Researches in Mathematics (Dec 2023)

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

  • D.I. Efimov,
  • M.S. Sydorov,
  • K.Ya. Sysak

DOI
https://doi.org/10.15421/242312
Journal volume & issue
Vol. 31, no. 2
pp. 26 – 34

Abstract

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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 $W_n(K)$.

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