New large-rank Nichols algebras over nonabelian groups with commutator subgroupZ2

Abstract

In this article, we explicitly construct new finite-dimensional, indecomposable Nichols algebras with Dynkin diagrams of type An, Cn, Dn, E6,7,8, F4 over any group G with commutator subgroup isomorphic to Z2. The construction is generic in the sense that the type just depends on the rank and center of G, and thus positively answers for all groups of this class a question raised by Susan Montgomery in 1995 [Mon95][AS02]. Our construction uses the new notion of a covering Nichols algebra as a special case of a covering Hopf algebra [Len12] and produces nonfaithful Nichols algebras. We give faithful examples of Doi twists for type A3, C3, D4, C4, F4 over several nonabelian groups of order 16 and 32. These are hence the first known examples of nondiagonal, finitedimensional, indecomposable Nichols algebras of rank > 2 over nonabelian groups.

A simplicial complex of Nichols algebras

“…Question 4.12. As already mentioned in the previous example, all such Nichols algebras of rank < 4 over nonabelian groups had been uniformly constructed in [Len14a] by folding; they have root systems of type A n , C n , E 6 , E 7 , E 8 , F 4 . The resulting root system was calculated by hand and can now be understood as a restriction.…”

“…The first finite-dimensional Nichols algebra of this type has been constructed first for the dihedral group G = D 4 in [MS00] in the context of Coxeter groups. Later, in [Len14a] it has been constructed by the second author as smallest example of a new family of large-rank finite-dimensional indecomposable Nichols algebras, introducing the folding construction of non-diagonal Nichols algebras from diagonal ones (in this case A 2 × A 2 with q = −1). The reader may compare the root system below with the root system in 3.14 b).…”

Root Systems In Finite Symplectic Vector Spaces

“…Commutativity Graphs and Nichols Algebras. In the application [Len13] of symplectic root systems to Nichols algebras over finite nilpotent groups G, a symplectic root system is used to determine a generating set of elements with commutators prescribed by a fixed graph G. Thereby the commutator map plays the role of the symplectic form. Note that again we so far only consider commutators of order 2.…”

“…In our recent study of Nichols algebras over certain nonabelian groups G of nilpotency class 2 in [Len13] we started with a root system over C with given Cartan matrix for a Nichols algebra over an abelian group G/ [G, G]. Then, this Nichols algebra was extended to G using an additional root system structure on a symplectic vector space V = G/G 2 over the finite field F 2 with the same Cartan matrix.…”

“…In [Len13] we constructed the first Nichols algebras of rank > 2 over nonabelian groups G. The construction starts with a known finite-dimensional Nichols algebra over an abelian group Γ of simply-laced Cartan type with a diagram automorphism Z 2 . Then, a symplectic root system over the field F 2 is used to construct a new link-indecomposable finite-dimensional covering Nichols algebra over a central extension Z 2 → G → Γ and again over C.…”