Размерность Гельфанда - Кириллова относительно свободных ассоциативных алгебр

What is it about?

In the paper the Gelʹfand-Kirillov dimension ${\rm GK}\dim(A)$ of the relatively free associative algebra $A$ is estimated over an arbitrary field. This dimension is defined by the complexity type of the algebra $A$ or by a set of semidirect products of matrix algebras over the ring of polynomials from the variety ${\rm Var}(A)$. At the beginning the author gives the necessary background concerning definitions and the apparatus of representable algebras, traces and forms, Kemmer's diagrams and internal traces, representable spaces and test algebras. He proves the following basic theorem: Let $A$ be an $s$-generated relatively free algebra. Then ${\rm GK}\dim(A)$ depends only on its complexity type, i.e. it is equal to the maximal Gelʹfand-Kirillov dimension of the semidirect products of the algebras of generic matrices from ${\rm Var}(A)$. The Gelʹfand-Kirillov dimension of such a semidirect product is equal to the sum of the Gelʹfand-Kirillov dimension of its factors. The complexity types of the algebra $A$ and its subalgebra with two generators are equal. The method for proving the theorem first reduces the relatively free case to the representable one using the technique of representable spaces, which are extensions via nilpotents of a representable algebra, and then discusses the semidirect products of matrix algebras. The technique connected with representable spaces can be transferred to the case of algebras with common signature. The author discusses the differences in comparison with the associative case and shows how to prove the theorem under consideration in this case as well. Reviewed by Tsetska Grigorova Rashkova

Featured Image