Divisibility relations for the dimensions and Hilbert series of Nichols algebras of non-Abelian group type. (Q375187)

The paper contributes to the theory of Nichols algebras of non-Abelian group type. The author proves the following two results.NEWLINENEWLINE 1) Let \(V\) be a Yetter-Drinfeld module derived from an indecomposable quandle \(X\) and a 2-cocycle \(\chi\) such that the Nichols algebra \(\mathcal B(V)\) is finite dimensional. Assume that the degree of \(X\) divides the order of the diagonal elements of \(\chi\), or the characteristic of the base field if the diagonal elements are 1. Then each \(\text{Inn\,}X\)-homogeneous component of \(\mathcal B(V)\) has the same dimension, in particular \(\#\text{Inn\,}X\) divides \(\dim\mathcal B(V)\). Moreover, if \(X'\) is a non-empty proper subrack of \(X\) with corresponding Yetter-Drinfeld submodule \(V'\), and \(X\setminus X'\) generates \(\text{Inn\,}X\), then \(\#\text{Inn\,}X\cdot\dim\mathcal B(V')\) divides \(\dim\mathcal B(V)\).NEWLINENEWLINE 2) Let \(\mathcal B(V)\) be a finite dimensional Nichols algebra over an indecomposable quandle \(X\) and a 2-cocycle with diagonal elements of order \(m\); if the diagonal elements are 1, set \(m\) to be the characteristic of the base field. Let \(X'\) be a non-empty proper subrack of \(X\) and let \(V'\) be the corresponding Yetter-Drinfeld submodule of \(V\). Then the Hilbert series \(\mathcal H_{\mathcal B(V)}(t)\) is divisible by \((m)_t\cdot\mathcal H_{\mathcal B(V)}(t)\).NEWLINENEWLINE The proofs use a freeness theorem of \textit{M. Graña} [J. Algebra 231, No. 1, 235-257 (2000; Zbl 0970.16017)] and refinements of a method of \textit{A. Milinski} and \textit{H.-J. Schneider} [Contemp. Math. 267, 215-236 (2000; Zbl 1093.16504)].