On Fomin--Kirillov Algebras for Complex Reflection Groups

TL;DR

This paper explores the structure and classification of Fomin--Kirillov algebras generalized to complex reflection groups, revealing finite-dimensional cases for cyclic groups and infinite-dimensionality for many others.

Contribution

It applies classification results of Nichols algebras to complex reflection groups, identifying finite and infinite-dimensional cases and analyzing their algebraic properties.

Findings

Finite-dimensional Nichols algebras exist only up to order four for cyclic groups.

Weyl groupoids and finite-dimensional Nichols subalgebras are present in certain cases.

Many non-exceptional complex reflection groups have infinite-dimensional Nichols algebras.

Abstract

This note is an application of classification results for finite-dimensional Nichols algebras over groups. We apply these results to generalizations of Fomin--Kirillov algebras to complex reflection groups. First, we focus on the case of cyclic groups where the corresponding Nichols algebras are only finite-dimensional up to order four, and we include results about the existence of Weyl groupoids and finite-dimensional Nichols subalgebras for this class. Second, recent results by Heckenberger--Vendramin [ArXiv e-prints, 1412.0857 (December 2014)] on the classification of Nichols algebras of semisimple group type can be used to find that these algebras are infinite-dimensional for many non-exceptional complex reflection groups in the Shephard--Todd classification.