Research ArticleOpen AccessGoogle Scholar indexed
The Morita Equivalence for the A ∞ -Algebras
Department of Mathematics, Al Jamum University College, Umm Al-Qura University, Makkah, Kingdom of Saudi Arabia
- 1 Department of Mathematics, Al Jamum University College, Umm Al-Qura University, Makkah, Kingdom of Saudi Arabia
Advances in Pure Mathematics·Volume 15 (2025)·Pages 483–490·Published 11 July 2025·DOI10.4236/apm.2025.157023
Copy link · social · email
Abstract
This paper explores the foundational and advanced aspects of A ∞ -algebras. We present a comprehensive study of A ∞ -algebras and their homology, emphasizing the construction of long exact sequences in simplicial homology and their implications for short exact sequences of A ∞ -algebras. Furthermore, we investigate the trace and inclusion maps in matrix A ∞ -algebras, proving their mutual invertibility and highlighting their role in preserving homological properties. These results underscore the utility of A ∞ -algebras in simplifying complex algebraic systems while maintaining essential structural invariants.
KeywordsA-InfinityHomologyInclusion MapTrace Map
- Stasheff, J.D. (1963) Homotopy Associativity of H-Spaces. II. Transactions of the American Mathematical Society , 108, 293-312. https://doi.org/10.2307/1993609
- Noreldeen Mohamed, A.H. (2020) Perturbation Differential A-Infinity Algebra. Applied Mathematics & Information Sciences , 14, 447-451. https://doi.org/10.18576/amis/140311
- Noreldeen, A.H. (2019) On the Hochschild Cohomology Theory of A ∞ -Algebra. Scientific African , 5, e00115. https://doi.org/10.1016/j.sciaf.2019.e00115
- Keller, B. (2006) A-Infinity Algebras, Modules and Functor Categories. Contemporary Mathematices , 406, 67-94. https://doi.org/10.1090/conm/406/07654
- Noreldeen, A.H. and Gh Gouda, Y. (2013) On the Simplicial Cohomology Theory of Algebra. Life Science Journal , 10, 2639-2644.
- Kozae, M., Quota, S.A.A. and Noreldeen, A.H. (2022) The Relative (Co)Homology Theory through Operator Algebras. Mathematics and Statistics , 10, 468-476. https://doi.org/10.13189/ms.2022.100302
- Noreldeen, A.H. (2012) On the Homology Theory of Operator Algebras. International Journal of Mathematics and Mathematical Sciences , 2012, Article ID: 368527. https://doi.org/10.1155/2012/368527
- Adhikari, M.R. (2016) Homology and Cohomology Theories. In: Adhikari, M.R., Ed., Basic Algebraic Topology and Its Applications , Springer, 347-406. https://doi.org/10.1007/978-81-322-2843-1_10
- Lapin, S.V. (2002) D ∞ -Differential A ∞ -Algebras and Spectral Sequences. Sbornik : Mathematics , 193, 119-142. https://doi.org/10.1070/sm2002v193n01abeh000623
- Keller, B. (2001) Introduction to A ∞ -Infinity Algebras and Modules. Homology , Homotopy and Applications , 3, 1-35. https://doi.org/10.4310/hha.2001.v3.n1.a1
- Hochschild, G. (1945) On the Cohomology Groups of an Associative Algebra. The Annals of Mathematics , 46, 58-67. https://doi.org/10.2307/1969145
- Kadeishvili, T.V. (1988) The-Algebra Structure and the Hochschild and Harrison Cohomologies. arXiv: math/0210331. https://doi.org/10.48550/arXiv.math/0210331
- Cartan, H. and Eilenberg, S. (1956) Homological Algebra. Princeton University Press. https://doi.org/10.1515/9781400883844