Integrable structure of the classical lattice \(W_n\) algebra (Q1809618)

The integrable structure of the lattice analogue of the \(W_N\) algebra is considered. The author reviews the lattice Virasoro algebra. The elementary dynamical variable is introduced and a generalization to the lattice \(W_N\) algebra is considered. The author constructs the \(N\times N\) Lax matrix in terms of the elementary dynamical variables, and shows that the monodromy matrix has a fundamental Poisson structure with the \(\mathbb{Z}_N\) invariant classical \(r\)-matrix.