Abstract
A ubiquitous observation for finite-dimensional Nichols algebras is that as a graded algebra the Hilbert series factorizes into cyclotomic polynomials. For Nichols algebras of diagonal type (e.g., Borel parts of quantum groups), this is a consequence of the existence of a root system and a Poincare-Birkhoff-Witt (PBW) basis basis, but, for nondiagonal examples (e.g., Fomin–Kirillov algebras), this is an ongoing surprise. In this article, we discuss this phenomenon and observe that it continues to hold for the graded character of the involved group and for automorphisms. First, we discuss thoroughly the diagonal case. Then, we prove factorization for a large class of nondiagonal Nichols algebras obtained by the folding construction. We conclude empirically by listing all remaining examples, which were in size accessible to the computer algebra system GAP and find that again all graded characters factorize.
Highlights
A Nichols algebra B( M) over a finite group is a certain graded algebra associated with a given Yetter–Drinfel’d module over this group
The root system theory and PBW-basis of Nichols algebras developed by [1,2] precisely explains a factorization of B( M ) as graded vector space into Nichols subalgebras of rank 1 associated with each root
Heckenberger (e.g., [2]) introduced q-decorated diagrams, with each node corresponding to a simple Yetter–Drinfel’d module decorated by qii, and each edge decorated by τ 2 = qij q ji and edges are drawn if the decoration is 6= 1; it turns out that this data is all that is needed to determine the respective Nichols algebra
Summary
A Nichols algebra B( M) over a finite group is a certain graded algebra associated with a given Yetter–Drinfel’d module over this group. For Nichols algebras over nonabelian groups, the Nichols subalgebras of rank 1 may still be large complicated algebras, so the root system cannot explain the complete factorization of the Hilbert series that we observe. The first author has obtained in [8] new families of Nichols algebras over nonabelian groups, where we can prove the factorization of graded characters: they are constructed from diagonal Nichols algebras by a folding technique and, from this, they retain a finer root system (e.g., A2n−1 inside Cn ). This includes the Fomin–Kirillov algebra over D4 , which has a finer root system A2 × A2 inside A2. Again, all graded characters factorize in these examples
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