Homomorphisms on infinite direct product algebras, especially Lie algebras

Details Homomorphisms on infinite direct product algebras, especially Lie algebras Homomorphisms on infinite direct product algebras, especially Lie algebras

2009-10-27

arxiv.org/abs/0910.5181v2

J. Alg. 333 (2011) 67-104

Mathematics

Rings and Algebras

33 pages. The lemma in section 12.1 of the previous version was incorrect, and has been removed. (Nothing else depended on it.

Scientific paper

10.1016/j.jalgebra.2011.02.035

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.

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