Pure Maths Colloquium: Tsunekazu Nishinaka

In this talk, we first introduce an application of (undirected) two edge-colored graphs to group algebras of non-Noetherian groups, where a group is Noetherian if every subgroup is finitely generated. Almost all infinite groups are non-Noetherian except for polycyclic by finite groups. We have used these graphs to study primitivity of group algebras of non-Noetherian groups, where generally a ring is right primitive if it has a maximal right ideal which contains no non-trivial ideals. As every simple ring is primitive, we can regard primitivity as a generalization of simplicity. As for group algebras, any group algebra of a nontrivial group is not simple, but some group algebras of non-abelian infinite groups are primitive. In general our method using two edge-colored graphs seems to be effective to find a non-scalar element appeared in products of elements of a group algebra if its group has non-abelian free subgroups. But there exist some non-Noetherian groups with no non-abelian free subgroups; for example Thompson’s group \(F\) and a free Burnside group of large exponent. Finally in order to be able to investigate group algebras of these groups, we use a ‘directed’ two edge-colored graph and improve our method.