Semi-infinite cohomology and Hecke algebras

Abstract

This paper provides a homological algebraic foundation for generalizations of classical Hecke algebras introduced in (1999, A. Sevostyanov, Comm. Math. Phys.204, 137). These new Hecke algebras are associated to triples of the form (A, A0, ε), where A is an associative algebra over a field k containing subalgebra A0 with augmentation ε:A0→k. These algebras are connected with cohomology of associative algebras in the sense that for every left A-module V and right A-module W the Hecke algebra associated to triple (A, A0, ε) naturally acts in the A0-cohomology and A0-homology spaces of V and W, respectively. We also introduce the semi-infinite cohomology functor for associative algebras and define modifications of Hecke algebras acting in semi-infinite cohomology spaces. We call these algebras semi-infinite Hecke algebras. As an example we realize the W-algebra Wk(g) associated to a complex semisimple Lie algebra g as a semi-infinite Hecke algebra. Using this realization we explicitly calculate the algebra Wk(g) avoiding the bosonization technique used in (1992, B. Feigin and E. Frenkel, Internat. J. Mod. Phys. A7, Suppl. 1A, 197-215). © 2001 Academic Press.