Quadratic Lie Algebras

TL;DR

This paper classifies universal enveloping algebras for low-dimensional braided vector spaces with quadratic Nichols algebras and explores the structure of related braided bialgebras.

Contribution

It specializes the notion of universal enveloping algebra to quadratic Nichols algebras and provides a classification for dimensions up to two.

Findings

Classification of universal enveloping algebras for dim ≤ 2 braided vector spaces.

Structural insights into primitively generated connected braided bialgebras.

Analysis of Nichols algebras that are quadratic.

Abstract

In this paper, the notion of universal enveloping algebra introduced in [A. Ardizzoni, \emph{A First Sight Towards Primitively Generated Connected Braided Bialgebras}, submitted. (arXiv:0805.3391v3)] is specialized to the case of braided vector spaces whose Nichols algebra is quadratic as an algebra. In this setting a classification of universal enveloping algebras for braided vector spaces of dimension not greater than 2 is handled. As an application, we investigate the structure of primitively generated connected braided bialgebras whose braided vector space of primitive elements forms a Nichols algebra which is quadratic algebra.