Lattice $W$ algebras and quantum groups

We present Feigin's construction [Lectures given in Landau Institute] of lattice $W$ algebras and give some simple results: lattice Virasoro and $W_3$ algebras. For the simplest case $g=sl(2)$, we introduce the whole $U_q(sl(2))$ 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 of $U_q(sl(2))$. Another generalization is connected with the lattice integrals of motion as the invariants of the quantum affine group $U_q(\hat {n}_{+})$. We show that Volkov's scheme leads to a system of difference equations for a function of non-commutative variables.