Burnside-type problems, theorems on height, and independence

What is it about?

The paper under review is devoted to some questions related to investigations of bases in PI-algebras. The central point is a generalization and refinement of the Shirshov height theorem, the Amitsur-Shestakov hypothesis and the independence theorem. The paper is mainly inspired by the fact that these topics shed some light on the analogy between structure theory and constructive combinatorial reasoning related to the "microlevel'', to relations in algebras and straightforward calculations. Together with the representation theory of monomial algebras, height and independence theorems are closely connected with combinatorics of words and of normal forms, as well as with properties of primary algebras and with combinatorics of matrix units. The first chapter is devoted to representations of primary automata monomial algebras and focuses on the exactness of semigroups' representations without monomial kernels. Algorithmic problems are stated concerning the minimal dimension of a representation—the process of covering the set of nonzero words of a monomial algebra $A$ by sets of nonzero words of $A_i$ for $A=\bigoplus A_i$. This problem seems to be NP-complete and connected with finding bounded solutions of systems of exponential differential equations. By the theorem of independence for representations of automata algebras, the author shows the effect of matrix units properties on word combinatorics. The structure scheme given by I. Herstein (division rings $\rightarrow$ primitive rings $\rightarrow$ semiprime rings $\rightarrow$ radical) has a combinatorial interpretation in the theorem of the bounded height of words with bounded degrees; i.e., if $|u|$ is a noncyclic word with length $> {\rm PI}\,{\rm deg}(A)$ for $k$ too big, then the word $u^k$ is a linear combination of lexicographically smaller words. Combinatorial results are also given concerning infinite words in both directions. It is shown that the Tikhonov topology on them and the periodicity of the infinite words are the base of the combinatorial view on many theorems . If a word $t=t_1\cdots t_n$ is $n$-divisible, $t_1\succ \dots \succ t_p$, for any nontrivial permutation $\sigma \in S_p$ the word $t_\sigma=t_{\sigma(1)}\cdots t_{\sigma(n)}$ is lexicographically smaller than $t$. Thus if the algebra satisfies a polynomial identity of degree $p$ then the word $t$ is linear representable by smaller words. A non-diminishing word is not $n$-divisible. As all non-$n$-divisible words of enough length contain a degree of a subword the Shirshov height theorem leads to the positive solution of Burnside-type problems such as the local finiteness of algebraic PI-algebras (for example, the local nilpotency of nil algebras with bounded index) and the bounded Gelʹfand-Kirillov dimension. The paper proposes a very promising parallelism between structural and combinatorial investigations in PI-theory. Reviewed by Tsetska Grigorova Rashkova

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