The Cohomology of Dihedral Nichols Algebras using Koszul Complexes

Abstract

In [ETW], the authors equate the cohomology of Hurwitz spaces to the cohomology of a braided Hopf algebra called the quantum shuffle algebra. This quantum shuffle algebra has a subalgebra called a Nichols algebra, and in this document, we study this subalgebra's cohomology. In particular, we study the quadratic covers of Nichols algebras generated by the set of reflections in the dihedral group of order 2p, where p is an odd prime. When p=3, this algebra is isomorphic to the third Fomin-Kirillov algebra, whose cohomology was computed in [SV]. We study the dual Koszul complex of this algebra and use its structure to provide an alternate proof of this result. We expand our study of the dual Koszul complex to the case where p>3, and we prove that these algebras are infinite dimensional. We further prove preliminary results about their cohomology groups and state a conjecture equating their dimension in a fixed degree to a family of recursive sequences, S_p(n), related to the Lucas numbers.