On Tame and Wild Automorphisms of Algebras
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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 R k /R k+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 R k for every given integer k > 1.
KeywordsTame Chevalley Group Polynomial Algebra Free Associative Algebra Jacobian Conjecture
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- 1.V. A. Artamonov, “Nilpotency. Projectivity. Freedom,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 5, 50–53 (1971).Google Scholar
- 3.A. G. Chernyakiewicz, “Automorphisms of a free associative algebra of rank 2,” Trans. Am. Math. Soc., 160, 393–401 (1971); 171, 309–315 (1972).Google Scholar
- 6.R. Crowell and R. Fox, Knot Theory [Russian translation], Mir, Moscow (1967).Google Scholar
- 10.N. Gupta, Free Group Rings, Contemp. Math., Vol. 66, Amer. Math. Soc., Providence.Google Scholar
- 11.C. K. Gupta, V. M. Levchuk, and Yu. Yu. Ushakov, “Hypercentral and monic automorphisms of classical algebras, rings and groups,” J. SFU. Phys. & Maths, 4, No. 1, 380–390 (2008).Google Scholar
- 13.Kourovka Notebook (Unsolved Problems of Group Theory), [in Russian], NSU, Novosibirsk (2010).Google Scholar
- 14.V. M. Levchuk, “Automorphisms of unipotent subgroups of Chevalley groups,” Algebra Logika, 29, No. 2, 141–161 (1990); 29, No. 3, 315–338 (1990).Google Scholar
- 16.Yu. I. Merzlyakov, ed., Automorphisms of Classical Groups [in Russian], Mir, Moscow (1976).Google Scholar
- 17.M. Nagata, On Automorphism Group of k[x, y], Lect. Math., No. 5, Dept. of Maths of Kyoto Univ., Tokyo (1972).Google Scholar
- 23.Yu. Yu. Ushakov, Automorphisms of Free Algebras and Functions on Lie Type Groups of Rank 1, Ph.D. Thesis, Sib. Federal Univ., Krasnoyarsk (2013).Google Scholar