiplace regular representations are similar to those of the regular multiactions. If i = 1 or i = k, the multiplace  POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 33 representation is called a right or left regular representation respectively. If k is even, the representa- tion with i = k2 + 1 is called a middle regular representation. The case k = 2 was considered in B OROWIEC ET AL . [2006] for ternary groups. 10. M ULTIPLACE REPRESENTATIONS OF TERNARY GROUPS Let us consider the case n = 3, k = 2 in more detail, paying attention to its special peculiarities, which corresponds to the 2-place (bi-element) representations of ternary groups B OROWIEC ET AL . [2006]. Let V be a vector space over K and End V be a set of linear endomorphisms of V . From now  bysquare brackets only, as follows µ3 [g1 , g2 , g3 ] ≡ [g1 g2 g3 ], on we denote the ternary multiplication g1 and use the “horizontal” notation Π ≡ Π (g1 , g2 ). g2 Definition 10-1. A left representation of a ternary group G, [ ]) in V is a map ΠL : G × G → End V such that ΠL (g1 , g2 ) ◦ ΠL (g3 , g4 ) = ΠL ([g1 g2 g3 ] , g4 ) , (10.1) ΠL (g, g) = idV , (10.2) where g, g1, g2 , g3 , g4 ∈ G. Replacing in (10.2) g by g we obtain ΠL (g, g) = idV , which means that in fact (10.2) has the form L Π (g, g) = ΠL (g, g) = idV , ∀g ∈ G. Note that the axioms considered in the above definition are the natural ones satisfied by left multiplications g 7→ [abg]. For all g1 , g2, g3 , g4 ∈ G we have ΠL ([g1 g2 g3 ] , g4 ) = ΠL (g1 , [g2 g3 g4 ]) . For all g, h, u ∈ G we have ΠL (g, h) = ΠL ([guu], h) = ΠL (g, u) ◦ ΠL (u, h) (10.3) and ΠL (g, u) ◦ ΠL (u, g) = ΠL (u, g) ◦ ΠL (g, u) = idV , (10.4)
−1 and therefore every ΠL (g, u) is invertible and ΠL (g, u) = ΠL (u, g). This means that any left representation gives a representation of a ternary group by a binary group B OROWIEC ET AL . [2006]. If the ternary group is medial, then ΠL (g1 , g2) ◦ ΠL (g3 , g4 ) = ΠL (g3 , g4 ) ◦ ΠL (g1 , g2 ) , i.e. the group so obtained is commutative. If the ternary group hG, [ ]i is commutative, then also ΠL (g, h) = ΠL (h, g), because ΠL (g, h) = ΠL (g, h) ◦ ΠL (g, g) = ΠL ([g h g] , g) = ΠL ([h g g] , g) = ΠL (h, g) ◦ ΠL (g, g) = ΠL (h, g) . In the case of a commutative and idempotent ternary group any of its left representations is idempotent
−1 and ΠL (g, h) = ΠL (g, h), so that commutative and idempotent ternary groups are represented by Boolean groups. Assertion 10-2. Let hG, [ ]i = der (G, ⊙) be a ternary group derived from a binary group hG, ⊙i, then there is one-to-one correspondence between representations of (G, ⊙) and left representations of (G, [ ]). 34 STEVEN DUPLIJ Indeed, because (G, [ ]) = der (G, ⊙), then g ⊙ h = [geh] and e = e, where e is unity of the binary group (G, ⊙). If π ∈ Rep (G, ⊙), then (as it is not difficult to see) ΠL (g, h) = π (g) ◦ π (h) is a left representation of hG, [ ]i. Conversely, if ΠL is a left representation of hG, [ ]i then π (g) = ΠL (g, e) is a representation of (G, ⊙). Moreover, in this case ΠL (g, h) = π (g) ◦ π (h), because we have ΠL (g, h) = ΠL (g, [ehe]) = ΠL ([geh], e) = ΠL (g, e) ◦ ΠL (h, e) = π (g) ◦ π (h) . Let (G, [ ]) be a ternary group and (G × G, ∗) be a semigroup used in the construction of left representations. According to Post P OST [1940] one says that two pairs (a, b), (c, d) of elements of G are equivalent, if there exists an element g ∈ G such that [abg] = [cdg]. Using a covering group we can see that if this equation holds for some g ∈ G, then it holds also for all g ∈ G. This means that ΠL (a, b) = ΠL (c, d) ⇐⇒ (a, b) ∼ (c, d), i.e. ΠL (a, b) = ΠL (c, d) ⇐⇒ [abg] = [cdg] for some g ∈ G. Indeed, if [abg] = [cdg] holds for some g ∈ G, then ΠL (a, b) = ΠL (a, b) ◦ ΠL (g, g) = ΠL ([abg], g) = ΠL ([cdg], g) = ΠL (c, d) ◦ ΠL (g, g) = ΠL (c, d). By analogy we can define Definition 10-3. A right representation of a ternary group (G, [ ]) in V is a map ΠR : G×G → End V such that ΠR (g3 , g4 ) ◦ ΠR (g1 , g2 ) = ΠR (g1 , [g2 g3 g4 ]) , (10.5) ΠR (g, g) = idV , (10.6) where g, g1, g2 , g3 , g4 ∈ G. From (10.5)-(10.6) it follows that ΠR (g, h) = ΠR (g, [u u h]) = ΠR (u, h) ◦ ΠR (g, u) . (10.7)

