The lazy homology of a Hopf algebra

TL;DR

This paper introduces two new commutative Hopf algebras called lazy homology Hopf algebras, linking them to lazy cohomology groups and providing explicit computations for specific cases like group algebras and the Sweedler algebra.

Contribution

It defines the first and second lazy homology Hopf algebras for any Hopf algebra and relates them to lazy cohomology, with explicit calculations for particular examples.

Findings

Lazy homology Hopf algebras are related to lazy cohomology groups via universal coefficient theorems.

Lazy homology of group algebras expressed in terms of group homology.

First lazy homology of cosemisimple Hopf algebras corresponds to the universal abelian grading group.

Abstract

To any Hopf algebra H we associate two commutative Hopf algebras, which we call the first and second lazy homology Hopf algebras of H. These algebras are related to the lazy cohomology groups based on the so-called lazy cocycles of H by universal coefficient theorems. When H is a group algebra, then its lazy homology can be expressed in terms of the 1- and 2-homology of the group. When H is a cosemisimple Hopf algebra over an algebraically closed field of characteristic zero, then its first lazy homology is the Hopf algebra of the universal abelian grading group of the category of corepresentations of H. We also compute the lazy homology of the Sweedler algebra.