×

The Freiheitssatz for generic Poisson algebras. (English) Zbl 1346.17003

In the theory of free algebraic structures “The Freiheitssatz” is a main and famous theorem. The Freiheitssatz came from group theory and was proved by Magnus in 1930. The theorem says that every equation over a free group is solvable in some extension.
In the theory of free associative and nonassociative algebras, some analogues of the Freiheitssatz were proved in several papers of Shirshov and Makar-Limanov. So, the Freiheitssatz was proved for anticommutative, commutative and Lie algebras (A. Shirshov 1962); for associative algebras (L. Makar-Limanov, 1985); for Novikov, Poisson and right-symmetric algebras (L. Makar-Limanov and U. Umirbaev 2008–2011) and many other.
In this paper the authors study free generic Poisson algebras and free generic Poisson algebras with polynomials. The class of generic Poisson algebras is a generalization of Poisson algebras and was introduced by I. Shestakov for studying universal enveloping of Malcev algebras. Some properties of free generic Poisson algebras were studied in a paper of the reviewer, I. Shestakov and U. Umirbaev [to appear in Commun. Algebra 2017].
In the paper under review the authors prove analogues of some earlier results of D. R. Farkas [Zbl 0892.17001, Zbl 0932.17021], L. Makar-Limanov and U. Umirbaev [Zbl 1285.17007] for Poisson algebras:
(1) It is proved that, if a generic Poisson algebra has a polynomial identity then it has an identity of a special form.
(2) The Freiheitssatz holds for the variety of generic Poisson algebras.

MSC:

17A30 Nonassociative algebras satisfying other identities
17B63 Poisson algebras
17A50 Free nonassociative algebras
PDFBibTeX XMLCite
Full Text: DOI arXiv EMIS