On the Zariski topology of $Ω$-groups

Ruvim Lipyanski

Published 2017-12-11Version 1

A number of geometric properties of algebras from a given variety can be characterized using the notions of anticommutativity and stability of algebras. Let $\Theta$ be a variety and $F$ be a finitely generated free algebra in $\Theta$. The stability of the $\Omega$-group $H$ in the variety $\Theta$ means that we can equip the space ${\rm Hom}(F,H)$ with the Zariski topology, whose closed sets are precisely algebraic sets. In three classical cases of $\Omega$-groups (Lie algebras, groups, and associative rings) necessary and sufficient conditions of their stability are given.