ON A SHIRSHOV BASIS OF RELATIVELY FREE ALGEBRAS OF COMPLEXITY $ n$

What is it about?

he author proves the following results: (i) Let $A$ be a finitely generated associative PI-algebra of complexity $n$. Then $A$ is an algebra of bounded height over the set of monomials of length $\leq n$. In particular, if $A$ satisfies a multilinear identity of degree $m$ then $A$ is an algebra of bounded height over the set of monomials of length $\leq[m/2]$. (ii) Let $B$ be an alternative or Jordan finitely generated PI-algebra of degree $m$. Then $B$ is an algebra of bounded height over the set of monomials of length $<m^2$. (iii) Let $B$ be a universal PI-alternative finitely generated algebra. Then the set of Shirshov bases of $B$ is equal to the set of Shirshov bases of $B/(B,B,B)$, where $(B,B,B)$ is an associator ideal of $B$. Reviewed by Yu. N. Malʹtsev

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