The Relation between Quantum W algebras and Lie algebras

Abstract

By quantizing the generalized Drinfeld-Sokolov reduction scheme for arbitrary $s{l}_2$ embeddings we show that a large set $\mathcal{W}$ of quantum W algebras can be viewed as (BRST) cohomologies of affine Lie algebras. The set $\mathcal{W}$ contains many known $W$ algebras such as $W_N$ and $W_3^{(2)}$. Our formalism yields a completely algorithmic method for calculating the W algebra generators and their operator product expansions, replacing the cumbersome construction of W algebras as commutants of screening operators. By generalizing and quantizing the Miura transformation we show that any $W$ algebra in $\mathcal{W}$ can be embedded into the universal enveloping algebra of a semisimple affine Lie algebra which is, up to shifts in level, isomorphic to a subalgebra of the original affine algebra. Therefore {\em any} realization of this semisimple affine Lie algebra leads to a realization of the $W$ algebra. In particular, one obtains in this way a general and explicit method for constructing the free field realizations and Fock resolutions for all algebras in $\mathcal{W}$. Some examples are explicitly worked out.

21 pages, THU-93/05, ITFA-93/2

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• Representation theory of finite W algebras
• On the General Structure of Hamiltonian Reductions of the Wznw Theory
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