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, A 0 , =), where A is an associative algebra over a field k containing subalgebra A 0 with augmentation =: A 0 Ä 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, A 0 , =) naturally acts in the A 0 -cohomology and A 0 -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 W k (g) associated to a complex semisimple Lie algebra g as a semi-infinite Hecke algebra. Using this realization we explicitly calculate the algebra W k (g) avoiding the bosonization technique used in (1992, B. Feigin and E. Frenkel, Internat. J. Mod. Phys. A 7, Suppl. 1A, 197 215). Academic Press

Drinfeld-Sokolov reduction for quantum groups and deformations of W-algebras

“…The proof of this fact is similar to that of Proposition 2.6.4 in [58]. In order to prove that H =0 (C • h (λ k )) = 0 we shall apply Lemma 5.2.4.…”

“…where S • is a semijective resolution of the complex V • (see [58], Section 3.1 for details). Now assume that the algebra A ♯ 0 is augmented, i.e.…”

“…In the terminology introduced in [58] the algebra (1) is the Hecke algebra associated to the triple (U (g), U (n + ), χ 0 ), and the W-algebra W k (g) is the semi-infinite Hecke algebra associated to the triple (U ( g ′ ) k , U (n + [z, z −1 ]), χ), where χ is a nontrivial character of the algebra U (n + [z, z −1 ]) (see Section 3.1 for the exact definition). In the simplest case g = sl 2 the algebra W k (g) is isomorphic to the restricted completion of the quotient of the universal enveloping algebra of the Virasoro algebra by the two-sided ideal generated by C − (1 − 6 (k + 1) 2 k + 2 ), where C is the central element of the Virasoro algebra.…”

Drinfeld–Sokolov reduction for quantum groups and deformations of W-algebras

“…In papers [53,58] the author developed the general theory of Hecke algebras, a deep generalization of the classical notion of Hecke-Iwahori algebras and of the algebraic BRST reduction technique for Lie algebras (see [47]). …”

“…In this paper we use the results of [56,58] to define the deformed W-algebras W k,h (g). The paper is organized as follows.…”