Enveloping algebras satisfying a polynomial identity. (English) Zbl 0713.16013
The main result is another approach to the proof of a theorem of V. M. Petrogradskij giving necessary and sufficient conditions on a restricted Lie algebra under which its restricted enveloping algebra satisfies a nontrivial polynomial identity [V. M. Petrogradskij, On restricted enveloping algebra for a Lie-p-algebra. Fifth Siberian workshop on varieties of algebraic systems, Barnaul, 52-55 (1988; Zbl 0685.00002)]. The main tool is the use of \(\Delta\)-methods of J. Bergen and D. S. Passman. A considerably simpler proof is given to a theorem of the reviewer [Usp. Mat. Nauk 27, No.4, 201-202 (1972; Zbl 0276.17003)]. All these results are analogous to certain results on group rings [the author, Pac. J. Math. 36, 457-483 (1971; Zbl 0195.045); M. K. Smith, J. Algebra 18, 477-499 (1971; Zbl 0219.20002)].
Reviewer: Yu.A.Bakhturin
MSC:
| 16S30 | Universal enveloping algebras of Lie algebras |
| 16R10 | \(T\)-ideals, identities, varieties of associative rings and algebras |
| 17B50 | Modular Lie (super)algebras |
| 17B35 | Universal enveloping (super)algebras |
Keywords:
universal enveloping algebras; restricted Lie algebra; restricted enveloping algebra; polynomial identityReferences:
| [1] | Bachturin, Yu. A., Identities in the universal envelopes of Lie algebras, J. Austral. Math. Soc., 18, 10-21 (1974) · Zbl 0298.17013 |
| [3] | Berkson, A. J., The \(u\)-algebra of a restricted Lie algebra is Frobenius, (Proc. Amer. Math. Soc., 15 (1964)), 14-15 · Zbl 0119.27401 |
| [4] | Latyšev, V. N., Two remarks on PI algebras (in Russian), Sibirsk. Mat. Zh., 4, 1120-1121 (1963) · Zbl 0128.25804 |
| [5] | Lichtman, A. I., PI-subrings and algebraic elements in enveloping algebras and their fields of fractions, J. Algebra, 121, 139-154 (1989) · Zbl 0672.17009 |
| [6] | Passman, D. S., Linear identities in group rings, Pacific J. Math., 36, 457-483 (1971) · Zbl 0211.34702 |
| [7] | Passman, D. S., Group rings satisfying a polynomial identity, J. Algebra, 20, 103-117 (1972) · Zbl 0226.16015 |
| [8] | Rowen, L. H., Polynomial Identities in Ring Theory (1980), Academic Press: Academic Press New York · Zbl 0461.16001 |
| [9] | Smith, M. K., Group algebras, J. Algebra, 18, 477-499 (1971) · Zbl 0219.20002 |
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