Classifying complements for Hopf algebras and Lie algebras

TL;DR

This paper addresses the classification of complements in extensions of Hopf and Lie algebras, establishing a cohomological framework and introducing a factorization index to measure the complexity of the classification problem.

Contribution

It introduces a cohomological classification method for complements in algebra extensions and defines a factorization index to quantify the classification complexity.

Findings

Established a bijective correspondence between complements and a cohomological object.

Introduced the factorization index as a numerical measure of the classification problem.

Constructed a Hopf algebra with arbitrarily large factorization index for specific roots of unity.

Abstract

Let $A \subseteq E$ be a given extension of Hopf (respectively Lie) algebras. We answer the \emph{classifying complements problem} (CCP) which consists of describing and classifying all complements of $A$ in $E$. If $H$ is a given complement then all the other complements are obtained from $H$ by a certain type of deformation. We establish a bijective correspondence between the isomorphism classes of all complements of $A$ in $E$ and a cohomological type object ${\mathcal H}{\mathcal A}^{2} (H, A \, | \, (\triangleright, \triangleleft) )$, where $(\triangleright, \triangleleft)$ is the matched pair associated to $H$. The factorization index $[E: A]^f$ is introduced as a numerical measure of the (CCP). For two $n$-th roots of unity we construct a $4n^2$-dimensional Hopf algebra whose factorization index over the group algebra is arbitrary large.