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The relation between quantum \(W\) algebras and Lie algebras. (English) Zbl 0796.17027

Summary: By quantizing the generalized Drinfeld-Sokolov reduction scheme for arbitrary \(\text{sl}_ 2\) 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 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.

MSC:

17B68 Virasoro and related algebras
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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