Let 𝕂 be an algebraically closed field of characteristic zero, let 𝕂[x1,…,xn] be the polynomial algebra, and let Wn(𝕂) be the Lie algebra of all 𝕂-derivations on 𝕂[x1,…,xn]. For any derivation D with linear components, we describe the centralizer of D in Wn(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 Weitzenböck derivation. In a more general case where a finitely generated integral domain A over the field 𝕂 is considered instead of the polynomial algebra 𝕂[x1,…,xn] and D is a locally nilpotent derivation on A, we prove that the centralizer CDerA(D) of D in the Lie algebra DerA of all 𝕂-derivations on A is a “large” subalgebra of Der A. Specifically, the rank of CDerA(D) over A is equal to the transcendence degree of the field of fractions Frac(A) over the field 𝕂.
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References
L. P. Bedratyuk, “Kernels of derivations of polynomial rings and Casimir elements,” Ukr. Mat. Zh., 62, No. 4, 435–452 (2010); English translation: Ukr. Math. J., 62, No. 4, 495–517 (2010).
Y. Chapovskyi, D. Efimov, and A. Petravchuk, “Centralizers of elements in Lie algebras of vector fields with polynomial coefficients,” Proc. Int. Geom. Cent., 14, No. 4, 257–270 (2021).
D. R. Finston and S. Walcher, “Centralizers of locally nilpotent derivations,” J. Pure Appl. Algebra, 120, No. 1, 39–49 (1997).
G. Freudenburg, Algebraic Theory of Locally Nilpotent Derivations, 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, Springer, New York (1972).
M. Miyanishi, “Normal affine subalgebras of a polynomial ring,” in: Algebraic and Topological Theories (Kinosaki, 1984), Kinokuniya, Tokyo (1986), pp. 37–51.
J. Nagloo, A. Ovchinnikov, and P. Thompson, “Commuting planar polynomial vector fields for conservative Newton systems,” Comm. Contemp. Math., 22, No. 4, Article 1950025 (2020).
A. Nowicki and M. Nagata, “Rings of constants for k-derivations in k[x1,…,xn],” J. Math. Kyoto Univ., 28, No. 1, 111–118 (1988).
D. I. Panyushev, “Two results on centralisers of nilpotent elements,” J. Pure Appl. Algebra, 212, No. 4, 774–779 (2008).
A. P. Petravchuk and O. G. Iena, “On centralizers of elements in the Lie algebra of the special Cremona group SA2(k),” J. Lie Theory, 16, No. 3, 561–567 (2006).
R. Weitzenböck, “Über die Invarianten von linearen Gruppen,” Acta Math., 58, No. 1, 231–293 (1932).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, No. 8, pp. 1043–1052, August, 2023. Ukrainian DOI: https://doi.org/10.37863/umzh.v75i8.7529.
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Bedratyuk, L., Petravchuk, A. & Chapovskyi, E. Centralizers of Linear and Locally Nilpotent Derivations. Ukr Math J 75, 1190–1202 (2024). https://doi.org/10.1007/s11253-023-02255-x
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DOI: https://doi.org/10.1007/s11253-023-02255-x