On the Lie algebra of skew-symmetric elements of an enveloping algebra
Let \(A\) be an algebra with an anti-involution, and let \(A^+\) (resp. \(A^-\)) denote the set of symmetric (resp. skew-symmetric) elements of \(A\). A classical question is, which properties of \(A\) can be retrieved from properties of \(A^\pm\), see for example [\textit{S. A. Amitsur}, Isr. J. Math. 7, 63--68 (1969; Zbl 0179.33701)] for general rings with involutions, or [\textit{A. Giambruno} and \textit{S. K. Sehgal}, Commun. Algebra 21, No.11, 4253--4261 (1993; Zbl 0797.16036)] for group rings. The paper under review deals with the following situation. Let \(L\) be a restricted Lie algebra over a field of characteristic \(p > 2\), and \(u(L)\) be the restricted enveloping algebra. The space \(u(L)^-\) is a Lie subalgebra of \(u(L)\) for the commutator bracket. The author investigates which properties of \(L\) can be read on the Lie structure of \(u(L)^-\). This extends previous results by \textit{D. M. Riley} and \textit{A. Shalev} relating the Lie structure of \(L\) and \(u(L)\) [J. Algebra 162, No. 1, 46--61 (1993; Zbl 0798.17011)]. A sample result is the following. Theorem 1: For a restricted Lie algebra \(L\) in characteristic \(p > 2\), the following conditions are equivalent: (1) \(u(L)^-\) is solvable; (2) \(u(L)\) is solvable; (3) \([L,L]\) is finite-dimensional and \(p\)-nilpotent. Similar characterisations hold for the \(n\)-Engel property, and for nilpotency. Examples are given to show the results do not extend to the case when the ground field has characteristic \(p = 2\).
scientific article; zbMATH DE number 5802620
scientific article