Papers by Dimitri Gourevitch
Journal of Geometry and Physics, Dec 1, 1998
It is known that symmetric orbits in g * for any simple Lie algebra g are equiped with a Poisson ... more It is known that symmetric orbits in g * for any simple Lie algebra g are equiped with a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the reduced Sklyanin bracket associated to the "canonical" R-matrix. We realize quantization of this Poisson pencil on CP n type orbits (i.e. orbits in sl(n + 1) * whose real compact form is CP n ) by means of q-deformed Verma modules.
Journal of Physics A, Jul 15, 2009
The main goal of this review is to compare different approaches to constructing geometry associat... more The main goal of this review is to compare different approaches to constructing geometry associated with a Hecke type braiding (in particular, with that related to the quantum group U q (sl(n))). We make an emphasis on affine braided geometry related to the so-called Reflection Equation Algebra (REA). All objects of such type geometry are defined in the spirit of affine algebraic geometry via polynomial relations on generators. We begin with comparing the Poisson counterparts of "quantum varieties" and describe different approaches to their quantization. Also, we exhibit two approaches to introducing q-analogs of vector bundles and defining the Chern-Connes index for them on quantum spheres. In accordance with the Serre-Swan approach, the q-vector bundles are treated as finitely generated projective modules over the corresponding quantum algebras. Besides, we describe the basic properties of the REA used in this construction and compare different ways of defining q-analogs of partial derivatives and differentials on the REA and algebras close to them. In particular, we present a way of introducing a q-differential calculus via Koszul type complexes. The elements of the q-calculus are applied to defining q-analogs of some relativistic wave operators.
Journal of physics, May 18, 2001
As was shown in [GPS] the matrix L = ||l j i || whose entries l j i are generators of the so-call... more As was shown in [GPS] the matrix L = ||l j i || whose entries l j i are generators of the so-called reflection equation algebra is subject to some polynomial identity looking like the Cayley-Hamilton identity for a numerical matrix. Here a similar statement is presented for a matrix whose entries are generators of a filtered algebra being a "noncommutative analogue" of the reflection equation algebra. In an appropriate limit we get a similar statement for the matrix formed by the generators of the algebra U (gl(n)). This property is used to introduce the notion of line bundles over quantum orbits in the spirit of the Serre-Swan approach. The quantum orbits in question are presented explicitly as some quotients of one of the mentioned above algebras both in the quasiclassical case (i.e. that related to the quantum group U q (sl(n))) and a non-quasiclassical one (i.e. that arising from a Hecke symmetry with non-standard Poincaré series of the corresponding symmetric and skewsymmetric algebras).
Journal of physics, 1999
A new approach is suggested to quantum differential calculus on certain quantum varieties. It con... more A new approach is suggested to quantum differential calculus on certain quantum varieties. It consists in replacing quantum de Rham complexes with differentials satisfying Leibniz rule by those which are in a sense close to Koszul complexes from [G1]. We also introduce the tangent space on a quantum hyperboloid equipped with an action on the quantum function space and define the notions of quantum (pseudo)metric and quantum connection (partially defined) on it. All objects are considered from the viewpoint of flatness of quantum deformations. A problem of constructing a flatly deformed quantum gauge theory is discussed as well.
Journal of Physics A, Jun 30, 2008
We introduce an analog of the Maxwell operator on a q-Minkowski space algebra (treated as a parti... more We introduce an analog of the Maxwell operator on a q-Minkowski space algebra (treated as a particular case of the so-called Reflection Equation Algebra) and on certain of its quotients. We treat the space of "quantum differential forms" as a projective module in the spirit of the Serre-Swan approach. Also, we use "braided tangent vector fields" which are q-analogs of Poisson vector fields associated to the Lie bracket sl(2).
Theoretical and Mathematical Physics, Jun 1, 1995
We describe two types of Poisson pencils generated by a linear bracket and a quadratic one arisin... more We describe two types of Poisson pencils generated by a linear bracket and a quadratic one arising from a classical R-matrix. A quantization scheme is discussed for each. The quantum algebras are represented as the enveloping algebras of "generalized Lie algebras".
Journal of Geometry and Physics, Apr 1, 2005
To some Hecke symmetries (i.e. Yang-Baxter braidings of Hecke type) we assign algebras called bra... more To some Hecke symmetries (i.e. Yang-Baxter braidings of Hecke type) we assign algebras called braided non-commutative spheres. For any such algebra, we introduce and compute a q-analog of the Chern-Connes index. Unlike the standard Chern-Connes index, ours is based on the so-called categorical trace specific for a braided category in which the algebra in question is represented.
