Tensor products and support varieties for some noncocommutative Hopf algebras

TL;DR

This paper investigates the behavior of tensor products and projectivity of modules over certain noncocommutative finite-dimensional Hopf algebras, introducing support varieties as a key analytical tool.

Contribution

It constructs new examples of noncocommutative Hopf algebras where tensor product properties differ based on order and develops a support variety theory for these algebras.

Findings

Tensor powers of nonprojective modules can become projective.

Tensor products may be projective in one order but not the other.

Support varieties effectively analyze module properties.

Abstract

We explore questions of projectivity and tensor products of modules for finite dimensional Hopf algebras. We construct many classes of examples in which tensor powers of nonprojective modules are projective and tensor products of modules in one order are projective but in the other order are not. Our examples are smash coproducts with duals of group algebras, some having algebra and coalgebra structures twisted by a cocycle. We develop a theory of support varieties for these Hopf algebras to use as a tool in our investigations.