Abelian and non-abelian second cohomologies of quantized enveloping algebras

TL;DR

This paper explores the second cohomology of quantized enveloping algebras, introducing a novel method to compute abelian cohomology using non-abelian cohomology insights, and proves a quantum analogue of Whitehead's lemma.

Contribution

It presents a new approach to compute abelian second cohomology via non-abelian cohomology for pointed Hopf algebras, including quantized enveloping algebras.

Findings

A non-standard method for computing abelian second cohomology.

A quantum analogue of Whitehead's second lemma proved.

Insights into cleft extensions and cocycle deformations.

Abstract

For a class of pointed Hopf algebras including the quantized enveloping algebras, we discuss cleft extensions, cocycle deformations and the second cohomology. We present such a non-standard method of computing the abelian second cohomology that derives information from the non-abelian second cohomology classifying cleft extensions. As a sample computation, a quantum analogue of Whitehead's second lemma for Lie-algebra cohomology is proved.