689 research outputs found
Homomorphisms on infinite direct product algebras, especially Lie algebras
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.Comment: 33 pages. The lemma in section 12.1 of the previous version was
incorrect, and has been removed. (Nothing else depended on it.) Other changes
are improvements in wording, et
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