Rank varieties for Hopf algebras

Abstract

We construct rank varieties for the Drinfeld double of the Taft algebra Λ n and for u q ( s l 2 ) . For the Drinfeld double when n = 2 this uses a result which identifies a family of subalgebras that control projectivity of Λ -modules whenever Λ is a Hopf algebra satisfying a certain homological condition. In this case we show that our rank variety is homeomorphic to the cohomological support variety. We also show that Ext ∗ ( M , M ) is finitely generated over the cohomology ring of the Drinfeld double for any finitely generated module M .