−1 It is easy to check that ΠR (g, h) = ΠL h, g = ΠL (g, h) . So it is sufficient to consider only left representations (as in the binary case). Consider the following example of a group algebra ternary generalization B OROWIEC ET AL . [2006]. Example 10-4. Let G be a ternary group and KG be a vector space P spanned by G, which means that any element of KG can be uniquely presented in the form t = ni=1 ki hi , ki ∈ K, hi ∈ G, n ∈ N (we do not assume that G has finite rank). Then left and right regular representations are defined by n X ΠLreg (g1 , g2 ) t = ki [g1 g2 hi ] , (10.8) i=1 n X ΠR reg (g1 , g2 ) t = ki [hi g1 g2 ] . (10.9) i=1 Let us construct the middle representations as follows. Definition 10-5. A middle representation of a ternary group hG, [ ]i in V is a map ΠM : G × G → End V such that ΠM (g3 , h3 ) ◦ ΠM (g2 , h2 ) ◦ ΠM (g1 , h1 ) = ΠM ([g3 g2 g1 ] , [h1 h2 h3 ]) , (10.10)

ΠM (g, h) ◦ ΠM g, h = ΠM g, h ◦ ΠM (g, h) = idV (10.11)  POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 35 It can be seen that a middle representation is a ternary group
homomorphism ΠM : G × Gop → der End V. Note that instead of (10.11) one can use ΠM g, h ◦ ΠM (g, h) = idV after changing g to g and taking into account that g = g. In the case of idempotent elements g and h we have ΠM (g, h) ◦ ΠM (g, h) = idV , which means that the matrices ΠM are Boolean. Thus all middle representation matrices of idempotent ternary groups are Boolean. The composition ΠM (g1 , h1 ) ◦ ΠM (g2 , h2 ) is not a middle representation, but the following proposition nevertheless holds. Let ΠM be a middle representation of a ternary group hG, [ ]i, then, if ΠLu (g, h) = ΠM (g, u) ◦ Π (h, u) is a left representation of hG, [ ]i, then ΠLu (g, h) ◦ ΠLu′ (g ′ , h′ ) = ΠLu′ ([ghu′] , h′ ), and, M ′ ′ if ΠR M M R R u (g, h) = Π (u, h) ◦ Π (u, g) is a right representation of hG, [ ]i, then Πu (g, h) ◦ Πu′ (g , h ) = R ′ ′ L R Πu (g, [hg h ]). In particular, Πu (Πu ) is a family of left (right) representations. If a middle representation ΠM of a ternary group hG, [ ]i satisfies ΠM (g, g) = idV for all g ∈ G, then it is a left and a right representation and ΠM (g, h) = ΠM (h, g) for all g, h ∈ G. Note that in general ΠM reg (g, g) 6= id. For regular representations we have the following commutation relations ΠLreg (g1 , h1 ) ◦ ΠR R L reg (g2 , h2 ) = Πreg (g2 , h2 ) ◦ Πreg (g1 , h1 ) .  Let hG, [ ]i be a ternary group and let G × G, [ ]′ be a ternary group used in the construction of ′ the middle representation. In hG, [ ]i, and consequently in G × G, [ ] , we define the relation (a, b) ∼ (c, d) ⇐⇒ [aub] = [cud] for all u ∈′  G. It is not difficult to see that this relation is a congruence in the ternary group G × G, [ ] . For regular representations ΠM reg (a, b) = ΠM reg (c, d) if (a, b) ∼ (c, d). We have the following relation a h a′ ⇐⇒ a = [ga′ g] for some g ∈ G or equivalently a h a′ ⇐⇒ a′ = [gag] for some g ∈ G. It is not difficult to see that it is an equivalence relation on hG, [ ]i, moreover, if hG, [ ]i is medial, then this relation is a congruence. Let G × G, [ ]′ be a ternary group used in a construction of middle representations, then (a, b) h (a′ , b) ⇐⇒ a′ = [gag] and b′ = [hbb ] for some (g, h) ∈ G × G  is an equivalence relation on G × G, [ ]′ . Moreover, if (G, [ ]) is medial, then this relation is a congruence. Unfortunately, however it is a weak relation. In a ternary group Z3 , where [ghu] = (g − h + u) (mod 3) we have only one class, i.e. all elements are equivalent. In Z4 with the operation [ghu] = (g + h + u + 1) (mod 4) we have a h a′ ⇐⇒ a = a′ . However, for this relation the following statement holds. If (a, b) h (a′ , b′ ), then tr ΠM (a, b) = tr ΠM (a′ , b′ ). We have tr(AB) = tr(BA) for all A, B ∈ EndV , and
tr ΠM (a, b) = tr ΠM ([ga′ g], [hb′ h] ) = tr ΠM (g, h) ◦ ΠM (a′ , b′ ) ◦ ΠM (g, h)

= tr ΠM (g, h) ◦ ΠM (g, h) ◦ ΠM (a′ , b′ ) = tr idV ◦ ΠM (a′ b′ ) = tr ΠM (a′ , b′ ) In our derived case the connection with standard group representations is given by the following. Let (G, ⊙) be a binary group, and the ternary derived group as hG, [ ]i = der (G, ⊙). There is one-to-one correspondence between a pair of commuting binary group representations and a middle 36 STEVEN DUPLIJ ternary derived group representation. Indeed, let π, ρ ∈ Rep (G, ⊙), π (g) ◦ ρ (h) = ρ (h) ◦ π (g) and ΠL ∈ Rep (G, [ ]). We take
