Cohomology of diagram algebras

TL;DR

This paper investigates the cohomology of new diagram algebras like blob and walled Brauer algebras, establishing connections with group cohomology and introducing an integer-graded cohomology theory linked to Tate cohomology.

Contribution

It introduces the cohomology of new diagram algebras and relates it to group cohomology, providing a unified framework and new phenomena in the field.

Findings

Cohomology of blob and walled Brauer algebras studied.

Cohomology of certain diagram algebras identified with group cohomology.

Established an integer-graded cohomology theory linked to Tate cohomology.

Abstract

The study of the homology of diagram algebras has emerged as an interesting and important field. In many cases, the homology of a diagram algebra can be identified with the homology of a group. In this paper we have two main aims. Firstly, we study the (co)homology of new families of diagram algebras such as the blob algebras and the walled Brauer algebras, both of which exhibit new phenomena in the field. Secondly, we show that in the cases where the homology of a diagram algebra can be identified with group homology one can also identify the cohomology of the algebra with the cohomology of a group. We use this to establish an integer-graded cohomology theory for these diagram algebras and identify this with the Tate cohomology of a group.