"Geometry and Analysis on Groups" Research Seminar

Time: 2016.04.16, 15:00–17:00
Location: Seminarraum 9, Oskar-Morgenstern-Platz 1, 2.Stock
Title: "Arithmetically-free group-gradings of algebras with the permutation-contraction property."
Speaker: Wolfgang Moens (Universitšt Wien)
Abstract: We will study under which conditions an algebra \(A\) that is graded by a group \(G\) must be nilpotent. Some partial results in the literature have obtained the nilpotency of \(A\) by imposing conditions on the algebra itself; other results have imposed conditions on the group. Some more results have used a mixed strategy by imposing conditions on both the algebra and the group. We will show that the correct emphasis is on the support \(X\) of the grading.

Concretely, we will introduce arithmetically-free gradings of (a broad family) of algebras by (arbitrary) groups, and show:

  1. If the support \(X\) of the grading is arithmetically-free, then \(A\) is nilpotent of \(|X|\)–bounded class.
  2. If \(X\) is not arithmetically-free, then it supports the grading of a non-nilpotent algebra.
  3. If \(X\) is arithmetically-free and admits a good-ordering, then a Lie algebra \(L\) supported by \(X\) is nilpotent of class at most \(|X|^{2^{|X|}}\).

The proof for 1. is combinatorial in nature and is based on an existence result by G. Higman in the special case \((G,\cdot) = (\mathbb{Z}_p,+)\). (It can also be stated in terms of walks in Cayley-graphs.) The proof for 3. uses some Lie theory and touches on several problems of Erdös in additive combinatorics. We conclude with some brief remarks about the connection between arithmetically-free gradings, periodic transformations, and the co-class conjectures for \(p\)–groups.