On Fomin–Kirillov algebras for complex reflection groups

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.

KEYWORDS:

• Braided Hopf algebras
• Fomin–Kirillov algebras
• Nichols algebras
• Weyl groupoids

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 17B37
• 16T05
• 16T30

Acknowledgments

The author likes to thank Yuri Bazlov for introducing him to Fomin–Kirillov algebras. Some of this work was conducted during a visit of the author at University of Marburg, and the author is grateful for the hospitality offered by the Department of Mathematics. Special thanks go to István Heckenberger for teaching the author about Weyl groupoids and providing many helpful insights and guidance which made this work possible. The visit to Marburg was supported by the DFG Schwerpunkt “Darstellungstheorie”.

Notes

¹Note that the original Fomin–Kirillov algebras ℰ_n are only quadratic covers of their Nichols algebra quotients. It is unknown whether the Nichols algebras themselves are quadratic in the symmetric group case (n≥6). However, for cyclic groups, most relations are not quadratic, so we do not consider the quadratic cover.

²In the table, ξ is a primitive n-th root of unity. We omit reflections that invert given ones, or remain at one generalized Dynkin diagram.

³For some computations, e.g., of dimensions, we use [9].

⁴It is a not known that finite Gelfand–Kirillov-dimension implies finiteness of the root system, and hence of the Nichols algebra (cf. [10, Section 3]). The Gelfand–Kirillov dimension for ${\mathcal{ℬ}}_{C_p}$ for primes p>3 might also be infinite.

⁵Note that it is an open question whether ${\mathcal{ℬ}}_{S_n}\cong {\mathcal{ℰ}}_{S_n}$ for n≥6.