SECONDARY QUANTUM HAMILTONIAN REDUCTIONS

Recently, it has been shown how to perform the quantum hamiltonian red uction in the case of general sl(2) embeddings into Lie (super)algebra s, and in the case of general osp(1\2) embeddings into Lie superalgebr as. In another development it has been shown that when H and H' are bo th subalgebras of a Lie algebra G with H' subset of H, then classicall y the W(G, H) algebra can be obtained by performing a secondary hamilt onian reduction on W(G, H'). In this paper we show that the correspond ing statement is true also for quantum hamiltonian reduction when the simple roots of H' can be chosen as a subset of the simple roots of H. As an application, we show that the quantum secondary reductions prov ide a natural framework to study and explain the linearization of the W algebras, as well as a great number of new realizations of W algebra s.