On Schur-type theorem for Leibniz 3-algebras

Pavlo Minaiev; Oleksandr Pypka

doi:10.17721/1812-5409.2024/1.3

Authors

Pavlo Minaiev Oles Honchar Dnipro National University, Dnipro https://orcid.org/0000-0002-5365-5014
Oleksandr Pypka Oles Honchar Dnipro National University, Dnipro https://orcid.org/0000-0003-0837-5395

DOI:

https://doi.org/10.17721/1812-5409.2024/1.3

Keywords:

Leibniz algebra, Leibniz 3-algebra, Lie algebra, center, derived ideal, Schur theorem

Abstract

One of the classic results of group theory is the so-called Schur theorem. It states that if the central factor-group G/ζ(G) of a group G is finite, then its derived subgroup [G,G] is also finite. This theorem was proved by B. Neumann in 1951. 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. In 2016, L.A. Kurdachenko, J. Otal and O.O. Pypka proved an analogue of Schur theorem for Leibniz algebras: if central factor-algebra L/ζ(L) of Leibniz algebra L has finite dimension, then its derived ideal [L,L] is also finite-dimensional. Moreover, they also proved a slightly modified analogue of Schur theorem: if the codimensions of the left ζ^l (L) and right ζ^r (L) centers of Leibniz algebra L are finite, then its derived ideal [L,L] is also finite-dimensional. One of the generalizations of Leibniz algebras is the so-called Leibniz n-algebras. It is worth noting that Leibniz n-algebra theory is currently much less developed than Leibniz algebra theory. One of the directions of development of the general theory of Leibniz n-algebras is the search for analogies with other types of algebras. Therefore, the question of proving analogs of the above results for this type of algebras naturally arises. In this article, we prove the analogues of the two mentioned theorems for Leibniz n-algebras for the case n=3. The obtained results indicate the prospects of further research in this direction.

Pages of the article in the issue: 22 - 25

Language of the article: English

References

Baer, R. (1952). Endlichkeitskriterien für kommutatorgruppen. Math. Ann., 124(1), 161-177. https://doi.org/10.1007/BF01343558

Blokh, A. (1965). On a generalization of the concept of Lie algebra. Dokl. Akad. Nauk SSSR, 165(3), 471-473.

Casas, J.M., Loday, J.-L., & Pirashvili, T. (2002). Leibniz n-algebras. Forum Math., 14, 189-207. https://doi.org/10.1515/form.2002.009

Chupordia, V.A., Pypka, A.A., Semko, N.N., & Yashchuk, V.S. (2019). Leibniz algebras: a brief review of current results. Carpathian Math. Publ., 11(2), 250-257. https://doi.org/10.15330/cmp.11.2.250-257

Filippov, V.T. (1985). n-Lie algebras. Sib. Math. J., 26(6), 879-891. https://doi.org/10.1007/BF00969110

Kirichenko, V.V., Kurdachenko, L.A., Pypka, A.A., & Subbotin, I.Ya. (2017). Some aspects of Leibniz algebra theory. Algebra Discrete Math., 24(1), 1-33.

Kurdachenko, L.A., Otal, J., & Pypka, A.A. (2016). Relationships between the factors of the canonical central series of Leibniz algebras. Eur. J. Math., 2(2), 565-577. https://doi.org/10.1007/s40879-016-0093-5

Loday, J.-L. (1992). Cyclic homology. Springer Verlag. https://doi.org/10.1007/978-3-662-11389-9

Loday, J.-L. (1993). Une version non commutative des algébres de Lie: les algébras de Leibniz. Enseign. Math., 39, 269-293.

Loday, J.-L., & Pirashvili, T. (1993). Universal enveloping algebras of Leibniz algebras and (co)homology. Math. Ann., 296(1), 139-158. https://doi.org/10.1007/BF01445099

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