ΠM (g, h) = π (g) ◦ ρ h−1 , π (g) = ΠM (g, e) , ρ (g) = ΠM (e, g) . Then using (10.10) we prove the needed representation laws. ′ Let hG, [ ]i be a fixed ternary group, G × G, [ ] a corresponding ternary groupused in the con- struction of middle representations, ((G × G) , ⊛) a covering group of G × G, [ ]′ , (G × G, ⋄) = ∗ ret(a,b) (G × G, h i). If ΠM (a, b) is a middle representation of hG, [ ]i, then π defined by π(g, h, 0) = ΠM (g, h), π(g, h, 1) = ΠM (g, h) ◦ ΠM (a, b) is a representation of the covering group P OST [1940]. Moreover ρ(g, h) = ΠM (g, h) ◦ ΠM (a, b) = π(g, h, 1) is a representation of the above retract induced by (a, b). Indeed, (a, b) is the identity of this retract and ρ(a, b) = ΠM (a, b) ◦ ΠM (a, b) = idV . Similarly ρ ((g, h) ⋄ (u, u)) = ρ (h(g, h), (a, b), (u, u)i) = ρ ([gau], [ubh]) = ΠM ([gau], [ubh])) ◦ ΠM (a, b) = ΠM (g, h) ◦ ΠM (a, b) ◦ ΠM (u, u) ◦ ΠM (a, b) = ρ(g, h) ◦ ρ(u, u)  But τ (g) = (g, g) is an embedding of (G, [ ]) into G × G, [ ]′ . Hence µ defined by µ(g, 0) = ΠM (g, g) and µ(g, 1) = ΠM (g, g) ◦ ΠM (a, a) is a representation of a covering group G∗ for (G, [ ]) (see the Post theorem P OST [1940] for a = c). On the other hand, β(g) = ΠM (g, g) ◦ ΠM (a, a) is a representation of a binary retract (G, · ) = reta (G, [ ]). Thus β can induce some middle representation of (G, [ ]) (by the Gluskin-Hosszú theorem G LUSKIN [1965]). Note that in the ternary group of quaternions hK, [ ]i (with norm 1), where [ghu] = ghu(−1) = −ghu and gh is the multiplication ofquaternions (−1 is a central element) we have 1 = −1, −1 = 1 and g = g for others. In K × K, [ ]′ we have (a, b) ∼ (−a, −b) and (a, −b) ∼ (−a, b), which gives 32 two-element equivalence classes. The embedding τ (g) = (g, g) suggest that ΠM (i, i) = π(i) 6= π(−i) = ΠM (−i, −i). Generally ΠM (a, b) 6= ΠM (−a, −b) and ΠM (a, −b) 6= ΠM (−a, b). The relation (a, b) ∼ (c, d) ⇐⇒ [abg] = [cdg] for all g ∈ G is a congruence on (G × G, ∗). Note that this relation can be defined as ”for some g”. Indeed, using a covering group we can see that if [abg] = [cdg] holds for some g then it holds also for all g. Thus π L (a, b) = ΠL (c, d) ⇐⇒ (a, b) ∼ (c, d). Indeed ΠL (a, b) = ΠL (a, b) ◦ ΠL (g, g) = ΠL ([a b g], g) = ΠL ([c d g], g) = ΠL (c, d) ◦ ΠL (g, g) = ΠL (c, d). We conclude, that every left representation of a commutative group hG, [ ]i is a middle representa- tion. Indeed, ΠL (g, h) ◦ ΠL (g, h) = ΠL ([g h g], h) = ΠL ([g g h], h) = ΠL (h, h) = idV and ΠL (g1 , g2 ) ◦ ΠL (g3 , g4 ) ◦ ΠL (g5 , g6) = ΠL ([[g1 g2 g3 ]g4 g5 ], g6 ) = ΠL ([[g1 g3 g2 ]g4 g5 ], g6 ) = ΠL ([g1 g3 [g2 g4 g5 ]], g6 ) = ΠL ([g1 g3 [g5 g4 g2 ]], g6 ) = ΠL ([g1 g3 g5 ], [g4 g2 g6 ]) = ΠL ([g1 g3 g5 ], [g6 g4 g2 ]). Note that the converse holds only for the special kind of middle representations such that M Π (g, g) = idV . Therefore,  POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 37 Assertion 10-6. There is one-one correspondence between left representations of hG, [ ]i and binary representations of the retract reta (G, [ ]). Indeed, let ΠL (g, a) be given, then ρ(g) = ΠL (g, a) is such representation of the retract, as can be directly shown. Conversely, assume that ρ(g) is a representation of the retract reta (G, [ ]). Define ΠL (g, h) = ρ(g) ◦ ρ(h)−1 , then ΠL (g, h) ◦ ΠL (u, u) = ρ(g) ◦ ρ(h)−1 ◦ ρ(u) ◦ ρ(u)−1 = ρ(g ⊛ (h)−1 ◦ ⊛u) ◦ ρ(u)−1 = ρ([ [ g a [ a h a ] ] a u ]) ◦ ρ(u)−1 = ρ([ g h g ]) ◦ ρ(u)−1 = ΠL ([ g h u ], u), 11. M ATRIX REPRESENTATIONS OF TERNARY GROUPS Here we give several examples of matrix representations for concrete ternary groups. Let G = Z3 ∋ {0, 1, 2} and the ternary multiplication be [ghu] = g − h + u. Then [ghu] = [uhg] and 0 = 0, 1 = 1, 2 = 2, therefore (G, [ ]) is an idempotent medial ternary group. Thus ΠL (g, h) = ΠR (h, g) and ΠL (a, b) = ΠL (c, d) ⇐⇒ (a − b) = (c − d)mod 3. (11.1) The calculations give the left regular representation in the manifest matrix form ΠLreg (0, 0) = ΠLreg (2, 2) = ΠLreg (1, 1) = ΠR reg (0, 0)   1 0 0 = ΠR R reg (2, 2) = Πreg (1, 1) =  0 1 0  = [1] ⊕ [1] ⊕ [1], (11.2) 0 0 1   0 1 0 ΠLreg (2, 0) = ΠLreg (1, 2) = ΠLreg (0, 1) = ΠR R R reg (2, 1) = Πreg (1, 0) = Πreg (0, 2) =  0 0 1  1 0 0  √  1 3      √2− − 1 1 √ 1 1 √ = [1] ⊕  2  = [1] ⊕ − + i 3 ⊕ − − i 3 ,  (11.3) 3 1 2 2 2 2 − 2 2   0 0 1 ΠLreg (2, 1) = ΠLreg (1, 0) = ΠLreg (0, 2) = ΠR R R reg (2, 0) = Πreg (1, 2) = Πreg (0, 1) =  1 0 0  0 1 0  √  1 3      −√2 1 1 2  = [1] ⊕ − − i 3 ⊕ − + i 3 .  √ 1 1 √ = [1] ⊕  (11.4) 3 1 2 2 2 2 − − 2 2 Consider next the middle representation construction. The middle regular representation is defined by X n ΠMreg (g , 1 2g ) t = ki [g1 hi g2 ] . i=1 For regular representations we have ΠM R R M reg (g1 , h1 ) ◦ Πreg (g2 , h2 ) = Πreg (h2 , h1 ) ◦ Πreg (g1 , g2 ) , (11.5) ΠM reg (g1 , h1 ) ◦ ΠLreg (g2 , h2 ) = ΠLreg (g1 , g2 ) ◦ ΠM reg (h2 , h1 ) . (11.6) 38 STEVEN DUPLIJ For the middle regular representation matrices we obtain   1 0 0 ΠM reg (0, 0) = ΠM reg (1, 2) = ΠM reg (2, 1) =  0 0 1 , 0 1 0   0 1 0 ΠM reg (0, 1) = ΠM reg (1, 0) = ΠM reg (2, 2) =  1 0 0 , 0 0 1   0 0 1 ΠM reg (0, 2) = ΠM reg (2, 0) = ΠM reg (1, 1) =  0 1 0 . 1 0 0 The above representation ΠM reg of hZ3 , [ ]i is equivalent to the orthogonal direct sum of two irre- ducible representations   M M M −1 0 Πreg (0, 0) = Πreg (1, 2) = Πreg (2, 1) = [1] ⊕ , 0 1  √  1 3 − ΠM M M  2 √ 2  reg (0, 1) = Πreg (1, 0) = Πreg (2, 2) = [1] ⊕  , 3 1 − − 2 √ 2   1 3  √2 2  ΠM M M reg (0, 2) = Πreg (2, 0) = Πreg (1, 1) = [1] ⊕  , 3 1 − 2 2 i.e. one-dimensional trivial [1] and two-dimensional irreducible. Note, that in this example ΠM (g, g) = ΠM (g, g) 6= idV , but ΠM (g, h) ◦ ΠM (g, h) = idV , and so ΠM are of the second degree. Consider a more complicated example of left representations. Let G = Z4 ∋ {0, 1, 2, 3} and the ternary multiplication be [ghu] = (g + h + u + 1) mod 4. (11.7) We have the multiplication table     1 2 3 0 2 3 0 1  2 3 0 1   3 0 1 2  [g, h, 0] =   3  [g, h, 1] =   0 1 2   0 1 2 3  0 1 2 3 1 2 3 0     3 0 1 2 0 1 2 3  0 1 2 3   1 2 3 0  [g, h, 2] =   1  [g, h, 3] =   2 3 0   2 3 0 1  2 3 0 1 3 0 1 2 Then the skew elements are 0 = 3, 1 = 2, 2 = 1, 3 = 0, and therefore (G, [ ]) is a (non-idempotent) commutative Pn ternary group. The left representation is defined by the expansion ΠLreg (g1 , g2 ) t = i=1 ki [g1 g2 hi ], which means that (see the general formula (9.10)) ΠLreg (g, h) |u >= | [ghu] > .  POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 39 Analogously, for right and middle representations ΠR M reg (g, h) |u >= | [ugh] >, Πreg (g, h) |u >= | [guh] > . Therefore | [ghu] >= | [ugh] >= | [guh] > and ΠLreg (g, h) = ΠR M reg (g, h) |u >= Πreg (g, h) |u >, so ΠLreg (g, h) = ΠR M reg (g, h) = Πreg (g, h). Thus it is sufficient to consider the left representation only. In this case the equivalence is ΠL (a, b) = ΠL (c, d) ⇐⇒ (a + b) = (c + d)mod 4, and we obtain the following classes   0 0 0 1  1 0 0 0  ΠLreg (0, 0) = ΠLreg (1, 3) = ΠLreg (2, 2) = ΠLreg (3, 1) =   0 1 0 0  = [1] ⊕ [−1] ⊕ [−i] ⊕ [i] ,  0 0 1 0   0 0 1 0  0 0 0 1  ΠLreg (0, 1) = ΠLreg (1, 0) = ΠLreg (2, 3) = ΠLreg (3, 2) =   1 0 0 0  = [1] ⊕ [−1] ⊕ [−1] ⊕ [−1] ,  0 1 0 0   0 1 0 0  0 0 1 0  ΠLreg (0, 2) = ΠLreg (1, 1) = ΠLreg (2, 0) = ΠLreg (3, 3) =   0 0 0 1  = [1] ⊕ [−1] ⊕ [i] ⊕ [−i] ,  1 0 0 0   1 0 0 0  0 1 0 0  ΠLreg (0, 3) = ΠLreg (1, 2) = ΠLreg (2, 1) = ΠLreg (3, 0) =   0 0 1 0  = [1] ⊕ [−1] ⊕ [1] ⊕ [1] .  0 0 0 1 It is seen that, due to the fact that the ternary operation (11.7) is commutative, there are only one- dimensional irreducible left representations. Let us “algebralize” the above regular representations in the following way. From (10.1) we have, for the left representation ΠLreg (i, j) ◦ ΠLreg (k, l) = ΠLreg (i, [jkl]) , (11.8) where [jkl] = j − k + l, i, j, k, l ∈ Z3 . Denote γiL = ΠLreg (0, i), i ∈ Z3 , then we obtain the algebra with the relations γiL γjL = γi+j L . (11.9) Conversely, any matrix representation of γi γj = γi+j leads to the left representation by ΠL (i, j) = M γj−i . In the case of the middle regular representation we introduce γk+l = ΠM reg (k, l), k, l ∈ Z3 , then we obtain γiM γjM γkM = γ[ijk] M , i, j, k ∈ Z3 . (11.10) In some sense (11.10) can be treated as a ternary analog of the Clifford algebra. As before, any matrix representation of (11.10) gives the middle representation ΠM (k, l) = γk+l . 40 STEVEN DUPLIJ 12. T ERNARY ALGEBRAS AND H OPF ALGEBRAS Let us consider associative ternary algebras C ARLSSON [1980], DE A ZCARRAGA AND I ZQUIERDO [2010]. One can introduce an autodistributivity property [[xyz] ab] = [[xab] [yab] [zab]] (see D UDEK [1993]). If we take 2 ternary operations { , , } and [ , , ], then distributivity is given by {[xyz] ab} = [{xab} {yab} {zab}]. If (+) is a binary operation (addition), then left linearity is [(x + z) , a, b] = [xab] + [zab] . (12.1) By analogy one can define central (middle) and right linearity. Linearity is defined, when left, middle and right linearity hold simultaneously.
