Non-matrix polynomial identity enveloping algebras. (English) Zbl 1287.17038
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.
Enveloping algebras satisfying polynomial identities were first constructed in [V. N. Latyshev, Sib. Mat. Zh. 4, 1120–1121 (1963; Zbl 0128.25804)]. In [D. S. Passman, J. Algebra 134, No. 2, 469–490 (1990; Zbl 0713.16013)]; [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.
The 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 \(p>2\).
The 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 \(x\in X\).
The following statements are equivalent:
1. The restricted enveloping algebra \(u(L)\) satisfies a non-matrix polynomial identity.
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 \((L_1,L_0)\subset M\).
3. The commutator ideal of \(u(L)\) is nil of bounded index.
Enveloping algebras satisfying polynomial identities were first constructed in [V. N. Latyshev, Sib. Mat. Zh. 4, 1120–1121 (1963; Zbl 0128.25804)]. In [D. S. Passman, J. Algebra 134, No. 2, 469–490 (1990; Zbl 0713.16013)]; [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.
The 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 \(p>2\).
The 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 \(x\in X\).
The following statements are equivalent:
1. The restricted enveloping algebra \(u(L)\) satisfies a non-matrix polynomial identity.
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 \((L_1,L_0)\subset M\).
3. The commutator ideal of \(u(L)\) is nil of bounded index.
Reviewer: Dmitri Artamonov (Moscow)
MSC:
| 17B50 | Modular Lie (super)algebras |
| 17B35 | Universal enveloping (super)algebras |
| 16R10 | \(T\)-ideals, identities, varieties of associative rings and algebras |
| 16R40 | Identities other than those of matrices over commutative rings |
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