Continuity of homomorphisms on pro-nilpotent algebras
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.
Keywords
Cite
@article{arxiv.1007.0223,
title = {Continuity of homomorphisms on pro-nilpotent algebras},
author = {George M. Bergman},
journal= {arXiv preprint arXiv:1007.0223},
year = {2021}
}
Comments
Apologies; in submitting version 2, I didn't realize I had to delete version 1; so the result was a mess. Here is the proper revised version. 23 pages, to appear, Ill.J.Math. Main changes in Aug.2010 revision: Re-formatted to fit journal-sized page. New Section 8 added, on question of when subalgebras of finite codimension must be open. Proofs of Lemma 5 and of Lemma 6(iii) shortened