On analogs of some classical group-theoretic results in Poisson algebras

Authors

DOI:

https://doi.org/10.15407/dopovidi2021.03.011

Keywords:

Poisson algebra, center, hypercenter, zero divisors, nilpotency

Abstract

We investigate the Poisson algebras, in which the n-th hypercenter (center) has a finite codimension. It was established that, in this case, the Poisson algebra P includes a finite-dimensional ideal K such that P/K is nilpotent (Abelian). Moreover, if the n-th hypercenter of a Poisson algebra P over some field has a finite codimension, and if P does not contain zero divisors, then P is Abelian.

References

Lichnerowicz, A. (1977). Les variétés de Poisson et leurs algèbres de Lie associées. J. Differential Geom., 12, No. 2, pp. 253-300. https://doi.org/10.4310/jdg/1214433987

Weinstein, A. (1977). Lecture on symplectic manifolds. CBMS regional conference series in mathematics, No. 29. Providence, R.I: Amer. Math. Soc. https://doi.org/10.1090/cbms/029

Berezin, F. A. (1967). Some remarks about the associated envelope of a Lie algebra. Funct. Anal. Appl., 1, No. 2, pp. 91-102. https://doi.org/10.1007/BF01076082

Vergne, M. (1972). La structure de Poisson sur l'algèbre symétrique d'une algèbre de Lie nilpotente. Bull. Soc. Math. France, 100, pp. 301-335.

https://doi.org/10.24033/bsmf.1740

Braconnier, J. (1977). Algèbres de Poisson, C.R. Acad. Sci., A, 284, No. 21, pp. 1345-1348.

Prosser, R. T. (1977). Poisson brackets and commutator brackets. I. Proc. Amer. Math. Soc., 62, No. 2, pp. 305-309. https://doi.org/10.1090/S0002-9939-1977-0443739-1

Prosser, R. T. (1977). Poisson brackets and commutator brackets. II. Proc. Amer. Math. Soc., 62, No. 2, pp. 310-315. https://doi.org/10.1090/S0002-9939-1977-0443740-8

Atkin, C. J. (1984). A note on the algebra of Poisson brackets. Math. Proc. Cambridge Philos. Soc., 96, No. 1, pp. 45-60. https://doi.org/10.1017/S0305004100061922

Neumann, B. H. (1951). Groups with finite classes of conjugate elements. Proc. Lond. Math. Soc., 3, No. 1, pp. 178-187. https://doi.org/10.1112/plms/s3-1.1.178

Baer, R. (1952). Endlichkeitskriterien für Kommutatorgruppen. Math. Ann., 124, No. 1., pp. 161-177. https://doi.org/10.1007/BF01343558

Kurdachenko, L. A. & Subbotin, I. Ya. (2016). On the relationships between the factors of upper and lower central series in groups and other algebraic structures. Note Mat., 36, No. 1, pp. 35-50. https://doi.org/10.1285/i15900932v36suppl1p35

Vaughan-Lee, M. R. (1972). Metabelian BFC p-groups. J. Lond. Math. Soc., 5, No. 4, pp. 673-680. https://doi.org/10.1112/jlms/s2-5.4.673

Kurdachenko, L. A., Pypka, A. A. & Subbotin, I. Ya. (2015). On some relations between the factors of the upper and lower central series in Lie algebras. Serdica Math. J., 60, No. 2-3, pp. 293-306.

Kurdachenko, L. A., Otal, J. & Pypka, A. A. (2016). Relationships between factors of canonical central series of Leibniz algebras. Eur. J. Math., 2, No. 2, pp. 565-577. https://doi.org/10.1007/ s40879-016-0093-5

Kurdachenko, L. A., Otal, J. & Subbotin, I. Ya. (2019). On some properties of the upper central series in Leibniz algebras. Comment. Math. Univ. Carolin., 60, No. 2, pp. 161-175. https://doi.org/10.14712/1213-7243.2019.009

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Published

06.07.2021

How to Cite

Kurdachenko, L. ., Pypka, A., & Subbotin, I. (2021). On analogs of some classical group-theoretic results in Poisson algebras. Reports of the National Academy of Sciences of Ukraine, (3), 11–16. https://doi.org/10.15407/dopovidi2021.03.011