A CLASS OF W-ALGEBRAS WITH INFINITELY GENERATED CLASSICAL LIMIT

There is a relatively well understood class of deformable W-algebras, resulting from Drinfeld-Sokolov (DS) type reductions of Kac-Moody alge bras, which are Poisson bracket algebras based on finitely, freely gen erated rings of differential polynomials in the classical limit. The p urpose of this paper is to point out the existence of a second class o f deformable W-algebras, which in the classical limit are Poisson brac ket algebras carried by infinitely, nonfreely generated rings of diffe rential polynomials. We present illustrative examples of coset constru ctions, orbifold projections, as well as first class hamiltonian reduc tions of DS type W-algebras leading to reduced algebras with such infi nitely generated classical limit. We also show in examples that the re duced quantum algebras are finitely generated due to quantum correctio ns arising upon normal ordering the relations obeyed by the classical generators. We apply invariant theory to describe the relations and to argue that classical cosets are infinitely, nonfreely generated in ge neral. As a by-product, we also explain the origin of the previously c onstructed and so far unexplained deformable quantum W(2, 4, 6)- and W (2, 3, 4, 5)-algebras.