Definition 12-1. An associative ternary algebra is a triple A, µ3 , η (3) , where A is a linear space over a field K, µ3 is a linear map A ⊗ A ⊗ A → A called ternary multiplication µ3 (a ⊗ b ⊗ c) = [abc] which is ternary associative [[abc] de] = [a [bcd] e] = [ab [cde]] or µ3 ◦ (µ3 ⊗ id ⊗ id) = µ3 ◦ (id ⊗µ3 ⊗ id) = µ3 ◦ (id ⊗ id ⊗µ3 ) . (12.2) There are two types D UPLIJ [2001] of ternary unit maps η (3) : K → A: 1) One strong unit map


µ3 ◦ η (3) ⊗ η (3) ⊗ id = µ3 ◦ η (3) ⊗ id ⊗η (3) = µ3 ◦ id ⊗η (3) ⊗ η (3) = id; (12.3) (3) (3) 2) Two sequential units η1 and η2 satisfying       (3) (3) (3) (3) (3) (3) µ3 ◦ η1 ⊗ η2 ⊗ id = µ3 ◦ η1 ⊗ id ⊗η2 = µ3 ◦ id ⊗η1 ⊗ η2 = id; (12.4) In first case the ternary analog of the binary relation η (2) (x) = x1, where x ∈ K, 1 ∈ A, is η (3) (x) = [x, 1, 1] = [1, 1, x] = [1, x, 1] . (12.5) Let (A, µA , ηA ), (B, µB , ηB ) and (C, µC , ηC ) be ternary algebras, then the ternary tensor product space A ⊗ B ⊗ C is naturally endowed with the structure of an algebra. The multiplication µA⊗B⊗C on A ⊗ B ⊗ C reads [(a1 ⊗ b1 ⊗ c1 )(a2 ⊗ b2 ⊗ c2 )(a3 ⊗ b3 ⊗ c3 )] = [a1 a2 a3 ] ⊗ [b1 b2 b3 ] ⊗ [c1 c2 c3 ] , (12.6) and so the set of ternary algebras is closed under taking ternary tensor products. A ternary algebra map (homomorphism) is a linear map between ternary algebras f : A → B which respects the ternary algebra structure f ([xyz]) = [f (x) , f (y) , f (z)] , (12.7) f (1A ) = 1B . (12.8) Let C be a linear space over a field K. Definition 12-2. A ternary comultiplication ∆3 is a linear map over a field K such that ∆3 : C → C ⊗ C ⊗ C. (12.9) Pn In the standard Sweedler notations S WEEDLER [1969] ∆3 (a) = i=1 a′i ⊗a′′i ⊗a′′′ i = a(1) ⊗a(2) ⊗ a(3) . Consider different possible types of ternary coassociativity D UPLIJ [2001], B OROWIEC ET AL . [2001]. (1) A standard ternary coassociativity (∆3 ⊗ id ⊗ id) ◦ ∆3 = (id ⊗∆3 ⊗ id) ◦ ∆3 = (id ⊗ id ⊗∆3 ) ◦ ∆3 , (12.10)  POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 41 (2) A nonstandard ternary Σ-coassociativity (Gluskin-type positional operatives) (∆3 ⊗ id ⊗ id) ◦ ∆3 = (id ⊗ (σ ◦ ∆3 ) ⊗ id) ◦ ∆3 , where σ ◦ ∆3 (a) = ∆3 (a) = a(σ(1)) ⊗ a(σ(2)) ⊗ a(σ(3)) and σ ∈ Σ ⊂ S3 . (3) A permutational ternary coassociativity (∆3 ⊗ id ⊗ id) ◦ ∆3 = π ◦ (id ⊗∆3 ⊗ id) ◦ ∆3 , where π ∈ Π ⊂ S5 . A ternary comediality is (∆3 ⊗ ∆3 ⊗ ∆3 ) ◦ ∆3 = σmedial ◦ (∆3 ⊗ ∆3 ⊗ ∆3 ) ◦ ∆3 ,
where σmedial = 123456789 147258369 ∈ S9 . A ternary counit is defined as a map ε(3) : C → K. In general, ε(3) 6= ε(2) satisfying one of the conditions below. If ∆3 is derived, then maybe ε(3) = ε(2) , but another counits may exist. There are two types of ternary counits: (1) Standard (strong) ternary counit (ε(3) ⊗ ε(3) ⊗ id) ◦ ∆3 = (ε(3) ⊗ id ⊗ε(3) ) ◦ ∆3 = (id ⊗ε(3) ⊗ ε(3) ) ◦ ∆3 = id, (12.11) (3) (3) (2) Two sequential (polyadic) counits ε1 and ε2 (3) (3) (3) (3) (3) (3) (ε1 ⊗ ε2 ⊗ id) ◦ ∆ = (ε1 ⊗ id ⊗ε2 ) ◦ ∆ = (id ⊗ε1 ⊗ ε2 ) ◦ ∆ = id, (12.12) Below we will consider only the first standard type of associativity (12.10). The σ-cocommutativity is defined as σ ◦ ∆3 = ∆3 .
Definition 12-3. A ternary coalgebra is a triple C, ∆3 , ε(3) , where C is a linear space and ∆3 is a ternary comultiplication (12.9) which is coassociative in one of the above senses and ε(3) is one of the above counits. Let (A, µ3 ) be a ternary algebra and (C, ∆3 ) be a ternary coalgebra and f, g, h ∈ HomK (C, A). Ternary convolution product is [f, g, h]∗ = µ3 ◦ (f ⊗ g ⊗ h) ◦ ∆3 (12.13) 


 or in the Sweedler notation [f, g, h]∗ (a) = f a(1) g a(2) h a(3) . Definition 12-4. A ternary coalgebra is called derived, if there exists a binary (usual, see e.g. S WEEDLER [1969]) coalgebra ∆2 : C → C ⊗ C such that ∆3,der = (id ⊗∆2 ) ⊗ ∆2 . (12.14)

Definition 12-5. A ternary
bialgebra B is B, µ3 , η (3) , ∆3 , ε (3) for which B, µ3 , η (3) is a ternary (3) algebra and B, ∆3 , ε is a ternary coalgebra and they are compatible ∆3 ◦ µ3 = µ3 ◦ (id ⊗τ12 ⊗ id) ◦ ∆3 , (12.15)
where τ12 = 12 21 . One can distinguish four kinds of ternary bialgebras with respect to a “being derived” property: (1) A ∆-derived ternary bialgebra ∆3 = ∆3,der = (id ⊗∆2 ) ◦ ∆2 (12.16) (2) A µ-derived ternary bialgebra µ3,der = µ2 ◦ (µ2 ⊗ id) (12.17) 42 STEVEN DUPLIJ (3) A derived ternary bialgebra is simultaneously µ-derived and ∆-derived ternary bialgebra. (4) A non-derived ternary bialgebra which does not satisfy (12.16) and (12.17). Possible types of ternary antipodes can be defined by analogy with binary coalgebras. Definition 12-6. A skew ternary antipode is (3) (3) (3) µ3 ◦ (Sskew ⊗ id ⊗ id) ◦ ∆3 = µ3 ◦ (id ⊗Sskew ⊗ id) ◦ ∆3 = µ3 ◦ (id ⊗ id ⊗Sskew ) ◦ ∆3 = id . (12.18) If only one equality from (12.18) is satisfied, the corresponding skew antipode is called left, middle or right. Definition 12-7. Strong ternary antipode is (3) (3) (µ2 ⊗ id) ◦ (id ⊗Sstrong ⊗ id) ◦ ∆3 = 1 ⊗ id, (id ⊗µ2 ) ◦ (id ⊗ id ⊗Sstrong ) ◦ ∆3 = id ⊗1, where 1 is a unit of algebra. If in a ternary coalgebra the relation ∆3 ◦ S = τ13 ◦ (S ⊗ S ⊗ S) ◦ ∆3 (12.19) 123
holds true, where τ13 = 321 , then it is called skew-involutive.
