Crossed modules with action

Selim Çetin; Utku Gürdal

doi:10.3842/umzh.v74i4.7371

Selim Çetin Department of Mathematics, Burdur Mehmet Akif Ersoy University, Turkey
Utku Gürdal Department of Mathematics, Burdur Mehmet Akif Ersoy University, Turkey

DOI: https://doi.org/10.3842/umzh.v74i4.7371

Keywords: Crossed module, Groups with action, Category of groups

Abstract

UDC 512.5

Following the idea of a group with action, in a higher dimension, we consider crossed modules with a given crossed-module action upon itself. A category of these objects $\mathbf{XMod}^*$ is given by replacing the set of morphisms in the category $\mathbf{Gr}^*$ of groups with action by the set of a modified form of $\varphi$-crossed homomorphisms. Moreover, the connections of $\mathbf{XMod}^*$ with some other known categories, such as $\mathbf{Gr}^*$ and $\mathbf{XMod},$ are given by describing some related functors.

References

Y. Boyaci, J. M. Casas, T. Datuashvili, E. Ö. Uslu, Actions in modified categories of interest with application to crossed modules, Dedicated to Teimuraz Pirashvili on his 60th birthday (2015).

J. M. Casas, M. Ladra, The actor of a crossed module in Lie algebras, Commun. Algebra, 26, № 7, 2065–2089 (1998).

J. M. Casas, T. Datuashvili, M. Ladra, Universal strict general actors and actors in categories of interest, Appl. Categ. Structures, 18, № 1, 85–114 (2010).

T. Datuashvili, Central series for groups with action and Leibniz algebras, Georgian Math. J., 9, № 4, 671–681 (2002).

T. Datuashvili, Witt's theorem for groups with action and free Leibniz algebras, Georgian Math. J., 11, № 4, 691–712 (2004).

T. Datuashvili, T. Sahan, Actions and semi-direct products in categories of groups with action, Hacet. J. Math. and Stat., 1–11 (2023).

J.-L. Loday, Une version non commutative des algebres de Lie: les algebres de Leibniz, Les rencontres physiciens-math-maticiens de Strasbourg-RCP25, 44, 127–151 (1993).

J.-L. Loday, Algebraic $K$-theory and the conjectural Leibniz $K$-theory, $K$-theory, 30, № 2, 105–127 (2003).

K. J. Norrie, Crossed modules and analogues of group theorems, PhD Thesis. King's College London (University of London) (1987).