论文标题
伯恩斯坦代数是代数和黑鲁斯问题
Bernstein algebras that are algebraic and the Kurosh problem
论文作者
论文摘要
我们研究了代数的伯恩斯坦代数类别，从某种意义上说，每个元素都会产生有限维度的亚代词。每个Bernstein代数都有最大代数理想，并且商代数为零多层代数。给出了伯恩斯坦代数为代数的几种等效条件。特别是，伯恩斯坦代数的训练训练的已知特征被推广到局部训练代数的情况下。在此过程中，我们表明，如果Banach Bernstein代数为代数（分别为局部训练），那么它分别是有限的（分别是火车）。 然后，我们研究了伯恩斯坦代数的黑鲁斯问题：有限的有限程度代数的伯恩斯坦代数是否是有限维度的。事实证明，这个问题与有关关联代数的问题具有封闭的链接。特别是，当巴尼亚是零的时候，黑鲁什问题询问有限生成的伯恩斯坦培训代数是否是有限的。我们证明，答案对某些特定情况和低度是积极的，并且在一般情况下构建反例。 根据Yagzhev的结果，Jacobian的猜想等同于有关恩格尔和多方代数的nilpotence身份的某些陈述。我们表明，二次映射的雅各布猜想适用于伯恩斯坦代数。
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.
