On Simplicial Leibniz Algebras

Year 2020, Volume: 10 Issue: 2, 1224 - 1232, 01.06.2020

Ali Aytekin

https://doi.org/10.21597/jist.657475

Abstract

Aim of this paper is to define simplicial object in category of Leibniz algebras and to show the equivalence between the category of simplicial Leibniz algebras and the category of crossed modules over Leibniz algebras.

Keywords

Leibniz algebra, simplicial object, crossed module

Simplisel Leibniz Cebirler Üzerine

Abstract

Bu makalenin temel amacı, Leibniz cebirler kategorisinde simplisel objeyi tanımlayarak, simplisel Leibniz cebirler kategorisi ile Leibniz cebirler üzerinde çaprazlanmış modüller kategorisinin denkliğini göstermektir.

Keywords

Leibniz cebir, simplisel obje, çaprazlanmış modül

References

• Atik M, Aytekin A, Uslu EÖ, 2017. Representability of actions in the category of (Pre)crossed modules in Leibniz algebras. Communications in Algebra, 45(5): 1825–1841.
• Aytekin A, Casas JM, Uslu EÖ, 2012. Semi-Complete Crossed Modules of Lie Algebras. Journal of Algebra and Its Applications, 11(5): 1–24.
• Bloh A, 1965. A generalization of the concept of a Lie algebra. Doklady Akademii Lauk, 165 (3): 471–473.
• Casas JM, 1999. Crossed extensions of Leibniz algebras. Communations in Mathematics, 27 (12): 6253–6272.
• Casas JM, Fernandez-Casado R, Garcia-Martinez, X, Khmaladze E, 2018. Actor of a Crossed Module of Leibniz Algebras. Theory and Applications of Categories, 33(2): 23–42.
• Casas JM, Khmaladze E, Ladra M, 2008. Crossed modules for Leibniz n-algebras. Forum Mathematicum, 20: 841–858.
• D.M. Kan, 1958. A Combinatorial Definition of Homotopy Groups. Annals of Mathematics, 67(2): 288–312.
• Ellis GJ, 1993. Homotopical aspects of Lie algebras. Journal of The Australian Mathematical Society, 54(3): 393–419.
• Emir K, Akay HG, Pullback crossed modules in the category of racks. Hacettepe Journal of Mathematics and Statistics, 48(1): 140–149.
• Loday JL, 1993. Une version non commutative des algebres de Lie: les algebres de Leibniz. L’Enseignement Mathhematique, 39: 269–293.
• Loday JL, Pirashvili T, 1993. Universal enveloping algebras of Leibniz algebras and (co)homology. Mathematische Annalen, 296(1): 139–158.
• Şahan T, 2019. Further remarks on liftings of crossed modules. Hacettepe Journal of Mathematics and Statistics, 48(3): 743–752.
• Şahan T, Erciyes A, 2019. Actions of internal groupoids in the category of Leibniz Algebra. Communications Series A1:Mathematics and Statistics, 68(1): 619–632.
• Whitehead JHC, 1949. Combinatorial Homotopy. Bulletin of the American Mathematical Society, 55: 453–496.