Poisson reduction of the lattice Kac-Moody algebra (Q1292020)

The Poisson reduction of the \(\text{SL}(N)\) lattice Kac-Moody algebra under the action of the maximal nilpotent subgroup (which is similar to the known Drinfeld-Sokolov reduction) is studied. The Poisson bracket on the reduced spave is obtained by restricting to it the bracket of \(G\)-invariant functions. In the lattice case the group \(G\) acts in a Poissonian, but not in a Hamiltonian way, i.e. the constraints do not form a Lie algebra globally and are deeply nonlocal with respect to the lattice variables. It is shown that in the continuous limit the constraints are related with the Gelfand-Fuks cocycle.