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Prime Lie algebras satisfying the standard Lie identity of degree 5. (English) Zbl 1406.17031

Summary: For every commutative differential algebra one can define the Lie algebra of special derivations. It is known for years that not every Lie algebra can be embedded into the Lie algebra of special derivations of some differential algebra. More precisely, the Lie algebra of special derivations of a commutative algebra always satisfies the standard Lie identity of degree 5. The problem of existence of such embedding is a long-standing problem, which is closely related to the Lie algebra of vector fields on the affine line. It was solved by Yu. P. Razmyslov in [Math. USSR, Izv. 26, 553–590 (1986); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 49, No. 3, 592–634 (1985; Zbl 0581.17008)] for simple Lie algebras satisfying this identity (see also L. Poinsot, Algebra 2013, Article ID 341631, 8 p. (2013; Zbl 1334.17005), Th. 16]). We extend this result to prime (and semiprime) Lie algebras over a field of zero characteristic satisfying the standard Lie identity of degree 5. As an application, we prove that for any semiprime Lie algebra the standard identity \(St_5\) implies all other identities of the Lie algebra of polynomial vector fields on the affine line.
We also generalize some previous results about primeness of the Lie algebra of special derivations of a prime differential algebra to the case of non-unitary differential algebra.

MSC:

17B66 Lie algebras of vector fields and related (super) algebras
17B01 Identities, free Lie (super)algebras
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References:

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