On the computation of Hopf 2-cocycles, with an example of diagonal type

TL;DR

This paper develops a framework for computing Hopf 2-cocycles in Nichols algebra deformations, providing explicit examples and revealing their structural properties, especially in the context of quantum groups of type A2.

Contribution

It introduces a recurrence formula for Hopf 2-cocycles and explores their connection with Hochschild cohomology, with detailed computations for a specific quantum group case.

Findings

Explicit description of Hopf 2-cocycles for Nichols algebra of type A2

Demonstration that these cocycles are generically pure

Establishment of a recurrence relation for cocycle computation

Abstract

We present a framework for the computation of the Hopf 2-cocycles involved in the deformations of Nichols algebras over semisimple Hopf algebras. We write down a recurrence formula and investigate the extent of the connection with invariant Hochschild cohomology in terms of exponentials. As an example, we present detailed computations leading to the explicit description of the Hopf 2-cocycles involved in the deformations of a Nichols algebra of Cartan type $A_2$ with $q=-1$, a.k.a. the positive part of the small quantum group $\mathfrak{u}^+_{\sqrt{\text{-1}}}(\mathfrak{sl}_3)$. We show that these cocycles are generically pure, that is they are not cohomologous to exponentials of Hochschild 2-cocycles.