Centralizers of linear and locally nilpotent derivations

  • L. Bedratyuk Khmelnitsky University
  • A. Petravchuk Kyiv National University named after Taras Shevchenko
  • Ie. Chapovskyi Kyiv National University named after Taras Shevchenko
Keywords: Lie algebra, locally nilpotent differentiation, basic Weitzenbeck differentiation, centralizer, kernel of differentiation

Abstract

UDC 512.715, 512.554.31

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero, $\mathbb{K}[x_1,\dots,x_n]$ be the polynomial algebra and $W_n(\mathbb{K})$ be the Lie algebra of all $\mathbb K$-derivations on $\mathbb{K}[x_1,\dots,x_n].$ For any derivation $D$ with linear components,  we describe the centralizer of $D$ in $W_n(\mathbb{K}),$ and  propose an algorithm for finding the generators of this centralizer regarded as a module over the ring of constants of the derivation $D$ in the case where $D$ is a basic Weitzenboeck derivation. In a more general case where a finitely generated integral domain $A$ over the field $\mathbb{K}$ is considered instead of the polynomial algebra $\mathbb{K}[x_1,\dots,x_n]$ and $D$ is a locally nilpotent derivation on $A,$ we prove that the centralizer ${\rm C}_{{\rm Der} A}(D)$ of $D$ in the Lie algebra ${\rm Der} A$ of all $\mathbb K$-differentiations on $A$ is a ``large'' subalgebra of ${\rm Der} A.$ Specifically, the rank of ${\rm C}_{{\rm Der} _A}(D)$ over $A$ is equal to the transcendence degree of the field of fractions $\mathrm{Frac}(A)$ over the field~$\mathbb K.$

References

L. P. Bedratyuk, Kernels of derivations of polynomial rings and Casimir elements, Ukr. Math. J., 62, № 4, 495–517 (2010). DOI: https://doi.org/10.1007/s11253-010-0367-x

Y. Chapovskyi, D. Efimov, A. Petravchuk, Centralizers of elements in Lie algebras of vector fields with polynomial coefficients, Proc. Int. Geom. Cent., 14, № 4, 257–270 (2021). DOI: https://doi.org/10.15673/tmgc.v14i4.2153

D. R. Finston, S. Walcher, Centralizers of locally nilpotent derivations, J. Pure and Appl. Algebra, 120, № 1, 39–49 (1997). DOI: https://doi.org/10.1016/S0022-4049(96)00064-3

G. Freudenburg, Algebraic theory of locally nilpotent derivations, Encyclopedia Math. Sci., vol. 136, Springer, Berlin (2006).

F. R. Gantmacher, The theory of matrices, vols. 1, 2, Chelsea Publ. Co., New York (1959).

J. E. Humphreys, Introduction to Lie algebras and representation theory, Grad. Texts in Math., vol. 9, Springer, New York (1972). DOI: https://doi.org/10.1007/978-1-4612-6398-2

M. Miyanishi, Normal affine subalgebras of a polynomial ring, Algebraic and Topological Theories (Kinosaki, 1984), Kinokuniya, Tokyo (1986), p.~37–51.

J. Nagloo, A. Ovchinnikov, P. Thompson, Commuting planar polynomial vector fields for conservative Newton systems, Commun. Contemp. Math., 22, № 4, Article~1950025 (2020). DOI: https://doi.org/10.1142/S0219199719500251

A. Nowicki, M. Nagata, Rings of constants for $k$-derivations in $k[x_1,... , x_n]$, J. Math. Kyoto Univ., 28, № 1, 111–118 (1988). DOI: https://doi.org/10.1215/kjm/1250520561

D. I. Panyushev, Two results on centralisers of nilpotent elements, J. Pure and Appl. Algebra, 212, № 4, 774–779 (2008). DOI: https://doi.org/10.1016/j.jpaa.2007.07.003

A. P. Petravchuk, O. G. Iena, On centralizers of elements in the Lie algebra of the special Cremona group $SA_2(k)$, J. Lie Theory, 16, № 3, 561–567 (2006).

R. Weitzenböck, Über die Invarianten von linearen Gruppen, Acta Math., 58, № 1, 231–293 (1932). DOI: https://doi.org/10.1007/BF02547779

Published
30.08.2023
How to Cite
BedratyukL., PetravchukA., and ChapovskyiI. “Centralizers of Linear and Locally Nilpotent Derivations”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, no. 8, Aug. 2023, pp. 1043 -52, doi:10.3842/umzh.v75i8.7529.
Section
Research articles