Hostname: page-component-6bf8c574d5-qdpjg Total loading time: 0 Render date: 2025-02-16T00:42:25.442Z Has data issue: false hasContentIssue false
  • English
  • Français

On the representability of actions of Leibniz algebras and Poisson algebras

Part of: Rings and algebras with additional structure Other classes of algebra General and miscellaneous Lie algebras and Lie superalgebras General nonassociative rings

Published online by Cambridge University Press:  22 November 2023

Alan S. Cigoli Open the ORCID record for Alan S. Cigoli [Opens in a new window] ,
Manuel Mancini Open the ORCID record for Manuel Mancini [Opens in a new window]  and
Giuseppe Metere Open the ORCID record for Giuseppe Metere [Opens in a new window]
Alan S. Cigoli
Affiliation:
Dipartimento di Matematica “Giuseppe Peano”, Università degli Studi di Torino, Torino, Italy ([email protected])
Manuel Mancini
Affiliation:
Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Palermo, Italy ([email protected]; [email protected])
Giuseppe Metere
Affiliation:
Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Palermo, Italy ([email protected]; [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

In a recent paper, motivated by the study of central extensions of associative algebras, George Janelidze introduces the notion of weakly action representable category. In this paper, we show that the category of Leibniz algebras is weakly action representable and we characterize the class of acting morphisms. Moreover, we study the representability of actions of the category of Poisson algebras and we prove that the subvariety of commutative Poisson algebras is not weakly action representable.

Keywords

action representable categorysplit extensionassociative algebraLeibniz algebraPoisson algebra

MSC classification

Primary: 08C05: Categories of algebras 17A32: Leibniz algebras 17B63: Poisson algebras
Secondary: 16B50: Category-theoretic methods and results (except as in ) 17A36: Automorphisms, derivations, other operators 16W25: Derivations, actions of Lie algebras
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 66 , Issue 4 , November 2023 , pp. 998 - 1021
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Borceux, F., Janelidze, G. and Kelly, G. M., Internal object actions, Comment. Math. Univ. Carolin. 46(2) (2005), 235–255.Google Scholar
Borceux, F., Janelidze, G. and Kelly, G. M., On the representability of actions in a semi-abelian category, Theory Appl. Categ. 14 (1) (2005), 244–286.Google Scholar
Bourn, D. and Janelidze, G., Centralizers in action accessible categories, Cah. Topol. Géom. Différ. Catég. 50(3) (2009), 211–232.Google Scholar
Casas, J. M., Datuashvili, T. and Ladra, M., Universal strict general actors and actors in categories of interest, Appl. Categ. Structures 18(1) (2010), 85–114.CrossRefGoogle Scholar
Cigoli, A. S. and Mantovani, S., Action accessibility via centralizers, J. Pure Appl. Algebra 216 (8-9) (2012), 1852–1865.CrossRefGoogle Scholar
Cigoli, A. S., Metere, G. and Montoli, A., Obstruction theory in action accessible categories, J. Algebra 385 (3) (2013), 27–46.CrossRefGoogle Scholar
Datuashvili, T., Cohomologically trivial internal categories in categories of groups with operations, Appl. Categ. Structures 3(3) (1995), 221–237.CrossRefGoogle Scholar
García-Martínez, X., Tsishyn, M., Van der Linden, T. and Vienne, C., Algebras with representable representations, Proc. Edinb. Math. Soc. 64(2) (2021), 555–573.CrossRefGoogle Scholar
Gray, J. R. A., A note on the relationship between action accessible and weakly action representable categories, (2022), Preprint, available at https://arXiv:2207.06149.Google Scholar
Janelidze, G., Central extensions of associative algebras and weakly action representable categories, Theory Appl. Categ. 38 (36) (2022), 1395–1408.Google Scholar
Janelidze, G., Márki, L. and Tholen, W., Semi-abelian categories, J. Pure Appl. Algebra 168(2) (2002), 367–386.CrossRefGoogle Scholar
Loday, J. -L., Une version non commutative des algèbres de Lie: les algebres de Leibniz, Enseign. Math. 39(3-4) (1993), 269–293.Google Scholar
Mac Lane, S., Extensions and obstructions for rings, Illinois J. Math. 2(3) (1958), 316–345.CrossRefGoogle Scholar
Mancini, M.. Biderivations of low-dimensional Leibniz algebras, in Non-Associative Algebras and Related Topics. NAART 2020, Springer Proceedings in Mathematics & Statistics, (eds. Albuquerque, H., Brox, H.J., Martínez, H.J.C., Saraiva, H.J.C.P.), (Cham: Springer, 2023), pp. pp. Vol 427(8).Google Scholar
Montoli, A., Action accessibility for categories of interest, Theory Appl. Categ. 23 (1) (2010), 7–21.Google Scholar
Orzech, G., Obstruction theory in algebraic categories, I, J. Pure Appl. Algebra 2(4) (1972), 287–314.CrossRefGoogle Scholar