A remark on semi-infinite cohomology

Abstract

We extend the notion of semi-infinite cohomology of Lie algebras to include cases where the Lie algebra does not admit a semi-infinite structure but satisfies a mild condition. Our construction clarifies the definition of affine W-algebras in general nilpotent elements case given by V. Kac, S. Roan and M. Wakimoto. We will also give a characterization of admissible pairs with respect to a nilpotent element in a semisimple Lie algebra and define affine W-algebras associated to admissible pairs, while finite W-algebras associated to admissible pairs were already introduced before.

15 pages, an adjusted version of semi-infinite cohomology was introduced in this paper, which clarifies the definition and affine W-algebras in the general case and give it a uniform definition. In this second version, a characterization of admissible pairs was added, as an application of adjusted semi-infinite cohomology, affine W-algebras associated to admissible pairs were introduced