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Quantum $${\mathcal W}$$-algebras and elliptic algebras. (English) Zbl 0871.17007
By E. Frenkel and N. Reshetikhin [Commun. Math. Phys. 178, 237-264 (1996; Zbl 0869.17014)] Poisson algebras $${\mathcal W}_q(\mathfrak g)$$ were introduced as $$q$$-deformations of classical $$\mathcal W$$-algebras. They appear as centers of certain quantized universal enveloping algebras. Here $$\mathfrak g$$ denotes a finite-dimensional simple Lie algebra. For $$\mathfrak g=\mathfrak s\mathfrak l_N$$ the Wakimoto realization provides a homomorphism from this center to a Heisenberg-Poisson algebra (a free field realization). For $$q=1$$ this is the Miura transformation. In the article under review a quantum $$\mathcal W$$-algebra depending on two parameters $$q$$ and $$p$$ associated to $$\mathfrak s\mathfrak l_N$$ is introduced. For special values it is shown that these algebras become the ordinary $$\mathcal W$$ algebra of $$\mathfrak s\mathfrak l_N$$ or its $$q$$-deformed version. Using Heisenberg algebras which depend on 2 parameters, free field realization of this quantum $$\mathcal W$$-algebra and screening currents are constructed. It is interesting to note that in their definition and in particular in the commutation relations of the generating functions, elliptic functions appear. This is the central part of the article. The relation with the “locality property” of vertex algebras is discussed. The article deals with $$\mathfrak s\mathfrak l_N$$. In a concluding paragraph generalizations to arbitrary simply-laced simple $$\mathfrak g$$ are discussed. It is expected that the corresponding results are also true here.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 17B68 Virasoro and related algebras
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