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Higher Dimensional Leibniz-Rinehart Algebras

Year 2024, Volume: 7 Issue: 1, 45 - 50, 08.05.2024

Mahmut Koçak Selim Çetin

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

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.

Year 2024, Volume: 7 Issue: 1, 45 - 50, 08.05.2024

Mahmut Koçak Selim Çetin

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

Abstract

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.
There are 21 citations in total.

Details

Primary Language English
Subjects Pure Mathematics (Other)
Journal Section Articles
Authors

Mahmut Koçak 0000-0001-7774-0144

Selim Çetin 0000-0002-9017-1465

Early Pub Date May 8, 2024
Publication Date May 8, 2024
Submission Date April 8, 2024
Acceptance Date May 8, 2024
Published in Issue Year 2024 Volume: 7 Issue: 1

Cite

APA Koçak, M., & Çetin, S. (2024). Higher Dimensional Leibniz-Rinehart Algebras. Journal of Mathematical Sciences and Modelling, 7(1), 45-50. https://doi.org/10.33187/jmsm.1466687
AMA Koçak M, Çetin S. Higher Dimensional Leibniz-Rinehart Algebras. Journal of Mathematical Sciences and Modelling. May 2024;7(1):45-50. doi:10.33187/jmsm.1466687
Chicago Koçak, Mahmut, and Selim Çetin. “Higher Dimensional Leibniz-Rinehart Algebras”. Journal of Mathematical Sciences and Modelling 7, no. 1 (May 2024): 45-50. https://doi.org/10.33187/jmsm.1466687.
EndNote Koçak M, Çetin S (May 1, 2024) Higher Dimensional Leibniz-Rinehart Algebras. Journal of Mathematical Sciences and Modelling 7 1 45–50.
IEEE M. Koçak and S. Çetin, “Higher Dimensional Leibniz-Rinehart Algebras”, Journal of Mathematical Sciences and Modelling, vol. 7, no. 1, pp. 45–50, 2024, doi: 10.33187/jmsm.1466687.
ISNAD Koçak, Mahmut - Çetin, Selim. “Higher Dimensional Leibniz-Rinehart Algebras”. Journal of Mathematical Sciences and Modelling 7/1 (May 2024), 45-50. https://doi.org/10.33187/jmsm.1466687.
JAMA Koçak M, Çetin S. Higher Dimensional Leibniz-Rinehart Algebras. Journal of Mathematical Sciences and Modelling. 2024;7:45–50.
MLA Koçak, Mahmut and Selim Çetin. “Higher Dimensional Leibniz-Rinehart Algebras”. Journal of Mathematical Sciences and Modelling, vol. 7, no. 1, 2024, pp. 45-50, doi:10.33187/jmsm.1466687.
Vancouver Koçak M, Çetin S. Higher Dimensional Leibniz-Rinehart Algebras. Journal of Mathematical Sciences and Modelling. 2024;7(1):45-50.
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