Cohomology of Diagram Algebras and Generalised Tate Cohomology of Hopf Algebras

This thesis is split into two parts. In Part I, we will prove that the composition product on generalised Tate cohomology of finite-dimensional Hopf algebras over a field is equivalent to the cup product. We then deduce that there exists an A∞-structure on generalised Tate cohomology of such Hopf algebras using Kadeishvili’s Theorem. We also show, by way of example, that Steenrod operations on the ordinary cohomology of such Hopf algebras detect the underlying coalgebra structure.

In Part II, which is based on joint work of the author and Daniel Graves, we prove cohomological versions of homological results in recent literature and apply these to families of diagram algebras. We apply frameworks developed by Boyde and Sroka for studying homology of such algebras to families of diagram algebras not yet appearing in homological algebra literature. These algebras include Tanabe algebras, walled Brauer algebras, dilute Temperley–Lieb algebras and blob algebras. We deduce a number of (co)homological stability results for diagram algebras. We also define a notion of Tate cohomology for the diagram algebras considered in this thesis.