Homomorphisms on infinite direct product algebras, especially Lie algebras
George M. Bergman, Nazih Nahlus
备注：
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, etc
本文研究了交换环k上的未必结合代数的满同态f：\prod_I A_i \to B，其中I是一般无限集，特别是当k是域且B是k上可数维的.当k是无限域，代数是李代数，B是有限维时，我们的结果有如下结果：如果所有的李代数A_i都是可解的，那么B也是可解的.如果所有的李代数A_i都是幂零的，那么B也是幂零的.若k不是特征2或特征3，且所有的李代数A_i都是有限维的且是单代数的直积，则（i）B也是有限维的，（ii）f是分裂的，（iii）在I上的弱基数界下，f在亲离散拓扑中是连续的.在得到（i）-（iii）的过程中用到的一个关键事实是：在任何这样的域上，每个有限维单李代数L都可以写成L=[x_1，L]+[x_2，L]，其中x_1，x_2是\in L，这是我们从J. M. Bois最近的一个结果中证明的.本文的一般技巧是研究\prod_I A_i上的同态必须通过A_i的100多个超积的直积分解的条件。给出了几个例子，并指出开放的问题。
英文摘要：
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.
Triple Derivations and Triple Homomorphisms of Perfect Lie Superalgebras
本文研究了交换环$R$上完全李超代数的三重导子和三重同态。证明了：若基环含有$\frac{1}{2}$，$L$是一个中心为零的完备李超代数，则$L$的每一个三元导子都是导子，导子代数$ Der (L)$的每一个三元导子都是内导子.设$L,~L^{'}$是交换环$R$上的李超代数，引入了从$L$到$L^{'}$的三重同态的概念.证明了在一定的条件下，同态、反同态以及同态与反同态的和都是三重同态。
