A. Stolin

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Papers by A. Stolin

Symmetric algebras and Yang-Baxter equation

summary:Let $U$ be an open subset of the complex plane, and let $L$ denote a finite-dimensional c... more summary:Let $U$ be an open subset of the complex plane, and let $L$ denote a finite-dimensional complex simple Lie algebra. {\it A. A. Belavin} and {\it V. G. Drinfel'd} investigated non-degenerate meromorphic functions from $U\times U$ into $L\otimes L$ which are solutions of the classical Yang-Baxter equation [Funct. Anal. Appl. 16, 159-180 (1983; Zbl 0504.22016)]. They found that (up to equivalence) the solutions depend only on the difference of the two variables and that their set of poles forms a discrete (additive) subgroup $\Gamma$ of the complex numbers (of rank at most 2). If $\Gamma$ is non-trivial, they were able to completely classify all possible solutions. If $\Gamma$ is trivial, the solutions are called rational and for $L= sl_n(\bbfC)$ they were classified by {\it A. Stolin} [in Math. Scand. 69, No. 1, 57-80 (1991; Zbl 0727.17005)]. \par A Lie algebra $L$ is called symmetric if there exists a non-degenerate symmetric invariant bilinear form on $L$. In the

format_quoteTheorem 2.1 establishes a unique correspondence between rational solutions of CYBE and subspaces related to symmetric Lie algebras.format_quote

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Algebraic Bethe ansatz for deformed Gaudin model

by A. Stolin and Nenad Manojlović

Journal of Mathematical Physics, 2011

The Gaudin model based on the sl2-invariant r-matrix with an extra Jordanian term depending on th... more The Gaudin model based on the sl2-invariant r-matrix with an extra Jordanian term depending on the spectral parameters is considered. The appropriate creation operators defining the Bethe states of the system are constructed through a recurrence relation. The commutation relations between the generating function t(λ) of the integrals of motion and the creation operators are calculated and therefore the algebraic Bethe Ansatz is fully implemented. The energy spectrum as well as the corresponding Bethe equations of the system coincide with the ones of the sl2-invariant Gaudin model. As opposed to the sl2-invariant case, the operator t(λ) and the Gaudin Hamiltonians are not hermitian. Finally, the inner products and norms of the Bethe states are studied.

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Equivariant quantization of Poisson homogeneous spaces and Kostant's problem

Let g be a finite dimensional split semisimple Lie algebra and λ a weight of g. Let F be the alge... more Let g be a finite dimensional split semisimple Lie algebra and λ a weight of g. Let F be the algebra of quantized regular functions on the connected simply connected group G corresponding to g. In the present paper we introduce a certain subspace F' of F (which is not necessary a subalgebra of F) and endow it with an associative -product using the so-called reduced fusion element. We prove that the algebra (F',) is isomorphic to (L(λ))_fin, where L(λ) is the irreducible highest weight Ǔ_q g-module and "fin" stands for the subalgebra of the locally finite elements with respect to the adjoint action of Ǔ_q g. The introduced -product has some limiting properties what enables us to prove Kostant's problem for Ǔ_q g in certain cases. We remind the reader that this means that (L(λ))_fin coincides with the image of Ǔ_q in L(λ). We also note that if λ is such that <λ,α_i^∨>=0 for some simple roots α_i and generic otherwise, then (F,) is a Ǔ_q g-invariant quantiza...

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Q A ] 1 7 Ju l 2 00 5 Irreducible highest-weight modules and equivariant quantization

The notion of deformation quantization, motivated by ideas coming from both physics and mathemati... more The notion of deformation quantization, motivated by ideas coming from both physics and mathematics, was introduced in classical papers [2, 7, 8]. Roughly speaking, a deformation quantization of a Poisson manifold (P, { , }) is a formal associative product on (Fun P)[[]] given by f 1 ⋆ f 2 = f 1 f 2 + c(f 1 , f 2) + O(2) for any f 1 , f 2 ∈ Fun P , where the skew-symmetric part of c is equal to { , }, and the coefficients of the series for f 1 ⋆ f 2 should be given by bi-differential operators. The fact that any Poisson manifold can be quantized in this sense was proved by Kontzevich in [15]. However, finding exact formulas for specific cases of Pois-son brackets is an interesting separate problem. There are several well-known examples of such explicit formulas. One of the first was the Moyal product quan-tizing the standard symplectic structure on R 2n. Another one is the standard quantization of the Kirilov-Kostant-Souriau bracket on the dual space g * to a Lie algebra g (see [10]...

