代数的Bernstein代数和Kurosh问题
Bernstein algebras that are algebraic and the Kurosh problem
摘要：
我们研究一类伯恩斯坦代数是代数的，在这个意义上说，每个元素产生一个有限维子代数。每个Bernstein代数都有一个极大代数理想，并且其商代数是零乘代数.给出了Bernstein代数是代数的几个等价条件。特别地，将训练Bernstein代数的幂零性刻画推广到局部训练代数.一路上，我们表明，如果一个Banach伯恩斯坦代数是代数（分别，本地训练），那么它是有界度（分别，训练）。然后研究了Bernstein代数的Kurosh问题：一个有界次代数的生成Bernstein代数是否是有限维的。这个问题与一个关于结合代数的问题有着密切的联系。特别地，当barideal为nil时，Kurosh问题会问一个生成的Bernstein-train代数是否是有限维的。我们证明了答案是积极的一些特定的情况下，低度，并构造反例在一般情况下。根据Yagzhev的结果，Jacobian猜想等价于关于多算子代数的Engel恒等式和nilerian恒等式的一个陈述.证明了二次映射的广义雅可比矩阵猜想对Bernstein代数成立。
英文摘要：
We study the class of Bernstein algebras that are algebraic, in the sense that each element generates a finite-dimensional subalgebra. Every Bernstein algebra has a maximal algebraic ideal, and the quotient algebra is a zero-multiplication algebra. Several equivalent conditions for a Bernstein algebra to be algebraic are given. In particular, known characterizations of train Bernstein algebras in terms of nilpotency are generalized to the case of locally train algebras. Along the way, we show that if a Banach Bernstein algebra is algebraic (respectively, locally train), then it is of bounded degree (respectively, train). Then we investigate the Kurosh problem for Bernstein algebras: whether a finitely generated Bernstein algebra which is algebraic of bounded degree is finite-dimensional. This problem turns out to have a closed link with a question about associative algebras. In particular, when the barideal is nil, the Kurosh problem asks whether a finitely generated Bernstein-train algebra is finite-dimensional. We prove that the answer is positive for some specific cases and for low degrees, and construct counter-examples in the general case. By results of Yagzhev, the Jacobian conjecture is equivalent to a certain statement about Engel and nilpotence identities of multioperator algebras. We show that the generalized Jacobian conjecture for quadratic mappings holds for Bernstein algebras.
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