Tate–Shafarevich groups and algebras

Abstract

The Tate–Shafarevich set of a group [Formula: see text] defined by Takashi Ono coincides, in the case where [Formula: see text] is finite, with the group of outer class-preserving automorphisms of [Formula: see text] introduced by Burnside. We consider analogs 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.