Homomorphisms on infinite direct product algebras, especially Lie algebras

We study surjective homomorphisms f:∏IAi→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 Ai are solvable, then so is B.If all the Lie algebras Ai are nilpotent, then so is B.If k is not of characteristic 2 or 3, and all the Lie algebras Ai 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=[x1,L]+[x2,L] for some x1,x2∈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 ∏IAi must factor through the direct product of finitely many ultraproducts of the Ai.Several examples are given, and open questions noted.