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Chaotic Geodesics in Carnot Groups

arXiv: Differential Geometry, 1997

Graded nilpotent Lie groups, or Carnot Groups are to subRiemannian geometry as Euclidean spaces a... more Graded nilpotent Lie groups, or Carnot Groups are to subRiemannian geometry as Euclidean spaces are to Riemannian geometry. They are the metric tangent cones for this geometry. Hoping that the analogy between subRiemannian and Riemannian geometry is a strong one, one might conjecture that the subRiemannian geodesic flow on any Carnot group is completely integrable. We prove this conjecture is false by showing that that the subRiemannian geodesic flow is not algebraically completely integrable in the case of the group whose Lie algebra N consists of 4 by 4 nilpotent triangular matrices. We use this to prove that the centralizer for the corresponding quadratic “quantum” Hamiltonian H in the universal enveloping algebra of N is “as small as possible”.

format_quoteThe example of a 3-step Carnot group shows subRiemannian geodesic flow is not algebraically completely integrable, contrary to some expectations.format_quote

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Spherical principal series of quantum Harish-Chandra modules

Zh Mat Fiz Anal Geom, 2007

The non-degenerate spherical principal series of quantum Harish-Chandra modules is constructed. T... more

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Classical limit of the scaled elliptic algebra A (̂sl2)

$Research paper thumbnail of Classical Limit of the Scaled Elliptic Algebra $A_{\hbar,\eta}(sl_2)$$

Classical Limit of the Scaled Elliptic Algebra $A_{\hbar,\eta}(sl_2)$

arXiv: Quantum Algebra, 1997

The classical limit of the scaled elliptic algebra $A_{\hbar,\eta}(sl_2)$ is investigated. The li... more The classical limit of the scaled elliptic algebra $A_{\hbar,\eta}(sl_2)$ is investigated. The limiting Lie algebra is described in two equivalent ways: as a central extension of the algebra of generalized automorphic $sl_2$ valued functions on a strip and as an extended algebra of decreasing automorphic $sl_2$ valued functions on the real line. A bialgebra structure and an infinite-dimensional representation in the Fock space are studied. The classical limit of elliptic algebra $A_{q,p}(sl_2)$ is also briefly presented.

format_quoteAn infinite-dimensional representation of the algebra a η(sl2) in the Fock space reveals unique features, differing from typical integrable representations of sl2.format_quote

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A Non-Integrable Subriemannian Geodesic Flow On A Carnot Group

Graded nilpotent Lie groups, or Carnot Groups are to sub-Riemannian geometry as Euclidean spaces ... more Graded nilpotent Lie groups, or Carnot Groups are to sub-Riemannian geometry as Euclidean spaces are to Riemannian geometry. They are the metric tangent cones for this geometry. Hoping that the analogy between sub-Riemannian and Riemannian geometry is a strong one, one might conjecture that the sub-Riemannian geodesic flow on any Carnot group is completely integrable. We prove this conjecture to be false by showing that the sub-Riemannian geodesic flow is not algebraically completely integrable in the case of the group whose Lie algebra consists of 4 by 4 upper triangular matrices. As a corollary, we prove that the centralizer for the corresponding quadratic "quantum" Hamiltonian in the universal enveloping algebra of this Lie algebra is "as small as possible".

Tarasov: Dynamical Yang-Baxter equations, quasi-Poisson homogeneous spaces, and quantization

This paper is a continuation of [10]. Let us recall the main result of [10]. Let G be a Lie group... more

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Differential Calculi on Quantum Prehomogeneous Vector Spaces

