FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2013, VOLUME 18, NUMBER 4, PAGES 79-88

On tame and wild automorphisms of algebras

C. K. Gupta
V. M. Levchuk
Yu. Yu. Ushakov

Abstract

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Let $B$_n be a polynomial algebra of $n$ variables over a field $F$. Considering a free associative algebra $A$_n of rank $n$ over $F$ as a polynomial algebra of noncommuting variables, we choose the ideal $R$ of all polynomials with a zero absolute term in $B$_n and $A$_n. The well-known concept of wild automorphisms of the algebras $A$_n and $B$_n is transferred to $R$; the study of wild automorphisms is reduced to monic automorphisms of the algebra $R$, i.e., those identical on each factor $Rk/Rk+1$. In particular, this enables us to study the properties of the known Nagata and Anik automorphisms in detail. For $n\; =\; 3$ we investigate the hypothesis that the Anik automorphism is tame modulo $Rk$ for every given integer $k\; >\; 1$.

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Last modified: May 13, 2014