Secondary Quantum Hamiltonian Reduction

Abstract

Recently, it has been shown how to perform the quantum hamiltonian reduction in the case of general $sl(2)$ embeddings into Lie (super)algebras, and in the case of general $osp(1|2)$ embeddings into Lie superalgebras. In another development it has been shown that when $H$ and $H^{\prime }$ are both subalgebras of a Lie algebra $G$ with $H^{\prime }\subset H$, then classically the $W(G,H)$ algebra can be obtained by performing a secondary hamiltonian reduction on $W(G,H^{\prime })$. In this paper we show that the corresponding statement is true also for quantum hamiltonian reduction when the simple roots of $H^{\prime }$ can be chosen as a subset of the simple roots of $H$. As an application, we show that the quantum secondary reductions provide a natural framework to study and explain the linearization of the $W$ algebras, as well as a great number of new realizations of $W$ algebras.

33 pages, LATEX. Final version, including proof of conjecture. Accepted for publication in Comm. Math. Phys

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