İlgili Yapılarla Simplisel Lie-Rinehart Cebirler

Year 2025, Volume: 15 Issue: 1, 56 - 66, 30.06.2025

Ali Aytekin , Mahmut Koçak

https://doi.org/10.54370/ordubtd.1581038

Abstract

Bu çalışmada, simplisel Lie-Rinehart cebirler ve Lie-Rinehart cat^1 -cebirler tanımlanacaktır. Bu tanımlamalar yardımıyla, Lie-Rinehart çaprazlanmış modüller, cat^1-cebirler ve simplisel Lie-Rinehart cebirler arasındaki ilişki açıklanacaktır.

Keywords

Lie-Rinehart cebir , simplisel obje , çaprazlanmış modül

Simplicial Lie-Rinehart Algebras with Related Structures

Abstract

In this paper, simplicial Lie-Rinehart algebras and Lie-Rinehart cat^1-algebras will be defined. With the help of these definitions, the relations between Lie-Rinehart crossed modules, cat^1-algebras, and simplicial Lie-Rinehart algebras will be explained.

Keywords

Lie-Rinehart algebra , simplicial object , crossed module

Ethical Statement

There are no ethical issues regarding the publication of this article.

References

• Alp, M. (1998). Pullbacks of crossed modules and cat1-groups. Turkish Journal of Mathematics, 22, 273 – 281. https://journals.tubitak.gov.tr/math/vol22/iss3/2
• Alp, M., & Gürmen Alansal, Ö. (2003). Pushouts of profinite crossed module and cat1 profinite groups. Turkish Journal of Mathematics, 27(4), 539–548. https://journals.tubitak.gov.tr/math/vol27/iss4/6
• Arvasi, Z. (1997). Crossed squares and 2 crossed modules of commutative algebras . Theory and Applications of Categories, 3, 160–181. ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1997/n7/n7
• Arvasi, Z., & Akça, İ. İ. (2002). Simplicial and Crossed Lie Algebras. Homology Homotopy and Applications, 4(1), 43–57. ftp://ftp.rmi.acnet.ge/pub/hha/volumes/2002/n1a4/v4n1a4
• Arvasi, Z., & Odabaş, A. (2016). Computing 2 dimensional algebras Crossed modules and Cat1-algebras. Journal of Algebra and Its Applications, 15(10), 1650185–0. https://doi.org/10.1142/S0219498816501851
• Aytekin Arıcı, G., & Şahan, T. (2022). Coverings and liftings of generalized crossed modules. Categories and General Algebraic Structures with Applications, 17(1), 117-140. https://doi.org/10.52547/cgasa.17.1.117
• Casas, J. M. (2011). Obstructions to Lie–Rinehart Algebra Extensions. In Algebra Colloquium, 18(01),83-104. https://doi.org/10.1142/S1005386711000046
• Casas, J. M., Ladra, M., & Pirashvili, T. (2004). Crossed modules for Lie–Rinehart algebras. Journal of Algebra, 274(1), 192-201. https://doi.org/10.1016/j.jalgebra.2003.10.001
• Casas, J. M., Ladra, M., & Pirashvili, T. (2005). Triple cohomology of Lie–Rinehart algebras and the canonical class of associative algebras. Journal of Algebra, 291(1), 144-163. https://doi.org/10.1016/j.jalgebra.2005.05.018
• Curtis, E. B. (1971). Simplicial homotopy theory. Advances in Mathematics, 6(2), 107-209.
• Goerss, P. G., & Jardine, J. F. (2009). Simplicial homotopy theory. Springer Science & Business Media.
• Gürmen Alansal, Ö. (2023). Ideal and factor conditions for crossed modules and algebra-algebroids. Hacettepe Journal of Mathematics and Statistics, 52(3), 698-707. https://doi.org/10.15672/hujms.1131802
• Herz, J. C. (1953). Pseudo-algebres de Lie. 1. Comptes Rendus Hebdomadaıres Des Seances De l Academıe Des Scıences, 236(20), 1935-1937.
• Huebschmann, J. (1990). Poisson cohomology and quantization. Journal Für Die Reine und Angewandte Mathematik, 408, 57-113.
• Loday, J. L. (1982). Spaces with finitely many non-trivial homotopy groups. Journal of Pure and Applied Algebra, 24, 179 – 202.
• Mackenzie, K. (1987). Lie groupoids and Lie algebroids in differential geometry (Vol. 124). Cambridge University press.
• Odabaş, A., Uslu, E. Ö., & Ilgaz Çağlayan, E. (2016). Isoclinism of crossed modules. Journal of Symbolic Computation, 74, 408–424. https://doi.org/10.1016/j.jsc.2015.08.006
• Şahan, T. (2019). Further remarks on liftings of crossed modules. Hacettepe Journal of Mathematics and Statistics, 48(3), 743–752. https://doi.org/10.15672/hjms.2018.554
• Şahan, T., & Kendir, E. (2023). Çaprazlanmış Cat1-Modüller. Journal of the Institute of Science and Technology, 13(4), 2958-2972. https://doi.org/10.21597/jist.1303212
• Temel, S. (2019). Crossed semimodules and cat^1-monoids. The Korean Journal of Mathematics, 27(2), 535–545. https://doi.org/10.11568/kjm.2019.27.2.535
• Whitehead, J.H.C. (1949). Combinatorial homotopy. II. Bulletin of the American Mathematical Society, 55, 213 – 245.