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The Relation between Quantum W algebras and Lie algebras  Jan de Boer
; Tjark Tjin
;  Date: 
2 Feb 1993  Journal:  Commun.Math.Phys. 160 (1994) 317332  Subject:  hepth  Abstract:  By quantizing the generalized DrinfeldSokolov reduction scheme for arbitrary $sl_2$ embeddings we show that a large set $cal W$ of quantum W algebras can be viewed as (BRST) cohomologies of affine Lie algebras. The set $cal W$ 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 $cal W$ 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 $cal W$. Some examples are explicitly worked out.  Source:  arXiv, hepth/9302006  Services:  Forum  Review  PDF  Favorites 


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