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arXiv · 1605.00227

On Fomin--Kirillov Algebras for Complex Reflection Groups

Robert Laugwitz
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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.

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Robert Laugwitz. 2016-05-01. On Fomin--Kirillov Algebras for Complex Reflection Groups. https://doi.org/10.1080/00927872.2016.1243698

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