A. P. Isaev

q-deformations of Virasoro algebra and conformal dimensions

Abstract We describe the q -deformations of the realisations of the conformal algebra depending on the conformal dimension parameter Δ . The particular role of the conformal dimensions Δ=0, 1 2 , 1 is pointed out. The q -deformed central extension terms, describing q -deformation of the Virasoro algebra, are derived. In the limit q → 1 one obtains the usual central term. The transformation properties of the q -deformed energy-momentum tensor ( Δ =2) consistent with the q -deformed central extension term are described.

On quantum matrix algebras satisfying the Cayley - Hamilton - Newton identities

The Cayley - Hamilton - Newton identities which generalize both the characteristic identity and the Newton relations have been recently obtained for the algebras of the RTT-type. We extend this result to a wider class of algebras defined by a pair of compatible solutions of the Yang - Baxter equation. This class includes the RTT algebras as well as the reflection equation algebras.

ParaGrassmann analysis and quantum groups

Paragrassmann algebras with one and many paragrassmann variables are considered from the algebraic point of view without using the Green ansatz. A differential operator with respect to paragrassmann variable and a covariant para-super-derivative are introduced giving a natural generalization of the Grassmann calculus to a paragrassmann one. Deep relations between paragrassmann algebras and quantum groups with deformation parameters being root of unity are established.Paragrassmann algebras with one and many paragrassmann variables are considered from the algebraic point of view without using the Green ansatz. Operators of differentiation with respect to paragrassmann variables and a covariant para-super-derivative are introduced giving a natural generalization of the Grassmann calculus to a paragrassmann one. Deep relations between paragrassmann algebras and quantum groups with deformation parameters being roots of unity are established.

A New Fusion Procedure for the Brauer Algebra and Evaluation Homomorphisms

We give a new fusion procedure for the Brauer algebra by showing that all primitive idempotents can be found by evaluating a rational function in several variables. The function takes values in the Brauer algebra and has the form of a product of R-matrix type factors. In particular, this provides a one-parameter version of the fusion procedure for the symmetric group. The R-matrices are solutions of the Yang–Baxter equation associated with the classical Lie algebras of types B, C, and D. Moreover, we construct an evaluation homomorphism from a reflection equation algebra to and show that the fusion procedure provides an equivalence between natural tensor representations of with the corresponding evaluation modules.

On baxterized solutions of reflection equation and integrable chain models

Abstract Nonpolynomial baxterized solutions of reflection equations associated with affine Hecke and affine Birman–Murakami–Wenzl algebras are found. Relations to integrable spin chain models with nontrivial boundary conditions are discussed.

TWISTED YANG-BAXTER EQUATIONS FOR LINEAR QUANTUM (SUPER)GROUPS

We consider the modified (or twisted) Yang - Baxter equations for the groups and supergroups. The general solutions for these equations are presented in the case of the linear quantum (super)groups. The introduction of spectral parameters in the twisted Yang - Baxter equation and its solutions are also discussed.We consider the modified (or twisted) Yang-Baxter equations for the

Quantum group gauge theories and covariant quantum algebras

SL_{q}(N)

Generalized Cayley-Hamilton-Newton identities

groups and

On the Idempotents of Hecke Algebras

SL_{q}(N|M)

Conformal algebra: R-matrix and star-triangle relation

supergroups. The general solutions for these equations are presented in the case of the linear quantum (super)groups. The introduction of spectral parameters in the twisted Yang-Baxter equation and its solutions are also discussed.