Locally Nilpotent Derivations of Free Algebra of Rank Two

Vesselin Drensky   ; Leonid Makar-Limanov

Symmetry, integrability and geometry: methods and applications, Tome 15 (2019)

Résumé

In commutative algebra, if $\delta$ is a locally nilpotent derivation of the polynomial algebra $K[x_1,\ldots,x_d]$ over a field $K$ of characteristic 0 and $w$ is a nonzero element of the kernel of $\delta$, then $\Delta=w\delta$ is also a locally nilpotent derivation with the same kernel as $\delta$. In this paper we prove that the locally nilpotent derivation $\Delta$ of the free associative algebra $K\langle X,Y\rangle$ is determined up to a multiplicative constant by its kernel. We show also that the kernel of $\Delta$ is a free associative algebra and give an explicit set of its free generators.

Détail

Keywords: free associative algebras, locally nilpotent derivations
Mots-clés : algebras of constants.

Bibliographie

[1] Bass H., “A nontriangular action of ${\mathbb G}_{a}$ on ${\mathbb A}^{3}$”, J. Pure Appl. Algebra, 33 (1984), 1–5 | DOI | MR | Zbl

[2] Cohn P. M., Free rings and their relations, London Mathematical Society Monographs, 19, 2nd ed., Academic Press, Inc., London, 1985 | MR | Zbl

[3] Czerniakiewicz A. J., “Automorphisms of a free associative algebra of rank $2$. I”, Trans. Amer. Math. Soc., 160 (1971), 393–401 | DOI | MR | Zbl

[4] Czerniakiewicz A. J., “Automorphisms of a free associative algebra of rank $2$. II”, Trans. Amer. Math. Soc., 171 (1972), 309–315 | DOI | MR | Zbl

[5] de W. Jooste T., “Primitive derivations in free associative algebras”, Math. Z., 164 (1978), 15–23 | DOI | MR | Zbl

[6] Drensky V., Free algebras and PI-algebras. Graduate course in algebra, Springer-Verlag, Singapore, 2000 | MR | Zbl

[7] Drensky V., Gupta C. K., “Constants of Weitzenböck derivations and invariants of unipotent transformations acting on relatively free algebras”, J. Algebra, 292 (2005), 393–428, arXiv: math.RA/0412399 | DOI | MR | Zbl

[8] Falk G., “Konstanzelemente in Ringen mit Differentiation”, Math. Ann., 124 (1952), 182–186 | DOI | MR | Zbl

[9] Freudenburg G., “A survey of counterexamples to Hilbert's fourteenth problem”, Serdica Math. J., 27 (2001), 171–192 | MR | Zbl

[10] Freudenburg G., Algebraic theory of locally nilpotent derivations, Encyclopaedia of Mathematical Sciences, 136, Springer-Verlag, Berlin, 2006 | DOI | MR | Zbl

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

[12] Algebra Logic, 17 (1978), 316–321 | DOI | MR

[13] Lane D. R., Free algebras of rank two and their automorphisms, Ph.D. Thesis, Bedford College, London, 1976

[14] Funct. Anal. Appl., 4 (1970), 262–264 | DOI | MR | Zbl

[15] Makar-Limanov L., Locally nilpotent derivations, a new ring invariant and applications, Lecture notes, http://www.math.wayne.edu/l̃ml/lmlnotes.pdf

[16] Makar-Limanov L., Turusbekova U., Umirbaev U., “Automorphisms and derivations of free Poisson algebras in two variables”, J. Algebra, 322 (2009), 3318–3330, arXiv: 0708.1148 | DOI | MR | Zbl

[17] Nagata M., On automorphism group of $k[x,y]$, Lectures in Math., 5, Kyoto University, Kinokuniya Book-Store Co., Ltd., Tokyo, 1972 | MR | Zbl

[18] Nowicki A., Polynomial derivations and their rings of constants, Uniwersytet Mikołaja Kopernika, Toruń, 1994 | MR | Zbl

[19] Nowicki A., “The fourteenth problem of Hilbert for polynomial derivations”, Differential Galois Theory (Bȩdlewo, 2001), Banach Center Publ., 58, Polish Acad. Sci. Inst. Math., Warsaw, 2002, 177–188 | DOI | MR | Zbl

[20] Rentschler R., “Opérations du groupe additif sur le plan affine”, C. R. Acad. Sci. Paris Sér. A-B, 267 (1968), 384–387 | MR | Zbl

[21] Arbres, amalgames, ${\rm SL}_2$, Astérisque, 46, Société Mathématique de France, Paris, 1977 | MR | MR | Zbl

[22] Seshadri C. S., “On a theorem of Weitzenböck in invariant theory”, J. Math. Kyoto Univ., 1 (1962), 403–409 | DOI | MR | Zbl

[23] Shestakov I. P., Umirbaev U. U., “Poisson brackets and two-generated subalgebras of rings of polynomials”, J. Amer. Math. Soc., 17 (2004), 181–196 | DOI | MR | Zbl

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

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

[26] Specht W., “Gesetze in Ringen. I”, Math. Z., 52 (1950), 557–589 | DOI | MR | Zbl

[27] Tyc A., “An elementary proof of the Weitzenböck theorem”, Colloq. Math., 78 (1998), 123–132 | DOI | MR | Zbl

[28] Umirbaev U. U., “The Anick automorphism of free associative algebras”, J. Reine Angew. Math., 605 (2007), 165–178, arXiv: math.RA/0607029 | DOI | MR | Zbl

[29] van den Essen A., Polynomial automorphisms and the Jacobian conjecture, Progress in Mathematics, 190, Birkhäuser Verlag, Basel, 2000 | DOI | MR | Zbl

[30] van der Kulk W., “On polynomial rings in two variables”, Nieuw Arch. Wiskunde, 1 (1953), 33–41 | MR | Zbl

[31] Weitzenböck R., “Über die Invarianten von linearen Gruppen”, Acta Math., 58 (1932), 231–293 | DOI | MR | Zbl