Definition 12-8. A ternary Hopf algebra H, µ3 , η (3) , ∆3 , ε(3) , S (3) is a ternary bialgebra with a ternary antipode S (3) of the corresponding above type . Let us consider concrete constructions of ternary comultiplications, bialgebras and Hopf algebras. A ternary group-like element can be defined by ∆3 (g) = g ⊗ g ⊗ g, and for 3 such elements we have ∆3 ([g1 g2 g3 ]) = ∆3 (g1 ) ∆3 (g2 ) ∆3 (g3 ) . (12.20) But an analog of the binary primitive element (satisfying ∆2 (x) = x ⊗ 1 + 1 ⊗ x) cannot be chosen simply as ∆3 (x) = x ⊗ e ⊗ e + e ⊗ x ⊗ e + e ⊗ e ⊗ x, since the algebra structure is not preserved. Nevertheless, if we introduce two idempotent units e1 , e2 satisfying “semiorthogonality” [e1 e1 e2 ] = 0, [e2 e2 e1 ] = 0, then ∆3 (x) = x ⊗ e1 ⊗ e2 + e2 ⊗ x ⊗ e1 + e1 ⊗ e2 ⊗ x (12.21) and now ∆3 ([x1 x2 x3 ]) = [∆3 (x1 ) ∆3 (x2 ) ∆3 (x3 )]. Using (12.21) ε (x) = 0, ε (e1,2 ) = 1, and S (3) (x) = −x, S (3) (e1,2 ) = e1,2 , one can construct a ternary universal enveloping algebra in full analogy with the binary case (see e.g. K ASSEL [1995]). One of the most important examples of noncocommutative Hopf algebras is the well known Sweedler Hopf algebra S WEEDLER [1969] which in the binary case has two generators x and y satisfying µ2 (x, x) = 1, (12.22) µ2 (y, y) = 0, (12.23) (2) (2) σ+ (xy) = −σ− (xy) . (12.24) It has the following comultiplication ∆2 (x) = x ⊗ x, (12.25) ∆2 (y) = y ⊗ x + 1 ⊗ y, (12.26)  POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 43 counit ε(2) (x) = 1, ε(2) (y) = 0, and antipode S (2) (x) = x, S (2) (y) = −y, which respect the algebra structure. In the derived case a ternary Sweedler algebra is generated also by two generators x and y obeying D UPLIJ [2001] µ3 (x, e, x) = µ3 (e, x, x) = µ3 (x, x, e) = e, (12.27) (3) σ+ ([yey]) = 0, (12.28) (3) (3) σ+ ([xey]) = −σ− ([xey]) . (12.29) The derived Hopf algebra structure is given by ∆3 (x) = x ⊗ x ⊗ x, (12.30) ∆3 (y) = y ⊗ x ⊗ x + e ⊗ y ⊗ x + e ⊗ e ⊗ y, (12.31) ε(3) (x) = ε(2) (x) = 1, (12.32) ε(3) (y) = ε(2) (y) = 0, (12.33) S (3) (x) = S (2) (x) = x, (12.34) (3) (2) S (y) = S (y) = −y, (12.35) and it can be checked that (12.30)-(12.32) are algebra maps, while (12.34) are antialgebra maps. To obtain a non-derived ternary Sweedler example we have the following possibilities: 1) one “even” generator x, two “odd” generators y1,2 and one ternary unit e; 2) two “even” generators x1,2 , one “odd” generator y and two ternary units e1,2 . In the first case the ternary algebra structure is (no summation, i = 1, 2) [xxx] = e, (12.36) [yi yi yi ] = 0, (12.37) (3) (3) σ+ ([yi xyi ]) = σ+ ([xyi x]) = 0, (12.38) [xeyi ] = − [xyi e] , [exyi ] = − [yi xe] , (12.39) [eyi x] = − [yi ex] , (12.40) (3) (3) σ+ ([y1 xy2 ]) = −σ− ([y1 xy2 ]) . (12.41) The corresponding ternary Hopf algebra structure is ∆3 (x) = x ⊗ x ⊗ x, ∆3 (y1,2 ) = y1,2 ⊗ x ⊗ x + e1,2 ⊗ y2,1 ⊗ x + e1,2 ⊗ e2,1 ⊗ y2,1 , (12.42) ε(3) (x) = 1, ε(3) (yi ) = 0, (12.43) S (3) (x) = x, S (3) (yi ) = −yi . (12.44) In the second case we have for the algebra structure [xi xj xk ] = δij δik δjk ei , [yyy] = 0, (12.45) (3) (3) σ+ ([yxi y]) = 0, σ+ ([xi yxi ]) = 0, (12.46) (3) (3) σ+ ([y1 xy2 ]) = 0, σ− ([y1 xy2 ]) = 0, (12.47) 44 STEVEN DUPLIJ and the ternary Hopf algebra structure is D UPLIJ [2001] ∆3 (xi ) = xi ⊗ xi ⊗ xi , ∆3 (y) = y ⊗ x1 ⊗ x1 + e1 ⊗ y ⊗ x2 + e1 ⊗ e2 ⊗ y, (12.48) ε(3) (xi ) = 1, (12.49) ε(3) (y) = 0, (12.50) S (3) (xi ) = xi , (12.51) S (3) (y) = −y. (12.52) 13. T ERNARY QUANTUM GROUPS A ternary commutator can be obtained in different ways B REMNER AND H ENTZEL [2000]. We will consider the simplest version called a Nambu bracket (see e.g. TAKHTAJAN [1994], (3) DE A ZCARRAGA AND I ZQUIERDO [2010]). Let us introduce two maps ω± : A⊗A⊗A → A⊗A⊗A by (3) ω+ (a ⊗ b ⊗ c) = a ⊗ b ⊗ c + b ⊗ c ⊗ a + c ⊗ a ⊗ b, (13.1) (3) ω− (a ⊗ b ⊗ c) = b ⊗ a ⊗ c + c ⊗ b ⊗ a + a ⊗ c ⊗ b. (13.2) (3) (3) (3) Thus, obviously µ3 ◦ ω± = σ± ◦ µ3 , where σ± ∈ S3 denotes a sum of terms having even and (2) (2) odd permutations respectively. In the binary case ω+ = id ⊗ id and ω− = τ is the twist operator (2) (2) τ : a ⊗ b → b ⊗ a, while µ2 ◦ ω− is permutation σ− (ab) = ba. So the Nambu product is (3) (3) (3) (3) (3) (3) ωN = ω+ − ω− , and the ternary commutator is [, , ]N = σN = σ+ − σ− , or TAKHTAJAN [1994] [a, b, c]N = [abc] + [bca] + [cab] − [cba] − [acb] − [bac] (13.3) An abelian ternary algebra is defined by the vanishing of the Nambu bracket [a, b, c]N = 0 or (3) (3) ternary commutation relation σ+ = σ− . By analogy with the binary case a deformed ternary algebra can be defined by (3) (3) σ+ = qσ− or [abc] + [bca] + [cab] = q ([cba] + [acb] + [bac]) , (13.4) where multiplication by q is treated as an external operation. Let us consider a ternary analog of the Woronowicz example of a bialgebra construction, which has (2) (2) two generators satisfying xy = qyx (or σ+ (xy) = qσ− (xy)), then the following coproducts ∆2 (x) = x ⊗ x (13.5) ∆2 (y) = y ⊗ x + 1 ⊗ y (13.6) are the algebra maps. In the derived ternary case using (13.4) we have (3) (3) σ+ ([xey]) = qσ− ([xey]) , (13.7) where e is the ternary unit and ternary coproducts are ∆3 (e) = e ⊗ e ⊗ e, (13.8) ∆3 (x) = x ⊗ x ⊗ x, (13.9) ∆3 (y) = y ⊗ x ⊗ x + e ⊗ y ⊗ x + e ⊗ e ⊗ y, (13.10)  POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 45 which are ternary algebra maps, i.e. they satisfy (3) (3) σ+ ([∆3 (x) ∆3 (e) ∆3 (y)]) = qσ− ([∆3 (x) ∆3 (e) ∆3 (y)]) . (13.11) Let us consider the group G = SL (n, K). Then the algebra generated by aij ∈ SL (n, K) can be endowed with the structure of a ternary Hopf algebra (see, e.g., M ADORE [1995] for the binary case) by choosing the ternary coproduct, counit and antipode as (here summation is implied)



i ∆3 aij = aik ⊗ akl ⊗ alj , ε aij = δji , S (3) aij = a−1 j . (13.12) This antipode is a skew one since from (12.18) it follows that


i µ3 ◦ (S (3) ⊗ id ⊗ id) ◦ ∆3 aij = S (3) aik akl alj = a−1 k akl alj = δli alj = aij . (13.13) This ternary Hopf algebra is derived since for ∆2 = aij ⊗ ajk we have


∆3 = (id ⊗∆2 ) ⊗ ∆2 aij = (id ⊗∆2 ) aik ⊗ akj = aik ⊗ ∆2 akj = aik ⊗ akl ⊗ alj . (13.14) In the most important case n = 2 we can obtain the manifest action of the ternary coproduct ∆3 on components. Possible non-derived matrix representations of the ternary product  ij can be done only by four-rank n × n × n × n twice covariant and twice contravariant tensors akl . Among all products ij ol ko the non-derived ones are only the following: aoi jl ko jk boo cil and aok bio cil (where o is any index). So using e.g. the first choice we can define the non-derived Hopf algebra structure by
∆3 aij iµ vσ kl = avρ ⊗ akl ⊗ aµσ , ρj (13.15)
1 i j
ε aij kl = δk δl + δl i j δ k , (13.16) 2
and the skew antipode sij kl = S (3) aij iµ vσ i µ σ kl which is a solution of the equation svρ akl = δρ δk δl . Next consider ternary dual pair kG (push-forward) and F (G) (pull-back) which are related by (kG)∗ ∼ = F (G) (see, e.g., KOGORODSKI AND S OIBELMAN [1998]). Here kG = span (G) is a (3) ternary group algebra (G has a ternary product [ , , ]G or µG ) over a field k. If u ∈ kG (u = ui xi , xi ∈ G), then [uvw]k = ui v j w l [xi xj xl ]G (13.17) is associative, and so (kG, [ , , ]k ) becomes a ternary algebra. Define a ternary coproduct ∆3 : kG → kG ⊗ kG ⊗ kG by ∆3 (u) = ui xi ⊗ xi ⊗ xi (13.18) (derived and associative), then ∆3 ([uvw]k ) = [∆3 (u) ∆3 (v) ∆3 (w)]k , and kG is a ternary bialgebra. (3) If we define a ternary antipode by Sk = ui x̄i , where x̄i is a skew element of xi , then kG becomes a ternary Hopf algebra. In the dual case of functions F (G) : {ϕ : G → k} a ternary product [ , , ]F or µ3,F (derived and associative) acts on ψ (x, y, z) as (µ3,F ψ) (x) = ψ (x, x, x) , (13.19) and so F (G) is a ternary algebra. Let F (G) ⊗ F (G) ⊗ F (G) ∼ = F (G × G × G), then we define a ternary coproduct ∆3 : F (G) → F (G) ⊗ F (G) ⊗ F (G) as (∆3 ϕ) (x, y, z) = ϕ ([xyz]F ) , (13.20) which is derive and associative. Thus we can obtain ∆3 ([ϕ1 ϕ2 ϕ3 ]F ) = [∆3 (ϕ1 ) ∆3 (ϕ2 ) ∆3 (ϕ3 )]F , and therefore F (G) is a ternary bialgebra. If we define a ternary antipode by (3) SF (ϕ) = ϕ (x̄) , (13.21) 46 STEVEN DUPLIJ where x̄ is a skew element of x, then F (G) becomes a ternary Hopf algebra. Let us introduce a ternary analog of the R-matrix D UPLIJ [2001]. For a ternary Hopf algebra H we consider a linear map R(3) : H ⊗ H ⊗ H → H ⊗ H ⊗ H.
