Lattice $W$ algebras and quantum groups

Abstract

We represent Feigin's construction [22] of lattice W algebras and give some simple results: lattice Virasoro and $W_3$ algebras. For simplest case $g=sl(2)$ we introduce whole $U_q(sl(2))$ quantum group on this lattice. We find simplest two-dimensional module as well as exchange relations and define lattice Virasoro algebra as algebra of invariants of $U_q(sl(2))$. Another generalization is connected with lattice integrals of motion as the invariants of quantum affine group $U_q({\hat{n}}_{+})$. We show that Volkov's scheme leads to the system of difference equations for the function from non-commutative variables.

13 pages, misprints have been corrected

Cited by in corpus (4)

• LATTICE CONFORMAL THEORIES AND THEIR INTEGRABLE PERTURBATIONS
• Differential-difference equations associated with the fractional Lax operators
• Geometrical Description of the Local Integrals of Motion of Maxwell-Bloch Equation
• Quantum principal commutative subalgebra in the nilpotent part of $U_q{\widehat{s\ell }}_2$ and lattice KdV variables