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Higher Dimensional Leibniz-Rinehart Algebras

Yıl 2024, Cilt: 7 Sayı: 1, 45 - 50, 08.05.2024

Mahmut Koçak Selim Çetin

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

Öz

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.

Anahtar Kelimeler

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

Kaynakça

  • [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.

Yıl 2024, Cilt: 7 Sayı: 1, 45 - 50, 08.05.2024

Mahmut Koçak Selim Çetin

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

Öz

Kaynakça

  • [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.
Toplam 21 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Temel Matematik (Diğer)
Bölüm Makaleler
Yazarlar

Mahmut Koçak 0000-0001-7774-0144

Selim Çetin 0000-0002-9017-1465

Erken Görünüm Tarihi 8 Mayıs 2024
Yayımlanma Tarihi 8 Mayıs 2024
Gönderilme Tarihi 8 Nisan 2024
Kabul Tarihi 8 Mayıs 2024
Yayımlandığı Sayı Yıl 2024 Cilt: 7 Sayı: 1

Kaynak Göster

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. Mayıs 2024;7(1):45-50. doi:10.33187/jmsm.1466687
Chicago Koçak, Mahmut, ve Selim Çetin. “Higher Dimensional Leibniz-Rinehart Algebras”. Journal of Mathematical Sciences and Modelling 7, sy. 1 (Mayıs 2024): 45-50. https://doi.org/10.33187/jmsm.1466687.
EndNote Koçak M, Çetin S (01 Mayıs 2024) Higher Dimensional Leibniz-Rinehart Algebras. Journal of Mathematical Sciences and Modelling 7 1 45–50.
IEEE M. Koçak ve S. Çetin, “Higher Dimensional Leibniz-Rinehart Algebras”, Journal of Mathematical Sciences and Modelling, c. 7, sy. 1, ss. 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 (Mayıs 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 ve Selim Çetin. “Higher Dimensional Leibniz-Rinehart Algebras”. Journal of Mathematical Sciences and Modelling, c. 7, sy. 1, 2024, ss. 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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