On maximality of some solvable and locally nilpotent subalgebras of the Lie algebra $$$W_n(K)$$$

D.I. Efimov (Taras Shevchenko National University of Kyiv)
M.S. Sydorov (Taras Shevchenko National University of Kyiv)
K.Ya. Sysak (National University of Life and Environmental Sciences of Ukraine)


Let $$$K$$$ be an algebraically closed field of characteristic zero,  $$$P_n=K[x_1,\ldots ,x_n]$$$  the polynomial ring, and  $$$W_n(K)$$$  the Lie algebra of all $$$K$$$-derivations on $$$P_n$$$.   One of the most important subalgebras of $$$W_n(K)$$$ is the triangular subalgebra $$$u_n(K) = P_0\partial_1+\cdots+P_{n-1}\partial_n$$$, where $$$\partial_i:=\partial/\partial x_i$$$ are partial derivatives on $$$P_n$$$ and $$$P_0=K.$$$ This subalgebra consists of locally nilpotent derivations on $$$P_n.$$$ Such derivations  define automorphisms of the ring $$$P_n$$$ and were studied by many authors. The  subalgebra $$$u_n(K) $$$ is contained in another interesting subalgebra $$$s_n(K)=(P_0+x_1P_0)\partial_1+\cdots +(P_{n-1}+x_nP_{n-1})\partial_n,$$$ which  is solvable of the derived length $$$ 2n$$$ that is the maximum derived length of solvable subalgebras of $$$W_n(K).$$$ It is proved that $$$u_n(K)$$$  is a maximal locally nilpotent subalgebra and $$$s_n(K)$$$ is a maximal solvable subalgebra of the Lie algebra $$$W_n(K)$$$.


Lie algebra; derivation; locally nilpotent; solvable; maximal subalgebra

MSC 2020

17B66; 17B05; 17B40

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Amemiya I., Masuda K., Shiga K. "Lie algebras of differential operators", Osaka J. Math., 1975; 12: pp. 139-172. doi:10.18910/9498

Bavula V.V. "Lie algebras of triangular polynomial derivations and an isomorphism criterion for their Lie factor algebras", Izv. Math., 2013; 77: pp. 1067-1104. doi:10.1070/IM2013v077n06ABEH002670

González-López A., Kamran N., Olver P.J. "Lie algebras of vector fields in the real plane", Proc. London Math. Soc., 1992; 64(2): pp. 339-368. doi:10.1112/plms/s3-64.2.339

Dynkin E.B. "Maximal subgroups of classical groups", Trudy Moskovskogo Matem. obshchestva, 1952; 1 (Russian); English transl. in: Moscow Math. Soc. Translations Ser. 2, v.6: pp. 245-378.

Lie S. Theorie der Transformationsgruppen, Bd. 3. Teubner, Leipzig, 1893. (in German)

Makedonskyi Ie.O., Petravchuk A.P. "On nilpotent and solvable Lie algebras of derivations", J. Algebra, 2014; 401: pp. 245-257. doi:10.1016/j.jalgebra.2013.11.021

Martello M., Ribon J. "Derived length of solvable groups of local diffeomorphisms", Math. Ann., 2014; 358: pp. 701-728. doi:10.1007/s00208-013-0975-5

Skutin A. "Maximal Lie algebras among locally nilpotent derivations", Mat. Sb., 2021; 212(2): pp. 138-146. doi:10.1070/SM9360

DOI: https://doi.org/10.15421/242312



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