On Poisson (2-3)-algebras which are finite-dimensional over the center

P.Ye. Minaiev; O.O. Pypka; I.V. Shyshenko

doi:10.15421/242411

On Poisson (2-3)-algebras which are finite-dimensional over the center

P.Ye. Minaiev (Oles Honchar Dnipro National University)
O.O. Pypka (Oles Honchar Dnipro National University), https://orcid.org/0000-0003-0837-5395
I.V. Shyshenko (Sumy State Pedagogical University named after A.S. Makarenko), https://orcid.org/0000-0002-1026-5315

Abstract

One of the classic results of group theory is the so-called Schur theorem. It states that if the central factor-group $$$G/\zeta(G)$$$ of a group $$$G$$$ is finite, then its derived subgroup $$$[G,G]$$$ is also finite. This result has numerous generalizations and modifications in group theory. At the same time, similar investigations were conducted in other algebraic structures, namely in modules, linear groups, topological groups, $$$n$$$-groups, associative algebras, Lie algebras, Lie $$$n$$$-algebras, Lie rings, Leibniz algebras. In 2021, L.A. Kurdachenko, O.O. Pypka and I.Ya. Subbotin proved an analogue of Schur theorem for Poisson algebras: if the center of the Poisson algebra $$$P$$$ has finite codimension, then $$$P$$$ includes an ideal $$$K$$$ of finite dimension such that $$$P/K$$$ is abelian. In this paper, we continue similar studies for another algebraic structure. An analogue of Schur theorem for Poisson (2-3)-algebras is proved.

Keywords

Poisson algebra; Poisson (2-3)-algebra; Schur theorem

MSC 2020

Pri 17B63, Sec 17A42

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References

Dixon M.R., Kurdachenko L.A., Otal J. "Linear analogues of theorems of Schur, Baer and Hall", Int. J. Group Theory, 2013; 2(1): pp. 79-89. doi:10.22108/IJGT.2013.2441

Dixon M.R., Kurdachenko L.A., Pypka A.A. "On Some Variants of Theorems of Schur and Baer", Milan J. Math., 2014; 82: pp. 233-241. doi:10.1007/s00032-014-0215-9

Dixon M.R., Kurdachenko L.A., Pypka A.A. "The theorems of Schur and Baer: a survey", Int. J. Group Theory, 2015; 4(1): pp. 21-32. doi:10.22108/IJGT.2015.7376

Galmak A.M. "On the Schur theorem for $$$n$$$-ary groups", Ukrainian. Math. J., 2006; 58(6): pp. 730-741. doi:10.1007/s11253-006-0106-5

Kurdachenko L.A., Otal J., Pypka A.A. "On the structure and some numerical properties of subgroups and factor-groups defined by automorphism groups", J. Algebra Appl., 2015; 14(5): no. 1550075. doi:10.1142/S0219498815500759

Kurdachenko L.A., Otal J., Pypka A.A. "Relationships between factors of canonical central series of Leibniz algebras", Eur. J. Math., 2016; 2: pp. 565-577. doi:10.1007/s40879-016-0093-5

Kurdachenko L.A., Otal J., Pypka A.A. "Relationships between the factors of the upper and the lower central series of a group", B. Malays. Math. Sci. So., 2016; 39(3): pp. 1115-1124. doi:10.1007/s40840-015-0222-1

Kurdachenko L.A., Pypka O.O. "On some generalizations of Baer's theorem", Carpathian Math. Publ., 2014; 6(2): pp. 310-316. doi:10.15330/cmp.6.2.310-316

Kurdachenko L.A., Pypka A.A., Subbotin I.Ya. "On some relations between the factors of the upper and lower central series in Lie algebras", Serdica Math. J., 2015; 41(2-3): pp. 293-306.

Kurdachenko L.A., Pypka A.A., Subbotin I.Ya. "On new analogs of some classical group theoretical results in Lie rings", in Baginski P., Fine B., Gaglione A.M. (eds.): Infinite Group Theory. From the Past to the Future, World Scientific, New Jersey, 2018; pp. 197-213. doi:10.1142/9789813204058_0011

Kurdachenko L.A., Pypka A.A., Subbotin I.Ya. "On extension of classical Baer results to Poisson algebras", Algebra Discrete Math., 2021; 31(1): pp. 84-108. doi:10.12958/adm1758

Kurdachenko L.A., Subbotin I.Ya. "A brief history of an important classical theorem", Adv. Group Theory Appl., 2016; 2: pp. 121-124. doi:10.4399/97888548970148

Kurdachenko L.A., Subbotin I.Ya., Chupordia V.A. "On the relations between the central factor-module and the derived submodule in modules over group rings", Comment. Math. Univ. Carolin., 2015; 56(4): pp. 433-445. doi:10.14712/1213-7243.2015.136

Neumann B.H. "Groups with finite classes of conjugate elements", Proc. London Math. Soc., 1951; 1(1): pp. 178-187. doi:10.1112/plms/s3-1.1.178

Pypka A.A. "Relationships between the factors of the central series and the nilpotent residual in some infinite groups", Adv. Group Theory Appl., 2017; 4: pp. 65-82. doi:10.4399/97888255086975

Riley D., Shahada M. "Relationships between the canonical ascending and descending central series of ideals of an associative algebra", Commun. Algebra, 2017; 45(5): pp. 1969-1982. doi:10.1080/00927872.2016.1226871

Stewart I.N. "Verbal and marginal properties of non-associative algebras in the spirit of infinite group theory", Proc. London Math. Soc., 1974; 28(1): pp. 129-140. doi:10.1112/plms/s3-28.1.129

Saeedi F., Veisi B. "On Schur theorem and it converses for n-Lie algebras", Lin. Multilin. Algebra, 2014; 62(9): pp. 1139-1145. doi:10.1080/03081087.2013.809871

Ushakov V.I. "Topological groups with bicompact classes of conjugate subgroups", Matem. Sbor., 1964; 63(105): pp. 277-283.