Braided racks, Hurwitz actions and Nichols algebras with many cubic relations

Abstract

We classify Nichols algebras of irreducible Yetter-Drinfeld modules over groups such that the underlying rack is braided and the homogeneous component of degree three of the Nichols algebra satisfies a given inequality. This assumption turns out to be equivalent to a factorization assumption on the Hilbert series. Besides the known Nichols algebras we obtain a new example. Our method is based on a combinatorial invariant of the Hurwitz orbits with respect to the action of the braid group on three strands.

v2: 35 pages, 6 tables, 14 figures

References in corpus (7)

• Finite-dimensional pointed Hopf algebras with alternating groups are trivial
• Nichols algebras of group type with many quadratic relations
• Pointed Hopf algebras over the sporadic simple groups
• Cohomology of finite dimensional pointed Hopf algebras
• Finite dimensional pointed Hopf algebras over S_4
• Bootstrap percolation and kinetically constrained models on hyperbolic lattices
• Cocycle deformations for Hopf algebras with a coalgebra projection

Cited by in corpus (3)

• Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type V. Mixed classes in Chevalley and Steinberg groups
• Finite-dimensional Nichols algebras of simple Yetter-Drinfeld modules (over groups) of prime dimension
• Representations of copointed Hopf algebras arising from the tetrahedron rack