Twisted derivations of Hopf algebras

TL;DR

This paper introduces twisted derivations as infinitesimal symmetries of bialgebras, forming a Lie algebra that relates to twisted automorphisms, with explicit calculations for specific Hopf algebras.

Contribution

It defines twisted derivations of bialgebras and explores their structure, linking them to twisted automorphisms and computing their crossed modules for key examples.

Findings

Twisted derivations form a Lie algebra.

The Lie algebra fits into a crossed module with twisted automorphisms.

Explicit calculations for universal enveloping algebras and Sweedler's Hopf algebra.

Abstract

In the paper we introduce the notion of twisted derivation of a bialgebra. Twisted derivations appear as infinitesimal symmetries of the category of representations. More precisely they are infinitesimal versions of twisted automorphisms of bialgebras. Twisted derivations naturally form a Lie algebra (the tangent algebra of the group of twisted automorphisms). Moreover this Lie algebra fits into a crossed module (tangent to the crossed module of twisted automorphisms). Here we calculate this crossed module for universal enveloping algebras and for the Sweedler's Hopf algebra.