Non-matrix polynomial identity enveloping algebras (Q392157)

A variety of associative algebras over a field \(\mathbb{F}\) is called non-matrix if it does not contain the algebra \(\mathrm{Mat}(2,\mathbb{F})\). A polynomial identity is non-matrix if \(\mathrm{Mat}(2,\mathbb{F})\) does not satisfy it.NEWLINENEWLINEEnveloping algebras satisfying polynomial identities were first constructed in [\textit{V.~N. Latyshev,} Sib. Mat. Zh. 4, 1120--1121 (1963; Zbl 0128.25804)]. In [\textit{D. S. Passman}, J. Algebra 134, No. 2, 469--490 (1990; Zbl 0713.16013)]; [\textit{V. M. Petrogradskiĭ}, Math. Notes 49, No. 1, 60--66 (1991); translation from Mat. Zametki 49, No. 1, 84--93 (1991; Zbl 0721.17011)] the analogous considerations were done for restricted Lie algebras and their envelopes.NEWLINENEWLINEThe main purpose of the paper under review is to characterize restricted Lie superalgebras whose enveloping algebras satisfy non-matrix polynomial identities. In the paper the field is suggested to have characteristic \(p>2\).NEWLINENEWLINENEWLINEThe main result of the paper under review is the following theorem. Let \(L=L_1\oplus L_2\) be a restricted Lie superalgebra over a perfect field with the bracket \((.,.)\), denote by \(M\) the subspace spanned by all \(y\in L_1\) such that \((y,y)\) is \(p\)-nilpotent. A subset \(X\subset L_0\) is \(p\)-nilpotent if there exist an integer \(s\) such that \(x^{p^s}=0\) for every \(x\in X\).NEWLINENEWLINEThe following statements are equivalent:NEWLINENEWLINE1. The restricted enveloping algebra \(u(L)\) satisfies a non-matrix polynomial identity.NEWLINENEWLINE 2. \(u(L)\) satisfies a polynomial identity, \((L_0,L_0)\) is \(p\)-nilpotent, \(\dim L_1/M\leq 1\), \((M,L_1)\) is \(p\)-nilpotent and \((L_1,L_0)\subset M\).NEWLINENEWLINE 3. The commutator ideal of \(u(L)\) is nil of bounded index.