On extension of classical Baer results to Poisson algebras

L. A. Kurdachenko ; A. A. Pypka ; I. Ya. Subbotin

Algebra and discrete mathematics, Tome 31 (2021) no. 1, pp. 84-108.

Résumé

In this paper we prove that if $P$ is a Poisson algebra and the $n$th hypercenter (center) of $P$ has a finite codimension, then $P$ includes a finite-dimensional ideal $K$ such that $P/K$ is nilpotent (abelian). As a corollary, we show that if the $n$th hypercenter of a Poisson algebra $P$ (over some specific field) has a finite codimension and $P$ does not contain zero divisors, then $P$ is an abelian algebra.

Détail

Keywords: Lie algebra, ideal, center, hypercenter, zero divisor, finite dimension, nilpotency.
Mots-clés : Poisson algebra, subalgebra

Bibliographie

[1] V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts Math., 60, Springer-Verlag, New York–Heidelberg, 1978 | DOI | MR | Zbl

[2] C. J. Atkin, “A note on the algebra of Poisson brackets”, Math. Proc. Cambridge Philos. Soc., 96:1 (1984), 45–60 | DOI | MR | Zbl

[3] R. Baer, “Endlichkeitskriterien für Kommutatorgruppen”, Math. Ann., 124 (1952), 161–177 | DOI | MR | Zbl

[4] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, “Deformation theory and quantization. I. Deformations of symplectic structures”, Ann. Physics, 111:1 (1978), 61–110 | DOI | MR | Zbl

[5] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, “Deformation theory and quantization. II. Physical applications”, Ann. Physics, 111:1 (1978), 111–151 | DOI | MR | Zbl

[6] J. Bell, S. Launois, O. L. Sánchez, R. Moosa, “Poisson algebras via model theory and differential-algebraic geometry”, J. Eur. Math. Soc., 19:7 (2017), 2019–2049 | DOI | MR | Zbl

[7] F. A. Berezin, “Some remarks about the associated envelope of a Lie algebra”, Funct. Anal. Appl., 1:2 (1967), 91–102 | DOI | MR | Zbl

[8] K. H. Bhaskara, K. Viswanath, Poisson algebras and Poisson manifolds, Longman, 1988 | MR | Zbl

[9] J. Braconnier, “Algebres de Poisson”, C.R. Acad. Sci., A, 284:21 (1977), 1345–1348 | MR | Zbl

[10] A. J. Calderón Martín, “On extended graded Poisson algebras”, Linear Algebra Appl., 439:4 (2013), 879–892 | DOI | MR | Zbl

[11] V. Chari, A. Pressley, A Guide to Quantum Groups, Cambridge University Press, Cambridge, 1994 | MR | Zbl

[12] P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Grad. Sch. Sci., Yeshiva University, New York, 1964 | MR

[13] J. Soviet Math., 41:2 (1988), 898–915 | DOI | MR | Zbl | Zbl

[14] D. R. Farkas, “Poisson polynomial identities”, Comm. Algebra, 26:2 (1998), 401–416 | DOI | MR | Zbl

[15] D. R. Farkas, “Poisson polynomial identities II”, Arch. Math., 72:4 (1999), 252–260 | DOI | MR | Zbl

[16] B. Fresse, “Théorie des opérades de Koszul et homologie des algèbres de Poisson”, Ann. Math. Blaise Pascal, 13:2 (2006), 237–312 | DOI | MR | Zbl

[17] A. Giambruno, V. M. Petrogradsky, “Poisson identities of enveloping algebras”, Arch. Math., 87:6 (2006), 505–515 | DOI | MR | Zbl

[18] V. Ginzburg, D. Kaledin, “Poisson deformations of symplectic quotient singularities”, Adv. Math., 186:1 (2004), 1–57 | DOI | MR | Zbl

[19] M. Goze, E. Remm, “Poisson algebras in terms of non-associative algebras”, J. Algebra, 320:1 (2008), 294–317 | DOI | MR | Zbl

[20] J. Huebschmann, “Poisson cohomology and quantization”, J. Reine Angew. Math., 408 (1990), 57–113 | MR | Zbl

[21] M. Kontsevich, “Deformation quantization of Poisson manifolds”, Lett. Math. Phys., 66:3 (2003), 157–216 | DOI | MR | Zbl

[22] L. A. Kurdachenko, J. Otal, A. A. Pypka, “Relationships between factors of canonical central series of Leibniz algebras”, Eur. J. Math., 2:2 (2016), 565–577 | DOI | MR | Zbl

[23] L. A. Kurdachenko, J. Otal, I. Ya. Subbotin, “On some properties of the upper central series in Leibniz algebras”, Comment. Math. Univ. Carolin., 60:2 (2019), 161–175 | MR | Zbl

[24] L. A. Kurdachenko, A. A. Pypka, I. Ya. Subbotin, “On some relations between the factors of the upper and lower central series in Lie algebras”, Serdica Math. J., 60:2–3 (2015), 293–306 | MR

[25] L. A. Kurdachenko, I. Ya. Subbotin, “On the relationships between the factors of upper and lower central series in groups and other algebraic structures”, Note Mat., 36:1 (2016), 35–50 | MR | Zbl

