Tate-Shafarevich groups and algebras

TL;DR

This paper explores analogues of the Tate-Shafarevich group for Lie and associative algebras, establishing new structural properties and discussing open problems and potential generalizations.

Contribution

It introduces and analyzes the Tate-Shafarevich analogue for Lie and associative algebras, extending concepts from group theory to these algebraic structures.

Findings

Established structural properties of Tate-Shafarevich analogues for Lie and associative algebras

Connected the Tate-Shafarevich set with outer class-preserving automorphisms in finite groups

Discussed open problems and potential generalizations to other algebraic structures

Abstract

The Tate-Shafarevich set of a group G defined by Takashi Ono coincides, in the case where G is finite, with the group of outer class-preserving automorphisms of G introduced by Burnside. We consider analogues of this important group-theoretic object for Lie algebras and associative algebras and establish some new structure properties thereof. We also discuss open problems and eventual generalizations to other algebraic structures.