On Nichols algebras with standard braiding

TL;DR

This paper classifies finite-dimensional Nichols algebras arising from standard braided vector spaces, providing explicit bases, dimension formulas, and presentations, thereby extending understanding beyond Cartan type structures.

Contribution

It offers a complete classification of standard braided vector spaces with finite-dimensional Nichols algebras and provides explicit algebraic descriptions.

Findings

Classification of standard braided vector spaces with finite-dimensional Nichols algebras

Explicit PBW-basis and dimension formulas

Generators and relations presentation of Nichols algebras

Abstract

The class of standard braided vector spaces, introduced by Andruskiewitsch and the author in \texttt{arXiv:math/0703924v2} to understand the proof of a theorem of Heckenberger \cite{H2}, is slightly more general than the class of braided vector spaces of Cartan type. In the present paper, we classify standard braided vector spaces with finite-dimensional Nichols algebra. For any such braided vector space, we give a PBW-basis, a closed formula of the dimension and a presentation by generators and relations of the associated Nichols algebra.