The classification of Nichols algebras over groups with finite root system of rank two

Abstract

We classify all groups G and all pairs (V,W) of absolutely simple Yetter-Drinfeld modules over G such that the support of the direct sum of V and W generates G, the square of the braiding between V and W is not the identity, and the Nichols algebra of the direct sum of V and W admits a finite root system. As a byproduct, we determine the dimensions of such Nichols algebras, and several new families of finite-dimensional Nichols algebras are obtained. Our main tool is the Weyl groupoid of pairs of absolutely simple Yetter-Drinfeld modules over groups.

41 pages. Final version. Accepted for publication in Journal of the European Mathematical Society

References in corpus (1)

• Nichols algebras of group type with many quadratic relations

Cited by in corpus (12)

• A classification of Nichols algebras of semi-simple Yetter-Drinfeld modules over non-abelian groups
• The cohomology ring of the 12-dimensional Fomin-Kirillov algebra
• Finite dimensional Nichols algebras over Suzuki algebra I: simple Yetter-Drinfeld modules of $A_{N2n}^{\mu \lambda }$
• On Fomin--Kirillov Algebras for Complex Reflection Groups
• Finite dimensional Nichols algebras over Suzuki algebra II: over simple Yetter-Drinfeld modules of $A_{N2n+1}^{\mu \lambda }$
• Finite-dimensional Nichols algebras of simple Yetter-Drinfeld modules (over groups) of prime dimension
• Pointed Hopf algebras over non abelian groups with decomposable braidings, I
• Multinomial expansion and Nichols algebras associated to non-degenerate involutive solutions of the Yang-Baxter equation
• Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type VII. Semisimple classes in PSL(n,q) and PSp(2n,q)
• On the dimension of the Fomin-Kirillov algebra and related algebras
• Twist equivalence for Nichols algebras over Coxeter groups
• Generalized nil-Coxeter algebras over discrete complex reflection groups