关于代数簇中的Kurosh问题
On Kurosh problem in varieties of algebras
摘要：
我们考虑了经典Kurosh问题（是否存在无穷维代数？）的几个版本对于域上的线性多算子代数的变种。我们表明，给定一个任意的签名，有各种各样的这个签名的代数，使该品种的自由代数包含多线性元素的任意大的程度，而克隆的每一个这样的元素满足一些非平凡的身份。此外，如果二元运算的数量至少为2，则可以保证每个这样的克隆是有限维的。我们的做法如下：我们将问题转化为操作语言，然后应用通常的同调结构，以采用Golod对原始Kurosh问题的解决方案。这篇论文是简略的，因此省略了一些证明。同时，广泛讨论了运算、代数和簇的一般关系。
英文摘要：
We consider a couple of versions of classical Kurosh problem (whether there is an infinite-dimensional algebraic algebra?) for varieties of linear multioperator algebras over a field. We show that, given an arbitrary signature, there is a variety of algebras of this signature such that the free algebra of the variety contains multilinear elements of arbitrary large degree, while the clone of every such element satisfies some nontrivial identity. If, in addition, the number of binary operations is at least 2, then one can guarantee that each such clone is finitely-dimensional. Our approach is the following: we translate the problem to the language of operads and then apply usual homological constructions, in order to adopt Golod's solution of the original Kurosh problem. The paper is expository, so that some proofs are omited. At the same time, the general relations of operads, algebras, and varieties are widely discussed.
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