On tame and wild automorphisms of algebras

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

Fundamentalʹnaâ i prikladnaâ matematika, Tome 18 (2013) no. 4, pp. 79-88.

Résumé

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 $k>1$.

Bibliographie

[1] Yu. I. Merzlyakov (red.), Avtomorfizmy klassicheskikh grupp, Sb. perevodov, Mir, M., 1976 | MR

[2] Artamonov V. A., “Nilpotentnost, proektivnost, svoboda”, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 1971, no. 5, 50–53 | MR | Zbl

[3] Bunina E. I., “Avtomorfizmy grupp Shevalle nekotorykh tipov nad lokalnymi koltsami”, Uspekhi mat. nauk, 62:5 (2007), 143–144 | DOI | MR | Zbl

[4] Kourovskaya tetrad (nereshënnye zadachi teorii grupp), NGU, Novosibirsk, 2010

[5] Krouel R., Foks R., Teoriya uzlov, Mir, M., 1967

[6] Levchuk V. M., “Avtomorfizmy unipotentnykh podgrupp grupp lieva tipa malykh rangov”, Algebra i logika, 29:2 (1990), 141–161 | MR | Zbl

[7] Makar-Limanov L. G., “Ob avtomorfizmakh svobodnoi algebry s dvumya obrazuyuschimi”, Funkts. analiz i ego pril., 4:3 (1970), 107–108 | MR | Zbl

[8] Romankov V. A., “Teorema ob obratnoi funktsii dlya svobodnykh assotsiativnykh algebr”, Sib. mat. zhurn., 45:5 (2004), 1178–1183 | MR | Zbl

[9] Umirbaev U. U., “Opredelyayuschie sootnosheniya gruppy ruchnykh avtomorfizmov algebry mnogochlenov i dikie avtomorfizmy svobodnykh assotsiativnykh algebr”, Dokl. RAN, 407:3 (2006), 319–324 | MR

[10] Ushakov Yu. Yu., Avtomorfizmy svobodnykh algebr i funktsii na gruppakh lieva tipa ranga 1, Dis. $\dots$ kand. fiz.-mat. nauk, Krasnoyarsk, 2013

[11] Czerniakiewicz A. J., “Automorphisms of a free associative algebra of rank 2. I”, Trans. Am. Math. Soc., 160 (1971), 393–401 ; “Automorphisms of a free associative algebra of rank 2. II”, Trans. Am. Math. Soc., 171 (1972), 309–315 | MR | Zbl | MR | Zbl

[12] Cohn P. M., Free Rings and Their Relations, Academic Press, London, 1985 | MR | Zbl

[13] Cohn P. M., “Subalgebras of free associative algebras”, Proc. London Math. Soc., 3:14 (1964), 618–632 | DOI | MR | Zbl

[14] Dicks W., Lewin J., “A Jacobian conjecture for free associative algebras”, Commun. Algebra, 10:12 (1982), 1285–1306 | DOI | MR | Zbl

[15] Drensky V., Gupta Ch. K., “Automorphisms of free nilpotent Lie algebras”, Can. J. Math., 42:2 (1990), 259–279 | DOI | MR | Zbl

[16] Dubish R., Perlis S., “On total nilpotent algebras”, Am. J. Math., 73:3 (1951), 439–452 | DOI | MR

[17] Gupta N., Free Group Rings, Contemp. Math., 66, Amer. Math. Soc., Providence, 1987 | DOI | MR | Zbl

[18] Gupta C. K., Levchuk V. M., Ushakov Yu. Yu., “Hypercentral and monic automorphisms of classical algebras, rings and groups”, J. SFU. Maths Phys., 4:1 (2008), 380–390

[19] Hahn A. J., James D. G., Weisfeiler B., “Homomorphisms of algebraic and classical groups: a survey”, Can. Math. Soc. Conf. Proc., 4 (1984), 249–296 | MR | Zbl

[20] Nagata M., On automorphism group of $k[x,y]$, Lectures in Mathematics, No 5, Dept. of Maths of Kyoto Univ., Tokyo, 1972 | MR | Zbl

[21] Schofield A. H., Representation of Rings over Skew Fields, London Math. Soc. Lect. Note Ser., 92, Cambridge Univ. Press, Cambridge, 1985 | MR | Zbl

[22] Shestakov I. P., Umirbaev U. U., “The Nagata automorphism is wild”, Proc. Nat. Acad. Sci. USA, 100:22 (2003), 12561–12563 | DOI | MR | Zbl

[23] Smith M. K., “Stably tame automorphisms”, J. Pure Appl. Algebra, 58 (1989), 209–212 | DOI | MR | Zbl