arXiv subjects

D. Gurevich

Publications and source records attributed to D. Gurevich.

36 records · Page 2Linked to original sources

arXiv2007

Representation theory of (modified) Reflection Equation Algebra of GL(m|n) type

Let R: V x V -> V x V be a Hecke type solution of the quantum Yang-Baxter equation (a Hecke symmetry). Then, the Hilbert-Poincre' series of the associated R-exterior algebra of the space V is a ratio of two polynomials of degree m (numerator) and n (denominator). Assuming R to be skew-invertible, we define a rigid quasitensor category SW(V) of vector spaces, generated by the space V and its dual V*, and compute certain numerical characteristics of its objects. Besides, we introduce a braided bialgebra structure in the modified Reflection Equation Algebra, associated with R, and equip objects of the category SW(V) with an action of this algebra. In the case related to the quantum group U_q(sl(m)), we consider the Poisson counterpart of the modified Reflection Equation Algebra and compute the semiclassical term of the pairing, defined via the categorical (or quantum) trace.

math.QA↗

arXiv2004

Geometry of non-commutative orbits related to Hecke symmetries

To some braiding R of Hecke type (a Hecke symmetry) we put into correspondence an associative algebra called the modified Reflection Equation Algebra (mREA). We construct a series of matrices L_(m), m=1,2,... with entries belonging to mREA such that each of them satisfies a version of the Cayley-Hamilton identity with central coefficients. We also consider some quotients of the mREA which are called the non-commutative orbits. For each of these orbits we construct a large family of projective modules. In this family we introduce an algebraic structure which is close to that of $K^0(\Fl(\C^n))$. The algebraic structure respects an equivalence relation motivated by a "quantum" trace compatible with the initial Hecke symmetry R. For a subclass of non-commutative orbits we compute the spectrum of central elements of the mREA Tr_R(L_(m)^k), k\in {\Bbb N}.

math.QA↗

arXiv2003

q-Index on braided non-commutative spheres

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.

math.QA↗

arXiv2002

Quantum line bundles on noncommutative sphere

Noncommutative (NC) sphere is introduced as a quotient of the enveloping algebra of the Lie algebra su(2). Using the Cayley-Hamilton identities we introduce projective modules which are analogues of line bundles on the usual sphere (we call them quantum line bundles) and define a multiplicative structure in their family. Also, we compute a pairing between certain quantum line bundles and finite dimensional representations of the NC sphere in the spirit of the NC index theorem. A new approach to constructing the differential calculus on a NC sphere is suggested. The approach makes use of the projective modules in question and gives rise to a NC de Rham complex being a deformation of the classical one.

math.QA↗

arXiv2002

Traces in braided categories

With any even Hecke symmetry R (that is a Hecke type solution of the Yang-Baxter equation) we associate a quasitensor category. We formulate a condition on R implying that the constructed category is rigid and its commutativity isomorphisms R_{U,V} are natural. We show that this condition leads to rescaling of the initial Hecke symmetry. We suggest a new way of introducing traces as properly normalized categorical morphisms End(V) --> K and deduce the corresponding normalization from categorical dimensions.

math.QA↗

arXiv2000

Quantum line bundles via Cayley-Hamilton identity

As was shown in \cite{GPS} the matrix $L=|| l_i^j||$ whose entries $l_i^j$ 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 "non-commutative 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).

math.QA↗

arXiv2000

Quantum Sphere via Reflection Equation Algebra

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 \cite{PS}, \cite{GPS}. A new way to introduce some elements of "braided geometry" on the quantum sphere is discussed.

math.QA↗

arXiv2000

Cotangent and tangent modules on quantum orbits

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".

math.QA↗

arXiv1999

Schur-Weyl Categories and Non-quasiclassical Weyl Type Formula

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 $\Vl$ and compute its dimension. The functor $V\mapsto \Vl$ is an analogue of the well-known Schur functor. The category generated by the objects $\Vl$ 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 $N(\la)=\{\sharp \la_i\leq\la\}$, where $\la_i$ are the eigenvalues of this operator.

math.QA↗

arXiv1999

Some aspects of braided geometry: differential calculus, tangent space, gauge theory

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 \cite{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.

math.QA↗

arXiv1998

Double quantization of $\cp$ type orbits by generalized Verma modules

It is known that symmetric orbits in ${\bf g}^*$ for any simple Lie algebra ${\bf g}$ 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 $\cp$ type orbits (i.e. orbits in $sl(n+1)^*$ whose real compact form is $ CP^n$) by means of q-deformed Verma modules.

math.QA↗

arXiv1997

Noncommutative analogues of q-special polynomials and q-integral on a quantum sphere

The q-Legendre polynomials can be treated as some special "functions in the quantum double cosets $U(1)\setminus SU_q(2)/U(1)$". They form a family (depending on a parameter $q$) of polynomials in one variable. We get their further generalization by introducing a two parameter family of polynomials. If the former family arises from an algebra which is in a sense "q-commutative", the latter one is related to its noncommutative counterpart. We introduce also a two parameter deformation of the invariant integral on a sphere.

q-alg↗

arXiv1997

Two types of Poisson pencils and related quantum objects

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.

q-alg↗

arXiv1997

Dual quasitriangular structures related to the Temperley-Lieb algebra

We consider nonquasiclassical solutions to the quantum Yang-Baxter equation and the corresponding quantum cogroups $\Fun(SL(S))$ constructed earlier by one of the authors . We give a criterion of the existence of a dual quasitriangular structure in the algebra $\Fun(SL(S))$ and describe a large class of such objects related to the Temperley-Lieb algebra satisfying this criterion. We show also that this dual quasitriangular structure is in some sense nondegenerate.

q-alg↗

arXiv1996

Braided modules and reflection equations

We introduce a representation theory of q-Lie algebras defined earlier in \cite{DG1}, \cite{DG2}, formulated in terms of braided modules. We also discuss other ways to define Lie algebra-like objects related to quantum groups, in particular, those based on the so-called reflection equations. We also investigate the truncated tensor product of braided modules.

q-alg↗

arXiv1995

Quantum hyperboloid and braided modules

When a quantum hyperboloid is realized, as a three - parameter algebra $\ahqc$, 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 $\ahqc$, and we discuss a braided version of the orbit method, i.e. a correspondence between orbits in $\gggg^*$ and $\gggg$-modules. A braided trace and a braided involution are discussed as well.

q-alg↗

arXiv1995

Quantization of Poisson pencils and generalized Lie algebras

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''.

q-alg↗

arXiv1994

Quantum orbits of R-matrix type

Given a simple Lie algebra $\gggg$, we consider the orbits in $\gggg^*$ 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.

hep-th↗