Some results on stem covers of crossed modules

Mohammadzadeh, Hamid; Shahrokhi, Saeed; Salemkar, Ali Reza

doi:10.1016/j.jpaa.2014.02.017

Cited by 7 publications

(4 citation statements)

References 22 publications

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“…−→ (n, q, δ) −→ 0 be a free presentation of (n, q, δ). By applying a similar argument to the proof of [26,Theorem 3.7], we can show that the crossed modules…”

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confidence: 82%

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Some Properties of the Schur Multiplier and Stem Covers of Leibniz Crossed Modules

Casas

Ravanbod

2019

Bull. Malays. Math. Sci. Soc.

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In this article we investigate the interplay between stem covers, the Schur multiplier of Leibniz crossed modules and the non-abelian exterior product of Leibniz algebras. In concrete, we obtain a six-term exact sequence associated to a central extension of Leibniz crossed modules, which is useful to characterize stem covers. We show the existence of stem covers and determine the structure of all stem covers of Leibniz crossed modules. Also, we give the connection between the stem cover of a Lie crossed module in the categories of Lie and Leibniz crossed modules respectively.

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“…−→ (n, q, δ) −→ 0 be a free presentation of (n, q, δ). By applying a similar argument to the proof of [26,Theorem 3.7], we can show that the crossed modules…”

mentioning

confidence: 82%

“…In the following, we determine the structure of stem covers of Leibniz crossed module which is analogous to the similar works in group and in Lie crossed modules [30,26]. so (e) is stem cover of (n, q, δ).…”

Section: Stem Extensions and Stem Covers Of Leibniz Crossed Modulesmentioning

confidence: 93%

Some Properties of the Schur Multiplier and Stem Covers of Leibniz Crossed Modules

Casas

Ravanbod

2019

Bull. Malays. Math. Sci. Soc.

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“…The algebraic study of the category of crossed modules was initiated by Norrie [18] and has led to a substantial algebraic theory contained essentially in the following papers: [1,7,11,12,13,15,20,21]. In particular, Pirashvili [19] presented the concept of the tensor product of two abelian crossed modules and investigated its relation to the low-dimensional homology of crossed modules.…”

Section: Introductionmentioning

confidence: 99%

The non-abelian tensor product of normal crossed submodules of groups

Salemkar

Taha

2020

CGASA

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In this article, the notions of non-abelian tensor and exterior products of two normal crossed submodules of a given crossed module of groups are introduced and some of their basic properties are established. In particular, we investigate some common properties between normal crossed modules and their tensor products, and present some bounds on the nilpotency class and solvability length of the tensor product, provided such information is given at least on one of the normal crossed submodules.

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“…Loday in [4] showed that the category of crossed modules is equivalent to that of cat 1 -groups. Also, many mathematicians investigated the categorical properties of the category CM and its important subcategory, the category of crossed P -modules, and characterized important objects of these categories such as products, equalizers, pullbacks, coproducts etc (see [2,3,6,8,9]). Let us recall the following.…”

Section: Introductionmentioning

confidence: 99%

The category of generalized crossed modules

Yavari

Salemkar

2018

CGASA

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In the definition of a crossed module (T, G, ρ), the actions of the group T and G on themselves are given by conjugation. In this paper, we consider these actions to be arbitrary and thus generalize the concept of ordinary crossed module. Therefore, we get the category GCM, of all generalized crossed modules and generalized crossed module morphisms between them, and investigate some of its categorical properties. In particular, we study the relations between epimorphisms and the surjective morphisms, and thus generalize the corresponding results of the category of (ordinary) crossed modules. By generalizing the conjugation action, we can find out what is the superiority of the conjugation to other actions. Also, we can find out that a generalized crossed module with which other actions (other than the conjugation) has the properties similar to a crossed module.

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Some results on stem covers of crossed modules

Cited by 7 publications

References 22 publications

Some Properties of the Schur Multiplier and Stem Covers of Leibniz Crossed Modules

Some Properties of the Schur Multiplier and Stem Covers of Leibniz Crossed Modules

The non-abelian tensor product of normal crossed submodules of groups

The category of generalized crossed modules

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