"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 (Universität 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.