Lattice W algebras and quantum groups

TL;DR

This paper constructs lattice W algebras, including Virasoro and W_3, and explores their relation to quantum groups and integrals of motion, providing new algebraic structures and their continuum limits.

Contribution

It introduces lattice W algebras based on quantum groups, including explicit constructions and invariants, extending Feigin's approach and connecting to integrals of motion.

Findings

Constructed lattice Virasoro and W_3 algebras.

Defined lattice Virasoro algebra as invariants of U_q(sl(2)).

Connected lattice algebras with quantum affine groups and difference equations.

Abstract

We represent Feigin's construction [11] 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.Continium limit of this lattice algebras are considered.