Meander graphs and Frobenius Seaweed Lie algebras

The index of a seaweed Lie algebra can be computed from its associated meander graph. We examine this graph in several ways with a goal of determining families of Frobenius (index zero) seaweed algebras. Our analysis gives two new families of Frobenius seaweed algebras as well as elementary proofs of known families of such Lie algebras.Comment: 5 figures, to appear in Journal of Generalized Lie Theor

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

Classification of finite dimensional simple Lie algebras in prime characteristics

We give a comprehensive survey of the theory of finite dimensional Lie algebras over an algebraically closed field of characteristic p>0 and announce that for p>3 the classification of finite dimensional simple Lie algebras is complete. Any such Lie algebra is up to isomorphism either classical (i.e. comes from characteristic 0) or a filtered Lie algebra of Cartan type or a Melikian algebra of characteristic 5.Comment: Revised version: a list of open problems has been added as suggested by the refere