Identities of Lie algebras with nilpotent commutator ideal over a field of finite characteristic. (English. Russian original) Zbl 0791.17006
Math. Notes 51, No. 3, 255-258 (1992); translation from Mat. Zametki 51, No. 3, 47-52 (1992).
All algebras considered are over a field \(K\) with char \(K=p \neq 0\). There exist examples of Lie algebras having no finite bases of identities [see V. Drenski, Algebra Logika 13, 265-290 (1974; Zbl 0298.17011)]. The commutator ideals of these examples turned out to be \(p\)-nilpotent. The author of the note under review has shown that if the commutator ideal of a Lie algebra is \((p-1)\)-nilpotent then this Lie algebra has a finite basis of identities. Note that this implies that each Lie algebra of triangular matrices \(n \times n\), \(n \leq p\), has a finite basis of identities.
Reviewer: P.Koshlukov (Sofia)
MSC:
| 17B01 | Identities, free Lie (super)algebras |
| 17B30 | Solvable, nilpotent (super)algebras |
| 16R10 | \(T\)-ideals, identities, varieties of associative rings and algebras |
| 16R99 | Rings with polynomial identity |
Citations:
Zbl 0298.17011References:
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