Lattice $W$ algebras and quantum groups

TL;DR

This paper constructs lattice W algebras, including Virasoro and W_3, using Feigin's approach, and explores their relation to quantum groups and integrals of motion, providing new algebraic and difference equation frameworks.

Contribution

It introduces lattice W algebras based on Feigin's construction and links them to quantum groups and integrals of motion, expanding the algebraic understanding of lattice conformal structures.

Findings

Construction of lattice Virasoro and W_3 algebras.

Representation of U_q(sl(2)) on the lattice.

Difference equations from non-commutative variables.

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.