Continuity of homomorphisms on pro-nilpotent algebras

Abstract

Let $\mathbf{V}$ be a variety of not necessarily associative algebras, and $A$ an inverse limit of nilpotent algebras $A_i\in\mathbf{V}$, such that some finitely generated subalgebra $S\subseteq A$ is dense in $A$ under the inverse limit of the discrete topologies on the $A_i$.

A sufficient condition on $\mathbf{V}$ is obtained for all algebra homomorphisms from $A$ to finite-dimensional algebras $B$ to be continuous; in other words, for the kernels of all such homomorphisms to be open ideals. This condition is satisfied, in particular, if $\mathbf{V}$ is the variety of associative, Lie, or Jordan algebras.

Examples are given showing the need for our hypotheses, and some open questions are noted.