Homomorphisms on infinite direct product algebras, especially Lie algebras

TL;DR

This paper investigates the structure and properties of surjective homomorphisms between infinite direct product algebras, especially Lie algebras, revealing conditions under which the image algebra inherits properties like solvability and nilpotency.

Contribution

It establishes new results on the behavior of homomorphisms from infinite product Lie algebras to finite-dimensional algebras, including splitting and continuity, using recent structural insights.

Findings

If all A_i are solvable, B is solvable.

If all A_i are nilpotent, B is nilpotent.

Under certain conditions, f splits and is continuous.

Abstract

We study surjective homomorphisms f:\prod_I A_i\to B of not-necessarily-associative algebras over a commutative ring k, for I a generally infinite set; especially when k is a field and B is countable-dimensional over k. Our results have the following consequences when k is an infinite field, the algebras are Lie algebras, and B is finite-dimensional: If all the Lie algebras A_i are solvable, then so is B. If all the Lie algebras A_i are nilpotent, then so is B. If k is not of characteristic 2 or 3, and all the Lie algebras A_i are finite-dimensional and are direct products of simple algebras, then (i) so is B, (ii) f splits, and (iii) under a weak cardinality bound on I, f is continuous in the pro-discrete topology. A key fact used in getting (i)-(iii) is that over any such field, every finite-dimensional simple Lie algebra L can be written L=[x_1,L]+[x_2,L] for some x_1, x_2\in L, which we prove from a recent result of J.M.Bois. The general technique of the paper involves studying conditions under which a homomorphism on \prod_I A_i must factor through the direct product of finitely many ultraproducts of the A_i. Several examples are given, and open questions noted.