Higher commutators, hypercubes, and the hierarchy of centralizer conditions

Abstract: The commutator had historically been studied for specific varieties of algebras until Smith found a general definition for a commutator that worked for any Mal'cev algebra. Since then the commutator has become an essential part of the general algebraist's toolkit. Bulatov discovered at the beginning of the century that the (binary) commutator can be extended to an infinite sequence of higher arity operations, no one of which are term definable from the others. This discovery has most importantly led to the distinction between a nilpotent algebra and a 'supernilpotent' algebra. While this distinction is invisible for groups, supernilpotent Mal'cev algebras share many strong properties with nilpotent groups, while nilpotent algebras need not. We will discuss the extent to which some of the known results of commutator theory can be viewed as a low-dimensional case of a general multidimensional theory.

logicrings and algebras

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic