Continuity of homomorphisms on pro-nilpotent algebras

TL;DR

This paper establishes conditions under which homomorphisms from pro-nilpotent algebras to finite-dimensional algebras are continuous, with applications to associative, Lie, and Jordan algebras, and discusses related examples and open questions.

Contribution

It provides a sufficient condition on the variety of algebras ensuring the continuity of homomorphisms from pro-nilpotent algebras to finite-dimensional algebras, extending known cases.

Findings

Homomorphisms are continuous under the given conditions.

The condition applies to associative, Lie, and Jordan algebras.

Examples illustrate the necessity of hypotheses.

Abstract

Let V be a variety of not necessarily associative algebras, and A an inverse limit of nilpotent algebras A_i\in 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 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 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.