Finite vs affine Walgebras
 Alberto De Sole,
 Victor G. Kac
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In Section 1 we review various equivalent definitions of a vertex algebra V. The main novelty here is the definition in terms of an indefinite integral of the λbracket. In Section 2 we construct, in the most general framework, the Zhu algebra Zhu _{Γ} V, an associative algebra which “controls” Γtwisted representations of the vertex algebra V with a given Hamiltonian operator H. An important special case of this construction is the Htwisted Zhu algebra Zhu _{ H } V. In Section 3 we review the theory of nonlinear Lie conformal algebras (respectively nonlinear Lie algebras). Their universal enveloping vertex algebras (resp. universal enveloping algebras) form an important class of freely generated vertex algebras (resp. PBW generated associative algebras). We also introduce the Htwisted Zhu nonlinear Lie algebra Zhu _{ H } R of a nonlinear Lie conformal algebra R and we show that its universal enveloping algebra is isomorphic to the Htwisted Zhu algebra of the universal enveloping vertex algebra of R. After a discussion of the necessary cohomological material in Section 4, we review in Section 5 the construction and basic properties of affine and finite Walgebras, obtained by the method of quantum Hamiltonian reduction. Those are some of the most intensively studied examples of freely generated vertex algebras and PBW generated associative algebras. Applying the machinery developed in Sections 3 and 4, we then show that the Htwisted Zhu algebra of an affine Walgebra is isomorphic to the finite Walgebra, attached to the same data. In Section 6 we define the Zhu algebra of a Poisson vertex algebra, and we discuss quasiclassical limits. In the Appendix, the equivalence of three definitions of a finite Walgebra is established.
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 Title
 Finite vs affine Walgebras
 Journal

Japanese Journal of Mathematics
Volume 1, Issue 1 , pp 137261
 Cover Date
 20060401
 DOI
 10.1007/s1153700605052
 Print ISSN
 02892316
 Online ISSN
 18613624
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 vertex algebra
 (nonlinear) Lie conformal algebra
 deformed vertex operators
 twisted module over a vertex algebra
 Zhu algebra
 finite and affine Walgebras
 quasiclassical limit
 17B69
 Authors

 Alberto De Sole ^{(1)} ^{(2)}
 Victor G. Kac ^{(3)}
 Author Affiliations

 1. Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA, 02138, USA
 2. Istituto Nazionale di Alta Matematica, Cittá Universitaria, 00185, Roma, Italy
 3. Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA