Universal Strict General Actors and Actors in Categories of Interest

Casas, J. M.; Datuashvili, Tamar; Ladra, M.

doi:10.1007/s10485-008-9166-z

Cited by 37 publications

(77 citation statements)

References 17 publications

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“…A similar phenomenon occurs in any category of interest C, that might not be action representative. Nevertheless, it is shown in [11] that, viewing C as a subvariety of a variety C G of groups with operations, for every object X in C there exists an object USGA(X) in C G (the "universal strict general actor" of X), such that every action of an object Y in C on X yields a morphism Y → USGA(X) in C G . Then, as above, we get a commutative diagram in C G :…”

Section: Coproduct Of Crossed Modulesmentioning

confidence: 99%

Peiffer product and Peiffer commutator for internal pre-crossed modules

Cigoli

Mantovani

2017

Homology, Homotopy and Applications

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In this work we introduce the notions of Peiffer product and Peiffer commutator of internal pre-crossed modules over a fixed object B, extending the corresponding classical notions to any semi-abelian category C. We prove that, under mild additional assumptions on C, crossed modules are characterized as those pre-crossed modules X whose Peiffer commutator X, X is trivial. Furthermore we provide suitable conditions on C (fulfilled by a large class of algebraic varietes, including among others groups, associative algebras, Lie and Leibniz algebras) under which the Peiffer product realizes the coproduct in the category of crossed modules over B.

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Section: Coproduct Of Crossed Modulesmentioning

confidence: 99%

Peiffer product and Peiffer commutator for internal pre-crossed modules

Cigoli

Mantovani

2017

Homology, Homotopy and Applications

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“…The idea of the definition comes from [15] and the axioms are from [30] and [31]. We formulate two more axioms on C (Axiom (7) and Axiom (8) in [30]).…”

Section: Example 22mentioning

confidence: 99%

Left-right noncommutative Poisson algebras

2014

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The notions of left-right noncommutative Poisson algebra (NP lr -algebra) and left-right algebra with bracket AWB lr are introduced. These algebras are special cases of NLP-algebras and algebras with bracket AWB, respectively, studied earlier. An NP lr -algebra is a noncommutative analogue of the classical Poisson algebra. Properties of these new algebras are studied. In the categories AWB lr and NP lr -algebras the notions of actions, representations, centers, actors and crossed modules are described as special cases of the corresponding wellknown notions in categories of groups with operations. The cohomologies of NP lr -algebras and AWB lr (resp. of NP r -algebras and AWB r ) are defined and the relations between them and the Hochschild, Quillen and Leibniz cohomologies are detected. The cases P is a free AWB r , the Hochschild or/and Leibniz cohomological dimension of P is ≤ are considered separately, exhibiting interesting possibilities of representations of the new cohomologies by the well-known ones and relations between the corresponding cohomological dimensions.

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“…Since Leib is a category of interest (see [10]), hence is a category of Ω-groups, and following Proposition 4.3.2 in [19] we can conclude that the collection of all nilpotent objects of class ≤ c in Leib form a variety. Now following [15,Proposition 7.8], it can be showed that M Lie (q) is the Schur Lie-multiplier of a Leibniz algebra q (see [9,12]).…”

Section: Background On Leibniz Algebrasmentioning

confidence: 88%