Symmetric identities and restricted Lie algebras (Q5956896)

Let \({\mathfrak g}\) be a finite-dimensional restricted Lie algebra over an algebraically closed field \(\mathbb{F}\) of prime characteristic \(p\). Then for every linear form \(\chi\) on \({\mathfrak g}\) there is a canonical map \(\phi_\chi\) from the \(\chi\)-reduced centre of \({\mathfrak g}\) into the \(\chi\)-reduced universal enveloping algebra of \({\mathfrak g}\). The main result of this paper is a sufficient condition for the injectivity of \(\phi_\chi\) involving the maximal possible dimension of a simple \({\mathfrak g}\)-module. The injectivity of \(\phi_\chi\) for semisimple \({\mathfrak g}\) of classical type and regular \(\chi\) was already proved by \textit{I. Mirković} and the author [Math. Z. 231, 123-132 (1999; Zbl 0932.17020)]. Recently, A. Premet extended this result under mild restrictions on \(p\) to all \(\chi\). In the paper under review no such structural assumptions are made, and therefore the result is applicable to arbitrary finite-dimensional restricted Lie algebras, as is illustrated by the author for solvable Lie algebras, the Witt algebra, and semisimple Lie algebras of classical type.