Isomorphism invariants of restricted enveloping algebras (Q975815)

Let \(L\) be a restricted Lie algebra over a field \(\mathbb{F}\) of characteristic \(p > 0\), and let \(u(L)\) be the restricted enveloping algebra. An invariant of \(L\) is said to be \textit{determined by \(u(L)\)} if every restricted Lie algebra \(H\) such that \(u(H) \simeq u(L)\) also has this invariant. This fits within the context of the (restricted) isomorphism problem which asks whether the isomorphism class of \(L\) is determined by \(u(L)\), cf. [\textit{D. Riley} and \textit{H. Usefi}, Algebr. Represent. Theory 10, No. 6, 517--532 (2007; Zbl 1192.17005)]. The isomorphism problem was originally stated for integral group rings, see [\textit{R. Sandling}, Lect. Notes Math. 1142, 256--288 (1985; Zbl 0565.20005)] for an overview, or [\textit{M. Hertweck}, Ann. Math. (2) 154, No. 1, 115--138 (2001; Zbl 0990.20002)] for a counterexample in the general case. The main result of the paper under review is as follows. Let \(L\) be a finite-dimensional restricted \(p\)-nilpotent Lie algebra over a perfect field \(\mathbb{F}\). Let \(\gamma_n(L)\) denote the \(n\)th term of the descending central series, and let \(L'{}^p\) be the restricted subalgebra generated by the \(p\)-th powers of elements in \(L'=\gamma_2(L)\). Let \(I = L'{}^p + \gamma_3(L)\). Then the restricted Lie algebra \(L / I\) is determined by \(u(L)\). One of the main tools in the author's analysis is a filtration of \(L\) by suitable ideals, called the \textit{dimension subalgebras}. More precisely, let \(\omega^n(L)\) denote the \(n\)th power of the augmentation ideal of \(u(L)\): the \(n\)th dimension subalgebra is \(D_n(L):=L \cap \omega^n(L)\). The author proves that the associated graded Lie algebra \(gr(L)\) is determined by \(u(L)\). Another interesting result obtained along the way is a partial answer to the restricted isomorphism problem: If \(L\) is an abelian restricted \(p\)-nilpotent Lie algebra, then the isomorphism class of \(L\) is determined by \(u(L)\). If moreover \(\mathbb{F}\) is algebraically closed, then any abelian restricted Lie algebra is determined by its restricted enveloping algebra.