Higher Dimensional Leibniz-Rinehart Algebras

Year 2024, , 45 - 50, 08.05.2024

Mahmut Koçak , Selim Çetin

https://doi.org/10.33187/jmsm.1466687

Cited By: 1

Abstract

In this article, we delve into the realm of higher dimensional Leibniz-Rinehart algebras, exploring the intricate structures of Leibniz algebroids and their applications. By generalizing the concept of Lie algebroids and incorporating a Leibniz rule for the anchor map, the study sheds light on the fundamental principles underlying connections and underscores their significance. Through a comprehensive analysis of Leibniz-Rinehart algebras, this study paves the way for advancements and applications, offering a deeper understanding of the intricate relationship between algebraic and geometric structures.

Keywords

Crossed module, Leibniz algebra, Leibniz algebroid, Leibniz-Rinehart algebra, Lie-Rineart algebra

References

• [1] J.-L. Loday, Une version non commutative des algebres de Lie: les algebres de Leibniz, L’Enseignement Mathematique 39 (1993), 269–292.
• [2] J.-L. Loday, T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann., 296 (1993), 139–158.
• [3] T. Jubin, Benoı, N. Poncin, K. Uchino, Free Courant and derived Leibniz pseudoalgebras, J. Geom. Mech., 8(1) (2016) 71–97.
• [4] A. Aytekin, Categorical structures of Lie-Rinehart crossed module, Turkish J. Math., 43(1) (2019), 511–522.
• [5] A. B. Hassine, T. Chtioui, M. Elhamdadi, S. Mabrouk, Extensions and Crossed Modules of n-Lie-Rinehart Algebras, Adv. Appl. Clifford Algebr., 32(3) (2022),31.
• [6] J. M. Casas, M. Ladra, T. Pirashvili, Crossed modules for Lie-Rinehart algebras, Cent. Eur. Journal of Algebra, 274(1) (2004) 192–201.
• [7] Chen, Liangyun, M. Liu, J. Liu, Cohomologies and crossed modules for pre-Lie Rinehart algebras, J. Geom. Phys., 176 (2022)
• [8] A. Çobankaya, S. Çetin, Homotopy of Lie-Rinehart Crossed Module Morphisms, Adıyaman University Journal of Science, 9(1) (2019) 202–212.
• [9] J. Huebschmann, Poisson cohomology and quantization, J. Reine Angew. Math., 408 (1990), 57–113.
• [10] J. M. Casas, T. Datuashvili, M. Ladra, Left-right noncommutative Poisson algebras, Cent. Eur. J. Math., 12(1) (2014) 57–78.
• [11] M. Alp, B. Davvaz, Crossed polymodules and fundamental relations, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 77(2) (2015), 129–140.
• [12] H. G. Akay, İ. İ. Akça, Completeness of the category of rack crossed modules, Ikonion J. Math., 4(2) (2022), 56–68.
• [13] S. Çetin, Utku Gürdal, A characterization of crossed self-similarity on crossed modules in L-algebras, Logic Journal of the IGPL, jzae003 (2024).
• [14] J. M. Casas, S. Çetin, E. Ö. Uslu, Crossed modules in the category of Loday QD-Rinehart algebras, Homology Homotopy Appl., 22(2) (2020) 347–366.
• [15] S. Çetin, Leibniz-Rinehart cebirleri ve genellemeleri, Phd Thesis, Eskis¸ehir Osmangazi U¨ niversitesi, Tu¨rkiye, (2017)
• [16] U. Gürdal, A Jordan-Hölder theorem for crossed squares, Kuwait J. Sci., 50(2) (2023) 83–90.
• [17] M. H. Gürsoy, H. Aslan, İ. İcen, Generalized crossed modules and group-groupoids, Turkish J. Math., 41(6) (2017) 1535–1551.
• [18] J. Huebschmann, On the history of Lie brackets, crossed modules, and Lie-Rinehart algebras, J. Geom. Mech., 13(3) (2021) 385–402.
• [19] O. Mucuk, T. Şahan, Coverings and crossed modules of topological groups with operations, Turkish J. Math., 38(5) (2014) 833–845.
• [20] A. Mutlu, Join for (Augmented) Simplicial Group, Math. Comput. App., 5(2) (2000) 105–112.
• [21] S. Öztunç, N. Bildik, A. Mutlu, The construction of simplicial groups in digital images, J. Inequal. Appl., (2013) 1–13.