Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 115, 15 pages      arXiv:1405.7122
Contribution to the Special Issue on Poisson Geometry in Mathematics and Physics

The Freiheitssatz for Generic Poisson Algebras

Pavel S. Kolesnikov a, Leonid G. Makar-Limanov b and Ivan P. Shestakov c, a
a) Sobolev Institute of Mathematics, 630090, Novosibirsk, Russia
b) Department of Mathematics, Wayne State University, Detroit, MI 48202, USA
c) Instituto de Matematica e Estatí stica, Universidade de São Paulo, 05508-090 São Paulo, Brasil

Received July 29, 2014, in final form December 22, 2014; Published online December 29, 2014

We prove the Freiheitssatz for the variety of generic Poisson algebras.

Key words: Freiheitssatz; Poisson algebra; generic Poisson algebra; algebraically closed algebra; polynomial identity; differential algebra.

pdf (383 kb)   tex (23 kb)


  1. Bokut L.A., Theorems of imbedding in the theory of algebras, Colloq. Math. 14 (1966), 349-353.
  2. Bokut L.A., Chen Y., Li Y., Lyndon-Shirshov basis and anti-commutative algebras, J. Algebra 378 (2013), 173-183, arXiv:1110.1264.
  3. Bokut L.A., Kolesnikov P.S., Gröbner-Shirshov bases: from inception to the present time, J. Math. Sci. 116 (2003), 2894-2916.
  4. Cherlin G., Model theoretic algebra - selected topics, Lecture Notes in Math., Vol. 521, Springer-Verlag, Berlin - New York, 1976.
  5. Farkas D.R., Poisson polynomial identities, Comm. Algebra 26 (1998), 401-416.
  6. Higman G., Scott E., Existentially closed groups, London Mathematical Society Monographs, New Series, Vol. 3, The Clarendon Press, Oxford University Press, New York, 1988.
  7. Kolchin E.R., Differential algebra and algebraic groups, Pure and Applied Mathematics, Vol. 54, Academic Press, New York - London, 1973.
  8. Kolesnikov P.S., Varieties of dialgebras, and conformal algebras, Sib. Math. J. 49 (2008), 257-272, math.QA/0611501.
  9. Kozybaev D., Makar-Limanov L., Umirbaev U., The Freiheitssatz and the automorphisms of free right-symmetric algebras, Asian-Eur. J. Math. 1 (2008), 243-254, arXiv:0807.0608.
  10. Magnus W., Über discontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssatz), J. Reine Angew. Math. 163 (1930), 141-165.
  11. Makar-Limanov L., The skew field of fractions of the Weyl algebra contains a free noncommutative subalgebra, Comm. Algebra 11 (1983), 2003-2006.
  12. Makar-Limanov L., Algebraically closed skew fields, J. Algebra 93 (1985), 117-135.
  13. Makar-Limanov L., Umirbaev U., The Freiheitssatz for Poisson algebras, J. Algebra 328 (2011), 495-503, arXiv:1004.2747.
  14. Mikhalev A.A., Shestakov I.P., PBW-pairs of varieties of linear algebras, Comm. Algebra 42 (2014), 667-687.
  15. Ritt J.F., Differential algebra, American Mathematical Society Colloquium Publications, Vol. 33, Amer. Math. Soc., New York, 1950.
  16. Romanovskii N.S., A theorem on freeness for groups with one defining relation in varieties of solvable and nilpotent groups of given degrees, Math. USSR Sb. 89 (1972), 93-99.
  17. Scott W.R., Algebraically closed groups, Proc. Amer. Math. Soc. 2 (1951), 118-121.
  18. Shestakov I.P., Quantization of Poisson superalgebras and the specialty of Jordan superalgebras of Poisson type, Algebra Logika 32 (1993), 309-317.
  19. Shestakov I.P., Speciality problem for Malcev algebras and Poisson Malcev algebras, in Nonassociative Algebra and its Applications (São Paulo, 1998), Lecture Notes in Pure and Appl. Math., Vol. 211, Dekker, New York, 2000, 365-371.
  20. Shirshov A.I., Some algorithmic problems for Lie algebras, Sib. Math. J. 3 (1962), 292-296.
  21. Shirshov A.I., Some algorithmic problems for $\varepsilon $-algebras, Sib. Math. J. 3 (1962), 132-137.
  22. Umirbaev U.U., On Schreier varieties of algebras, Algebra Logika 33 (1994), 180-193.
  23. Zhevlakov K.A., Slin'ko A.M., Shestakov I.P., Shirshov A.I., Rings that are nearly associative, Pure and Applied Mathematics, Vol. 104, Academic Press, Inc., New York - London, 1982.

Previous article  Next article   Contents of Volume 10 (2014)