Bialgebra cohomology, pointed Hopf algebras, and deformations

TL;DR

This paper provides explicit formulas linking bialgebra and Hochschild cohomology, offers conditions for surjectivity of certain maps, and computes all bialgebra two-cocycles for specific Radford biproducts, illuminating their deformation theory.

Contribution

It introduces explicit formulas for cohomology maps, establishes surjectivity conditions, and classifies bialgebra two-cocycles for Radford biproducts related to pointed Hopf algebras.

Findings

Explicit formulas for cohomology maps

Surjectivity conditions for connecting homomorphism

Classification of bialgebra two-cocycles for Radford biproducts

Abstract

We give explicit formulas for maps in a long exact sequence connecting bialgebra cohomology to Hochschild cohomology. We give a sufficient condition for the connecting homomorphism to be surjective. We apply these results to compute all bialgebra two-cocycles of certain Radford biproducts (bosonizations). These two-cocycles are precisely those associated to the finite dimensional pointed Hopf algebras in the recent classification of Andruskiewitsch and Schneider, in an interpretation of these Hopf algebras as graded bialgebra deformations of Radford biproducts.