Cohomology of Hopf Algebras

Group algebras are Hopf algebras, and their Hopf structure plays crucial roles in representation theory and cohomology of groups. A Hopf algebra is an algebra A (say over a field k) that has a comultiplication (∆ : A → A ⊗k A) generalizing the diagonal map on group elements, an augmentation (ε : A → k) generalizing the augmentation on a group algebra, and an antipode (S : A → A) generalizing the inverse map on group elements. Hopf algebras of interest include group algebras, universal enveloping algebras of Lie algebras, restricted enveloping algebras, quantum groups, coordinate rings of groups, and more. The category of modules of a Hopf algebra A is a tensor category with unit object and duals; this extra structure on the category arises from the comultiplication, augmentation, and antipode. Denote the unit object (that is, trivial module) by k. The cohomology of A is H∗(A) := ExtA(k, k). The cohomology of A generally has some of the same properties as group cohomology: It is an algebra under a cup product arising from the tensor product of resolutions, equivalently Yoneda composition. It is graded commutative. If M is an A-module, then ExtA(M,M) is an H ∗(A)-module by tensoring with M . These are some of the ingredients required for a support variety theory of modules. My research involves the structure of Hopf algebra cohomology, support variety theory, and applications for various types of finite dimensional Hopf algebras. The following conjecture motivates some of my work.