Lattice W algebras and quantum groups

We present Feigin's construction [Lectures given in Landau Institute] of latticeW algebras and give some simple results: lattice Virasoro andW₃ algebras. For the simplest caseg=sl(2), we introduce the wholeU_q(2)) quantum group on this lattice. We find the simplest two-dimensional module as well as the exchange relations and define the lattice Virasoro algebra as the algebra of invariants ofU_q(sl(2)). Another generalization is connected with the lattice integrals of motion as the invariants of the quantum affine groupU_q(ñ₊). We show that Volkov's scheme leads to a system of difference equations for a function of non-commutative variables.