Definition 13-1. A ternary Hopf algebra H, µ3 , η (3) , ∆3 , ε(3) , S (3) is called quasifiveangular4 if it satisfies (3) (3) (3) (∆3 ⊗ id ⊗ id) = R145 R245 R345 , (13.22) (3) (3) (3) (id ⊗∆3 ⊗ id) = R125 R145 R135 , (13.23) (3) (3) (3) (id ⊗ id ⊗∆3 ) = R125 R124 R123 , (13.24) where as usual the index of R denotes action component positions. Using the standard procedure (see, e.g., K ASSEL [1995], C HARI AND P RESSLEY [1996], M AJID [1995]) we obtain a set of abstract ternary quantum Yang-Baxter equations, one of which has the form D UPLIJ [2001] (3) (3) (3) (3) (3) (3) (3) (3) (3) (3) R243 R342 R125 R145 R135 = R123 R132 R145 R245 R345 , (13.25) and others can be obtained by corresponding permutations. The classical ternary Yang-Baxter equa- tions form a one parameter family of solutions R (t) can be obtained by the expansion
R(3) (t) = e ⊗ e ⊗ e + rt + O t2 , (13.26) where r is a ternary classical R-matrix, then e.g. for (13.25) we have r342 r125 r145 r135 + r243 r125 r145 r135 + r243 r342 r145 r135 + r243 r342 r125 r135 + r243 r342 r125 r145 = r132 r145 r245 r345 + r123 r145 r245 r345 + r123 r132 r245 r345 + r123 r132 r145 r345 + r123 r132 r145 r245 .   (3) (3) (3) (3) (3) For three ternary Hopf algebras HA,B,C , µA,B,C , ηA,B,C , ∆A,B,C , εA,B,C , SA,B,C we can introduce a non-degenerate ternary “pairing” (see, e.g., C HARI AND P RESSLEY [1996] for the binary case) h , , i(3) : HA × HB × HC → K, trilinear over K, satisfying D UPLIJ [2001] D E(3) D E(3) D E(3) D E(3) (3) (3) (3) (3) ηA (a) , b, c = a, εB (b) , c , a, ηB (b) , c = εA (a) , b, c , D E(3) D E(3) D E(3) D E(3) (3) (3) (3) (3) b, ηB (b) , c = a, b, εC (c) , a, b, ηC (c) = a, εB (b) , c , D E(3) D E(3) D E(3) D E(3) (3) (3) (3) (3) a, b, ηC (c) = εA (a) , b, c , ηA (a) , b, c = a, b, εC (c) , 4 The reason for such notation is clear from (13.25).  POLYADIC SYSTEMS, REPRESENTATIONS AND QUANTUM GROUPS 47 D E(3) D E(3) (3) (3) µA (a1 ⊗ a2 ⊗ a3 ) , b, c = a1 ⊗ a2 ⊗ a3 , ∆B (b) , c , D E(3) D E(3) (3) (3) ∆A (a) , b1 ⊗ b2 ⊗ b3 , c = a, µB (b1 ⊗ b2 ⊗ b3 ) , c , D E(3) D E(3) (3) (3) a, µB (b1 ⊗ b2 ⊗ b3 ) , c = a, b1 ⊗ b2 ⊗ b3 , ∆C (c) , D E(3) D E(3) (3) (3) a, ∆B (b) , c1 ⊗ c2 ⊗ c3 = a, b, µC (c1 ⊗ c2 ⊗ c3 ) , D E(3) D E(3) (3) (3) a, b, µC (c1 ⊗ c2 ⊗ c3 ) = ∆A (a) , b, c1 ⊗ c2 ⊗ c3 , D E(3) D E(3) (3) (3) a1 ⊗ a2 ⊗ a3 , b, ∆C (c) = µA (a1 ⊗ a2 ⊗ a3 ) , b, c , D E(3) D E(3) D E(3) (3) (3) (3) SA (a) , b, c = a, SB (b) , c = a, b, SC (c) , where a, ai ∈ HA , b, bi ∈ HB . The ternary “paring” between HA ⊗ HA ⊗ HA and HB ⊗ HB ⊗ HB is given by ha1 ⊗ a2 ⊗ a3 , b1 ⊗ b2 ⊗ b3 i(3) = ha1 , b1 i(3) ha2 , b2 i(3) ha3 , b3 i(3) . These constructions can naturally lead to ternary generalizations of the duality concept and the quantum double which are key ingredients in the theory of quantum groups D RINFELD [1987], K ASSEL [1995], M AJID [1995]. 14. C ONCLUSIONS In this paper we presented a review of polyadic systems and their representations, ternary algebras and Hopf algebras. We have classified general polyadic systems and considered their homomorphisms and their multiplace generalizations, paying attention to their associativity. Then, we defined multi- place representations and multiactions and have given examples of matrix representations for some ternary groups. We defined and investigated ternary algebras and Hopf algebras, and have given some examples. We then considered some ternary generalizations of quantum groups and the Yang-Baxter equation. 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