167 citations · 1.9k across the 134 of their papers we have counts for
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Solutions of the qKZB Equation in Tensor Products of finite dimensional modules over the elliptic quantum group
E. Mukhin, A. Varchenko
We consider the quantized Knizhnik-Zamolodchikov-Bernard difference equation (qKZB) with step and values in a tensor product of finite dimensional evaluation modules over the e…
On algebraic equations satisfied by hypergeometric solutions of the qKZ equation
E. Mukhin, A. Varchenko
We consider the quantized Knizhnik-Zamolodchikov equation (qKZ), defined in terms of rational R-matrices. The properties of the equation change when the step of the equatio…
Quantization of the space of conformal blocks
E. Mukhin, A. Varchenko
We consider the discrete Knizhnik-Zamolodchikov connection (qKZ) associated to , defined in terms of rational R-matrices. We prove that under certain resonance conditions, t…
Solutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups
Pavel Etingof, Alexander Varchenko
The quantum dynamical Yang-Baxter (QDYB) equation is a useful generalization of the quantum Yang-Baxter (QYB) equation introduced by Gervais, Neveu, and Felder. The QDYB equation a…
Monodromy of solutions of the elliptic quantum Knizhnik-Zamolodchikov-Bernard difference equations
Giovanni Felder, Vitaly Tarasov, Alexander Varchenko
The elliptic quantum Knizhnik-Zamolodchikov-Bernard (qKZB) difference equations associated to the elliptic quantum group is a system of difference equations with va…
Elliptic quantum groups and Ruijsenaars models
Giovanni Felder, Alexander Varchenko
We construct symmetric and exterior powers of the vector representation of the elliptic quantum groups . The corresponding transfer matrices give rise to various int…
Canonical basis and homology of local systems
Igor Frenkel, Alexander Kirillov, Alexander Varchenko
Using the isomorphism between highest weight U_q(sl_2)-modules and homologies of certain local systems on the configuration spaces, constructed by Varchenko, we give a geometric co…
Geometry of q-Hypergeometric Functions, Quantum Affine Algebras and Elliptic Quantum Groups
Vitaly Tarasov, Alexander Varchenko
The trigonometric quantized Knizhnik-Zamolodchikov equation (qKZ equation) associated with the quantum group is a system of linear difference equations with values in a…
Geometry and classification of solutions of the Classical Dynamical Yang-Baxter Equation
Pavel Etingof, Alexander Varchenko
The classical Yang-Baxter equation (CYBE) is an algebraic equation central in the theory of integrable systems. Its solutions were classified by Belavin and Drinfeld. Quantization…
Algebraic integrability of the two-body Ruijsenaars operator
Giovanni Felder, Alexander Varchenko
We study the algebra of difference operators that commute with the two-body Ruijsenaars operator, a -deformation of the Lamé differential operator, for generic values of the def…
Quantization of geometry associated to the quantized Knizhnik-Zamolodchikov equations
Alexander Varchenko
It is known that solutions of the Knizhnik-Zamolodchikov differential equations are given by integrals of closed differential forms over suitable cycles. In this paper a quantizati…
Solutions of the elliptic qKZB equations and Bethe ansatz I
Giovanni Felder, Alexander Varchenko, Vitaly Tarasov
We give an integral representation for solutions of the elliptic quantum Knizhnik-Zamolodchikov-Bernard difference equations, in the case of sl(2). The result is based on a geometr…
Algebraic Bethe ansatz for the elliptic quantum group
Giovanni Felder, Alexander Varchenko
To each representation of the elliptic quantum group is associated a family of commuting transfer matrices. We give common eigenvectors by a version of the algebrai…
On representations of the elliptic quantum group
Giovanni Felder, Alexander Varchenko
We describe representation theory of the elliptic quantum group . It turns out that the representation theory is parallel to the representation theory of the Yangian…
Bases of Bethe Vectors and Difference Equations with Regular Singular Points
Vitaly Tarasov, Alexander Varchenko
We prove that Bethe vectors generically form a base in a tensor product of irreducible heighest weight -modules or -modules. We apply this result to difference equ…