Estimates related to Shirshov height theorem (PhD Thesis)
Mikhail Kharitonov

TL;DR
This thesis proves that the nilpotency degree of certain associative algebras with identities is bounded by a function related to word combinatorics, answering Zelmanov's question about exponential growth.
Contribution
It provides a definitive bound on the nilpotency degree for algebras with identities, based on combinatorial properties of words, improving previous recursive and exponential estimates.
Findings
Nilpotency degree is smaller than a specific function Ψ(d,d,l).
Words longer than Ψ(n,d,l) are either n-divided or contain d-th powers.
Bound on the height of non n-divided words is less than Φ(n,l).
Abstract
In 1993 E. I. Zelmanov asked the following question in Dniester Notebook: Suppose that is a -generated associative ring with the identity . Is it true, that the nilpotency degree of has exponential growth? We show that the nilpotency degree of -generated associative algebra with the identity is smaller than , where and is a constant. We give the definitive answer to E. I. Zelmanov by this result. It is the consequence of one fact, which is based on combinatorics of words. Let , and be positive integers. Then all the words over alphabet of cardinality which length is greater than are either -divided or contain -th power of subword, where a word is -divided, if it can be represented in the following form such that $W_n\succ…
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Taxonomy
TopicsFinite Group Theory Research · Pediatric Hepatobiliary Diseases and Treatments · Connective tissue disorders research
