, Volume 254, Issue 3, pp 659-694
Date: 11 Jan 2005

Freely Generated Vertex Algebras and Non–Linear Lie Conformal Algebras

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We introduce the notion of a non–linear Lie conformal superalgebra and prove a PBW theorem for its universal enveloping vertex algebra. We also show that conversely any graded freely generated vertex algebra is the universal enveloping algebra of a unique, up to isomorphism, non–linear Lie conformal superalgebra. This correspondence will be applied in the subsequent work to the problem of classification of finitely generated simple graded vertex algebras.

Communicated by L. Takhtajan
Acknowledgement. We would like to thank M. Artin, B. Bakalov, A. D’andrea and P. Etingof for useful discussions. This research was conducted by A. De Sole for the Clay Mathematics Institute. The paper was partially supported by the NSF grant DMS0201017.