Tame and wild automorphisms of differential polynomial algebras of rank~$2$

B. A. Duisengaliyeva ; A. S. Naurazbekova ; U. U. Umirbaev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 22 (2019) no. 4, pp. 101-114.

Résumé

It is proved that the tame automorphism group of a differential polynomial algebra $k\{x,y\}$ over a field $k$ of characteristic $0$ in two variables $x$, $y$ with $m$ commuting derivations $\delta_1, \ldots, \delta_m$ is a free product with amalgamation. An example of a wild automorphism of the algebra $k\{x,y\}$ in the case of $m\geq 2$ derivations is constructed.

Bibliographie

[1] Kurosh A. G., “Neassotsiativnye algebry i svobodnye proizvedeniya algebr”, Matem. sb., 20 (1947), 239–262 | MR | Zbl

[2] Magnus V., Karras A., Soliter D., Kombinatornaya teoriya grupp, Nauka, M., 1974

[3] Makar-Limanov L., “Avtomorfizmy svobodnoi algebry ot dvukh porozhdayuschikh”, Funkts. analiz i ego pril., 4 (1970), 107–108 | Zbl

[4] Mikhalev A. A., “Podalgebry svobodnykh tsvetnykh superalgebr Li”, Matem. zametki, 37:5 (1985), 653–661 | MR | Zbl

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

[6] Umirbaev U. U., Shestakov I. P., “Podalgebry i avtomorfizmy kolets mnogochlenov”, Dokl. RAN, 386:6 (2002), 745–748 | MR | Zbl

[7] Shirshov A. I., “Podalgebry svobodnykh lievykh algebr”, Matem. sb., 33:2 (1953), 441–452 | Zbl

[8] Shirshov A. I., “Podalgebry svobodnykh kommutativnykh i svobodnykh antikommutativnykh algebr”, Matem. sb., 34:1 (1954), 81–88 | Zbl

[9] Shirshov A. I., Koltsa i algebry, Nauka, M., 1984 | MR

[10] Shtern A. S., “Svobodnye superalgebry Li”, Sib. matem. zhurn., 27 (1986), 170–174 | MR | Zbl

[11] Cohn P. M., “Subalgebras of free associative algebras”, Proc. London Math. Soc., 56 (1964), 618–632 | MR

[12] Cohn P. M., Free Ideal Rings and Localization in General Rings, New Math. Monogr., 3, Cambridge Univ. Press, Cambridge, 2006 | MR | Zbl

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

[14] Jung H. W. E., “Über ganze birationale Transformationen der Ebene”, J. Reine Angew. Math., 184 (1942), 161–174 | MR

[15] Kolchin E. R., Differential Algebra and Algebraic Groups, Pure Appl. Math., 54, Academic Press, New York, 1973 | MR | Zbl

[16] Kondratieva M. V., Levin A. B., Mikhalev A. V., Pankratiev E. V., Differential and Difference Dimension Polynomials, Math. Its Appl., 461, Kluwer Academic, Dordrecht, 1999 | MR | Zbl

[17] Kozybaev D., Makar-Limanov L., Umirbaev U., “The Freiheitssatz and the automorphisms of free right-symmetric algebras”, Asian-Eur. J. Math., 1 (2008), 243–254 | MR | Zbl

[18] Van der Kulk W., “On polynomial rings in two variables”, Nieuw Arch. Wisk., 3:1 (1953), 33–41 | MR

[19] Lewin J., “On Schreier varieties of linear algebras”, Trans. Am. Math. Soc., 132 (1968), 553–562 | MR | Zbl

[20] Makar-Limanov L., Turusbekova U., Umirbaev U. U., “Automorphisms and derivations of free Poisson algebras in two variables”, J. Algebra, 322:9 (2009), 3318–3330 | MR | Zbl

[21] Nagata M., On Automorphism Group of $k[x,y]$, Lect. Math., Kyoto Univ., 5, Kinokuniya, Tokyo, 1972 | MR | Zbl

[22] Ritt J. F., Differential Algebra, Dover, New York, 1966 | MR

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

[24] Shestakov I. P., Umirbaev U. U., “Tame and wild automorphisms of rings of polynomials in three variables”, J. Amer. Math. Soc., 17 (2004), 197–227 | MR | Zbl

[25] Umirbaev U. U., “The Anick automorphism of free associative algebras”, J. Reine Angew. Math., 605 (2007), 165–178 | MR | Zbl

[26] Witt E., “Die Unterringe der freien Lieschen Ringe”, Math. Z., 64 (1956), 195–216 | MR | Zbl