Hochschild cochains as a Frobenius algebra

TL;DR

This paper constructs a Frobenius algebra structure on Hochschild cochains of a group ring, extending topological quantum field theory structures and exploring algebraic properties and degeneracies.

Contribution

It introduces a Frobenius algebra structure on Hochschild cochains of group rings, linking it to topological quantum field theories and Hochschild homology.

Findings

Frobenius algebra structure extends known TQFT on HH^0

Convolution product extends to Gerstenhaber product

Pairing degenerates on a subcomplex

Abstract

We construct a Frobenius algebra structure on the Hochschild cochains of a group ring k[G] that extends the known structure of a <1, 2> topological quantum field theory on HH^0(k[G]; k[G]), k a field and G a finite group. The convolution product extends to the homotopy commutative Gerstenhaber product on cochains, the Frobenius coproduct extends to a coproduct on the chain complex for Hochschild homology, and there is a pairing on Hochschild cocahins satisfying Frobenius associativity. The pairing, however, degenerates on a certain subcomplex of Hochschild cochains. The cochain complex for group cohomology under the simplicial cup product occurs as a homotopy commutative subalgebra of the Hochschild cochain complex under the Gerstenhaber product.