This work deals with differential calculi over quadratic algebras which arise in quantum theory o... more This work deals with differential calculi over quadratic algebras which arise in quantum theory of bounded symmetric domains [35]. We prove that the dimensions of homogeneous components for graded vector spaces of k-forms coincide with their classical values. In a special case this result is well known [31, p. 296]. The quadratic algebras in question constitute q-analogues for polynomial algebras on prehomogeneous vector spaces of a special class [34]. This allows one to prove that the dual de Rham complex is isomorphic to a quantum analogue of the generalized Bernstein-Gelfand-Gelfand resolution [25]. Authors are grateful to S. Kolb for helpful discussions. 1 The Uqg-module algebra C[p −]q Consider a simple complex Lie algebra g with the Chevalley generators {Hi, Ei, Fi}i=1,2,...,l. Let {αj}j=1,2,...,l be a system of simple roots and a = (aij)i,j=1,2,...,l the Cartan matrix: aij = αj(Hi). The maximal root of g is given by a linear combination of its simple roots l∑ njαj, nj ∈ Z+. (...

Algebraic Bethe Ansatz for deformed

The Gaudin model based on the sl2-invariant r-matrix with an extra Jordanian term depending on th... more The Gaudin model based on the sl2-invariant r-matrix with an extra Jordanian term depending on the spectral parameters is considered. The appropriate creation operators defining the Bethe states of the system are constructed through a recurrence relation. The commutation relations between the generating function t(λ) of the integrals of motion and the creation operators are calculated and therefore the algebraic Bethe Ansatz is fully implemented. The energy spectrum as well as the corresponding Bethe equations of the system coincide with the ones of the sl2-invariant Gaudin model. As opposed to the sl2-invariant case, the operator t(λ) and the Gaudin Hamiltonians are not Hermitian. Finally, the inner products and norms of the Bethe states are studied.

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A Quantum Analogue of the Bernstein Functor

We consider Knapp-Vogan Hecke algebras in the quantum group setting. This allows us to produce a ... more We consider Knapp-Vogan Hecke algebras in the quantum group setting. This allows us to produce a quantum analogue of the Bernstein functor as a first step towards the cohomological induction for quantum groups. The cohomological induction is one of the most important tools for producing unitarizable Harish-Chandra modules [7]. The present paper demonstrates that this method can be applied to modules over quantum universal enveloping algebras [6] as well. Our way to this circle of problems was as follows. The papers [11, 9] introduce quantum analogues of the bounded symmetric domains and the associated Harish-Chandra modules. One special case considered in [10] leads to two geometric realizations for the ladder representation of the quantum universal enveloping algebra Uqsu2,2 and to the quantum Penrose transform [1]. One of the main tools used in [10] was studying q-analogues of the Čech cohomology. Unfortunately this method can be hardly generalized because it relates to non-trivia...

25 Years of Quantum Groups: from Definition to Classification

Acta Polytechnica, 2008

In mathematics and theoretical physics, quantum groups are certain non-commutative, non-cocommuta... more In mathematics and theoretical physics, quantum groups are certain non-commutative, non-cocommutative Hopf algebras, which first appeared in the theory of quantum integrable models and later they were formalized by Drinfeld and Jimbo. In this paper we present a classification scheme for quantum groups, whose classical limit is a polynomial Lie algebra. As a consequence we obtain deformed XXX and XXZ Hamiltonians.

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ITEP-TH-1/97 RIMS-1139 q-alg/9703043 Classical Limit of the Scaled Elliptic Algebra A¯h,η ( ̂ sl2)

The classical limit of the scaled elliptic algebra A¯h,η ( ̂ sl2) is investigated. The limiting L... more The classical limit of the scaled elliptic algebra A¯h,η ( ̂ sl2) is investigated. The limiting Lie algebra is described in two equivalent ways: as a central extension of the algebra of generalized automorphic sl2 valued functions on a strip and as an extended algebra of decreasing automorphic sl2 valued functions on the real line. A bialgebra structure and an infinite-dimensional representation in the Fock space are studied. The classical limit of elliptic algebra Aq,p ( ̂ sl2) is also briefly presented.

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Quantum seaweed algebras and quantization of affine Cremmer-Gervais r-matrices

We propose a method of quantization of certain Lie bialgebra structures on the polynomial Lie alg... more We propose a method of quantization of certain Lie bialgebra structures on the polynomial Lie algebras related to quasi-trigonometric solutions of the classical Yang--Baxter equation. The method is based on an affine realization of certain seaweed algebras and their quantum analogues. We also propose a method of $\omega$-affinization, which enables us to quantize rational $r$-matrices of $\mathfrak{sl}(3)$.