[26] L.-C. Li, “Classical $r$-matrices and compatible Poisson structures for Lax equations on Poisson algebras”, Comm. Math. Phys., 203:3 (1999), 573–592 | DOI | MR | Zbl

[27] A. Lichnerowicz, “Les variétés de Poisson et leurs algèbres de Lie associées”, J. Differential Geom., 12:2 (1977), 253–300 | DOI | MR | Zbl

[28] L. Makar-Limanov, I. Shestakov, “Polynomial and Poisson dependence in free Poisson algebras and free Poisson fields”, J. Algebra, 349:1 (2012), 372–379 | DOI | MR | Zbl

[29] L. Makar-Limanov, U. Turusbekova, U. Umirbaev, “Automorphisms and derivations of free Poisson algebras in two variables”, J. Algebra, 322:9 (2009), 3318–3330 | DOI | MR | Zbl

[30] L. Makar-Limanov, U. Umirbaev, “Centralizers in free Poisson algebras”, Proc. Amer. Math. Soc., 135:7 (2007), 1969–1975 | DOI | MR | Zbl

[31] L. Makar-Limanov, U. Umirbaev, “The Freiheitssatz for Poisson algebras”, J. Algebra, 328:1 (2011), 495–503 | DOI | MR | Zbl

[32] M. Markl, E. Remm, “Algebras with one operation including Poisson and other Lie-admissible algebras”, J. Algebra, 229:1 (2006), 171–189 | DOI | MR

[33] S. P. Mishchenko, V. M. Petrogradsky, A. Regev, “Poisson PI algebras”, Trans. Amer. Math. Soc., 359:10 (2007), 4669–4694 | DOI | MR | Zbl

[34] I. Z. Monteiro Alves, V. M. Petrogradsky, “Lie structure of truncated symmetric Poisson algebras”, J. Algebra, 488 (2017), 244–281 | DOI | MR | Zbl

[35] B. H. Neumann, “Groups with finite classes of conjugate elements”, Proc. Lond. Math. Soc., 3:1 (1951), 178–187 | DOI | MR | Zbl

[36] A. Odzijewicz, “Hamiltonian and quantum mechanics”, Geom. Topol. Monogr., 17 (2011), 385–472 | MR | Zbl

[37] A. I. Ooms, “The Poisson center and polynomial, maximal Poisson commutative subalgebras, especially for nilpotent Lie algebras of dimension at most seven”, J. Algebra, 365 (2012), 83–113 | DOI | MR | Zbl

[38] A. Polishchuk, “Algebraic geometry of Poisson brackets”, J. Math. Sci., 84:5 (1997), 1413–1444 | DOI | MR | Zbl

[39] R. T. Prosser, “Poisson brackets and commutator brackets. I”, Proc. Amer. Math. Soc., 62:2 (1977), 305–309 | DOI | MR

[40] R. T. Prosser, “Poisson brackets and commutator brackets. II”, Proc. Amer. Math. Soc., 62:2 (1977), 310–315 | DOI | MR

[41] S. M. Ratseev, “Growth in Poisson algebras”, Algebra Logic, 50:1 (2011), 46–61 | MR | Zbl

[42] S. M. Ratseev, “Poisson algebras of polynomial growth”, Sib. Math. J., 54:3 (2013), 555–565 | DOI | MR | Zbl

[43] S. M. Ratseev, “Correlation of Poisson algebras and Lie algebras in the language of identities”, Math. Notes, 96:3–4 (2014), 538–547 | DOI | MR | Zbl

[44] S. M. Ratseev, “On minimal Poisson algebras”, Russian Math., 59:11 (2015), 54–61 | DOI | MR | Zbl

[45] I. P. Shestakov, “Quantization of Poisson superalgebras and speciality of Jordan Poisson superalgebras”, Algebra Logic, 32:5 (1993), 309–317 | DOI | MR | Zbl

[46] S. Siciliano, “Solvable symmetric Poisson algebras and their derived lengths”, J. Algebra, 543 (2020), 98–110 | DOI | MR | Zbl

[47] U. Umirbaev, “Universal enveloping algebras and universal derivations of Poisson algebras”, J. Algebra, 354:1 (2012), 77–94 | DOI | MR | Zbl

[48] I. Vaisman, Lectures on the geometry of Poisson manifolds, Progress in Mathematics, 118, Birkhäuser, Basel and Boston, 1994 | MR | Zbl

[49] M. Van den Bergh, “Double Poisson algebras”, Trans. Amer. Math. Soc., 360:11 (2008), 5711–5769 | DOI | MR | Zbl

[50] M. R. Vaughan-Lee, “Metabelian BFC $p$-groups”, J. Lond. Math. Soc., 5:4 (1972), 673–680 | DOI | MR | Zbl

[51] M. Vergne, “La structure de Poisson sur l'algèbre symétrique d'une algèbre de Lie nilpotente”, Bull. Soc. Math. France, 100 (1972), 301–335 | DOI | MR | Zbl

[52] A. Weinstein, Lecture on Symplectic Manifolds, CBMS Regional Conference Series in Mathematics, 29, Amer. Math. Soc., Providence, R.I., 1979 | MR