On Fomin–Kirillov algebras for complex reflection groups

Laugwitz, Robert

Abstract

In this note, we apply classification results for finite-dimensional Nichols algebras 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.

Citation

Laugwitz, R. (2017). On Fomin–Kirillov algebras for complex reflection groups. Communications in Algebra, 45(8), 3653-3666. https://doi.org/10.1080/00927872.2016.1243698

• Journal Article Type: Article
• Online Publication Date: Jan 19, 2017
• Publication Date: Aug 3, 2017
• Deposit Date: Apr 25, 2025
• Journal: Communications in Algebra
• Print ISSN: 0092-7872
• Electronic ISSN: 1532-4125
• Publisher: Taylor and Francis
• Peer Reviewed: Peer Reviewed
• Volume: 45
• Issue: 8
• Pages: 3653-3666
• DOI: https://doi.org/10.1080/00927872.2016.1243698
• Public URL: https://nottingham-repository.worktribe.com/output/48097432
• Publisher URL: https://www.tandfonline.com/doi/full/10.1080/00927872.2016.1243698