Letters in Mathematical Physics, Nov 1, 1995
Given a simple Lie algebra g, we consider the orbits in g * which are of R-matrix type, i.e., whi... more Given a simple Lie algebra g, we consider the orbits in g * which are of R-matrix type, i.e., which possess a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the so-called R-matrix bracket. We call an algebra quantizing the latter bracket a quantum orbit of R-matrix type. We describe some orbits of this type explicitly and we construct a quantization of the whole Poisson pencil on these orbits in a similar way. The notions of q-deformed Lie brackets, braided coadjoint vector fields and tangent vector fields are discussed as well.
arXiv (Cornell University), Nov 18, 1999
Quantum sphere is introduced as a quotient of the so-called Reflection Equation Algebra. This ena... more Quantum sphere is introduced as a quotient of the so-called Reflection Equation Algebra. This enables us to construct some line bundles on it by means of the Cayley-Hamilton identity whose a quantum version was discovered in [PS], [GPS]. A new way to introduce some elements of "braided geometry" on the quantum sphere is discussed.
arXiv (Cornell University), Jul 29, 2002
To some Hecke symmetries (i.e. Yang-Baxter braidings of Hecke type) we assign algebras called bra... more To some Hecke symmetries (i.e. Yang-Baxter braidings of Hecke type) we assign algebras called braided non-commutative spheres. For any such algebra, we introduce and compute a q-analog of the Chern-Connes index. Unlike the standard Chern-Connes index, ours is based on the so-called categorical trace specific for a braided category in which the algebra in question is represented.
arXiv (Cornell University), Nov 18, 1999
To a vector space V equipped with a non-quasiclassical involutary solution of the quantum Yang-Ba... more To a vector space V equipped with a non-quasiclassical involutary solution of the quantum Yang-Baxter equation and a partition λ, we associate a vector space V λ and compute its dimension. The functor V → V λ is an analogue of the well-known Schur functor. The category generated by the objects V λ is called the Schur-Weyl category. We suggest a way to construct some related twisted varieties looking like orbits of semisimple elements in sl(n) * . We consider in detail a particular case of such "twisted orbits", namely the twisted non-quasiclassical hyperboloid and we define the twisted Casimir operator on it. In this case, we obtain a formula looking like the Weyl formula, and describing the asymptotic behavior of the function N (λ) = {♯ λ i ≤ λ}, where λ i are the eigenvalues of this operator.
arXiv (Cornell University), May 29, 2007
St Petersburg Mathematical Journal, Jan 30, 2009
Let R : V ⊗2 → V ⊗2 be a Hecke type solution of the quantum Yang-Baxter equation (a Hecke symmetr... more Let R : V ⊗2 → V ⊗2 be a Hecke type solution of the quantum Yang-Baxter equation (a Hecke symmetry). Then, the Hilbert-Poincaré series of the associated R-exterior algebra of the space V is the ratio of two polynomials of degrees m (numerator) and n (denominator). Under the assumption that R is skew-invertible, a rigid quasitensor category SW(V (m|n) ) of vector spaces is defined, generated by the space V and its dual V * , and certain numerical characteristics of its objects are computed. Moreover, a braided bialgebra structure is introduced in the modified reflection equation algebra associated with R, and the objects of the category SW(V (m|n) ) are equipped with an action of this algebra. In the case related to the quantum group U q (sl(m)), the Poisson counterpart of the modified reflection equation algebra is considered and the semiclassical term of the pairing defined via the categorical (or quantum) trace is computed.
Two types of Poisson pencils connected to classical R-matrices and their quantum counterparts are... more Two types of Poisson pencils connected to classical R-matrices and their quantum counterparts are considered. A representation theory of the quantum algebras related to some symmetric orbits in sl(n) * is constructed. A twisted version of quantum mechanics is discussed.
International Journal of Modern Physics B, Sep 20, 2000
Let k(S 2 q ) be the "coordinate ring" of a quantum sphere. We introduce the cotangent module on ... more Let k(S 2 q ) be the "coordinate ring" of a quantum sphere. We introduce the cotangent module on the quantum sphere as a one-sided k(S 2 q )-module and show that there is no Yang-Baxter type operator converting it into a k(S 2 q )-bimodule which would be a flatly deformed object w.r.t. its classical counterpart. This implies non-flatness of any covariant differential calculus on the quantum sphere making use of the Leibniz rule. Also, we introduce the cotangent and tangent modules on generic quantum orbits and discuss some related problems of "braided geometry".
arXiv (Cornell University), Nov 1, 1995
When a quantum hyperboloid is realized, as a three -parameter algebra A c h,q , in the usual mann... more When a quantum hyperboloid is realized, as a three -parameter algebra A c h,q , in the usual manner, the following problem arises: what is a "representation theory" of this algebra? We construct the series of all spin representations of A c h,q , and we discuss a braided version of the orbit method, i.e. a correspondence between orbits in g * and g-modules. A braided trace and a braided involution are discussed as well.
Schur-Weyl Categories and Non-quasiclassical Weyl Type Formula
Lie Groups and Lie Algebras, 1998
Two types of Poisson pencils connected to classical R-matrices and their quantum counterparts are... more Two types of Poisson pencils connected to classical R-matrices and their quantum counterparts are considered. A representation theory of the quantum algebras related to some symmetric orbits in sl(n) * is constructed. A twisted version of quantum mechanics is discussed.
We discuss the consistency of the axioms which the definition of quantum Lie algebras is usually ... more We discuss the consistency of the axioms which the definition of quantum Lie algebras is usually based on.
Theoretical and Mathematical Physics, 1995
We describe two types of Poisson pencils generated by a linear bracket and a quadratic one arisin... more We describe two types of Poisson pencils generated by a linear bracket and a quadratic one arising from a classical R-matrix. A quantization scheme is discussed for each. The quantum algebras are represented as the enveloping algebras of "generalized Lie algebras".
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Papers by Dimitri Gourevitch