Quantization and representation theory of finite W algebras

algebras' are constructed as Poisson reductions of Kirillov Poisson structures on simple Lie algebras. The inequivalent reductions are labeled by the inequivalent embeddings of

sl_2

into the simple Lie algebra in question. For arbitrary embeddings a coordinate free formula for the reduced Poisson structure is derived. We also prove that any finite

W

algebra can be embedded into the Kirillov Poisson algebra of a (semi)simple Lie algebra (generalized Miura map). Furthermore it is shown that generalized finite Toda systems are reductions of a system describing a free particle moving on a group manifold and that they have finite

W

symmetry. In the second part we BRST quantize the finite

W

algebras. The BRST cohomology is calculated using a spectral sequence (which is different from the one used by Feigin and Frenkel). This allows us to quantize all finite

W

algebras in one stroke. Explicit results for

sl_3

and

sl_4

are given. In the last part of the paper we study the representation theory of finite

W

algebras. It is shown, using a quantum version of the generalized Miura transformation, that the representations of finite

W

algebras can be constructed from the representations of a certain Lie subalgebra of the original simple Lie algebra. As a byproduct of this we are able to construct the Fock realizations of arbitrary finite

W

algebras.

algebra: W(N)
algebra: Lie
algebra: finite
quantum algebra
algebra: SL(2)
cohomology: Becchi-Rouet-Stora
algebra: representation
Miura transformation
lattice: Toda
bibliography

References(39)

Figures(0)

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