Semiinfinite cohomology of quantum groups II

TL;DR

This paper establishes that the semi-infinite cohomology of a quantum group analogue of an affine Lie algebra's nilpotent subalgebra matches the classical case for various parameters, extending known results.

Contribution

It introduces a quantum analogue of the semi-infinite cohomology for affine Lie algebras and proves their equivalence with classical cohomology for general parameters.

Findings

Semi-infinite cohomology spaces are characterized by affine Weyl group elements.

Quantum analogue of the universal enveloping algebra is constructed.

Cohomology results hold for general parameter values.

Abstract

It is known that the semi-infinite cohomology spaces of the infinitely twisted nilpotent subalgebra in an affine Lie algebra $g$ with coefficients in an integrable simple module over the affine Lie algebra have a base enumerated by elements of the corresponding affine Weyl group graded by the semiinfinite length function. Let $U$ be the affine quantum group corresponding to $g$. It is possible to define a subalgebra in $U$ being the quantum analogue of the universal enveloping algebra of the infinitely twisted nilpotent subalgebra in $g$. In this paper we prove that for general values of the parameter $v$ the semiinfinite cohomology of this associative algebra with coefficients in an integrable simple module over $U$ coincides with the one of the corresponding Lie subalgebra in $g$ with coefficients in the corresponding $g$-module.