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On the representability of actions of Leibniz algebras and Poisson algebras

Part of: General and miscellaneous Other classes of algebra Rings and algebras with additional structure 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 (alan.cigoli@unito.it)
Manuel Mancini
Affiliation:
Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Palermo, Italy (manuel.mancini@unipa.it; giuseppe.metere@unipa.it)
Giuseppe Metere
Affiliation:
Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Palermo, Italy (manuel.mancini@unipa.it; giuseppe.metere@unipa.it)
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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
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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