Pure and Applied Mathematics Journal

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Classification, α-Inner Derivations and α-Centroids of Finite-Dimensional Complex Hom-Trialgebras

Received: 6 November 2023    Accepted: 5 December 2023    Published: 25 December 2023
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Abstract

This article focuses on the classification, α-inner-derivations and α-centroids of complex Hom-trialgebras up to dimension three. The initial research on these algebras was conducted by Loday and Ronco, and this paper builds upon their work by utilizing computer algebra software (Mathematica) to analyze the equations that define the structure constants. Furthermore, we explore the concept of α-inner-derivations and α-centroids of complex Hom-trialgebras. The findings reveal that for 2- and 3-dimensional algebras, there is only one trivial α-inner-derivation. However, there exist 23 non-isomorphic α-inner-derivations for 2- and 3-dimensional algebras. Regarding α-centroids, we identify trivial isomorphism classes for 2- and 3-dimensional Hom-trialgebras. Additionally, there are 11 non-isomorphic classes for 2-dimensional Hom-trialgebras and 19 for 3-dimensional algebras. The range of dimensions for both α-inner-derivations and α-centroids spans from 0 to 3.

DOI 10.11648/j.pamj.20231205.12
Published in Pure and Applied Mathematics Journal (Volume 12, Issue 5, October 2023)
Page(s) 86-97
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Hom-Associative Trialgebra, Classification, α-Inner-derivation, α-Centroid

References
[1] Bai, Y. X., Bokut, L. A., Chen, Y. Q., Zhang, Z. R. (2023). Centroid Hom-associative Algebras and Centroid Hom- Lie Algebras. Acta Mathematica Sinica, English Series, 1-27.
[2] Bakayoko, I. and Diallo, O. W. (2015) Some generalized Hom-algebra structures. J. Gen. Lie Theory Appl. 9, 226.
[3] Basri. W, Rakhimov I. S and Rokhsiboev I. M Four-Dimension Nilpotent Diassociative algebras, J. Generalised Lie Theory Appl. 28. doi: 10.4172/1736- 4337.1000218.
[4] Ebrahimi-Fard, K. (2002). Loday-type algebras and the RotaâBaxter relation. Letters in Mathematical Physics, 61, 139-147.
[5] Fiidow, M. A., Mohammed, N. F., Rakhimov, I. S., Husain, S. K. S., Husain, S. K. S., and Basri, W. (2010). On inner derivations of finite dimensional associative algebras. Education, 2012.
[6] Fiidow, M. A., Rakhimov, I. S., Husain, S. S. (2015, August). Centroids and derivations of associative algebras. In 2015 International Conference on Research and Education in Mathematics (ICREM7) (pp. 227-232). IEEE.
[7] Loday, J. L., Chapoton, F., Frabetti, A., Goichot, F., Loday, J. L. (2001). Dialgebras (pp. 7-66). Springer Berlin Heidelberg.
[8] Loday, J. L., Ronco, M. (2004). Trialgebras and families of polytopes. Contemporary Mathematics, 346, 369-398.
[9] Makhlouf, A., Silvestrov, S. (2006). On Hom-algebra structures. arXiv preprint math/0609501.
[10] Makhlouf, A., Zahari, A. (2020). Structure and classification of Hom-associative algebras. Acta et Commentationes Universitatis Tartuensis de Mathematica, 24 (1), 79-102.
[11] Mosbahi, B., Asif, S., and Zahari, A. (2023). Classification of tridendriform algebra and related structures. arXiv preprint arXiv: 2305.08513.
[12] Pozhidaev A. P, (2008). Dialgebras and related triple systems. Siberian Mathematical Journal, 49 (4), 696-708.
[13] Rakhimov, I. S and Fiidov M. A. (2015) Centroids of finite dimensional associative dialgebras, Far East Journal of Mathematical Sciences, 98 (4), 427-443.
[14] Zahari. A and Bakayoko. I (2023) On BiHom- Associative dialgebras, Open J. Math. Sci. vol (7), 96- 117.
[15] Zahari, A., Mosbahi, B., and Basdouri, I. (2023). Classification, Derivations and Centroids of Low- Dimensional Real Trialgebras. arXiv preprint arXiv: 2304.04251. [math.RA].
[16] Zahari, A., Mosbahi, B., and Basdouri, I. (2023). Classification, Derivations and Centroids of Low- Dimensional Complex BiHom-Trialgebras. arXiv preprint arXiv: 2304.06781. [math.RA].
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    Mosbahi, B., Zahari, A., Basdouri, I. (2023). Classification, α-Inner Derivations and α-Centroids of Finite-Dimensional Complex Hom-Trialgebras. Pure and Applied Mathematics Journal, 12(5), 86-97. https://doi.org/10.11648/j.pamj.20231205.12

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    ACS Style

    Mosbahi, B.; Zahari, A.; Basdouri, I. Classification, α-Inner Derivations and α-Centroids of Finite-Dimensional Complex Hom-Trialgebras. Pure Appl. Math. J. 2023, 12(5), 86-97. doi: 10.11648/j.pamj.20231205.12

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    AMA Style

    Mosbahi B, Zahari A, Basdouri I. Classification, α-Inner Derivations and α-Centroids of Finite-Dimensional Complex Hom-Trialgebras. Pure Appl Math J. 2023;12(5):86-97. doi: 10.11648/j.pamj.20231205.12

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  • @article{10.11648/j.pamj.20231205.12,
      author = {Bouzid Mosbahi and Ahmed Zahari and Imed Basdouri},
      title = {Classification, α-Inner Derivations and α-Centroids of Finite-Dimensional Complex Hom-Trialgebras},
      journal = {Pure and Applied Mathematics Journal},
      volume = {12},
      number = {5},
      pages = {86-97},
      doi = {10.11648/j.pamj.20231205.12},
      url = {https://doi.org/10.11648/j.pamj.20231205.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20231205.12},
      abstract = {This article focuses on the classification, α-inner-derivations and α-centroids of complex Hom-trialgebras up to dimension three. The initial research on these algebras was conducted by Loday and Ronco, and this paper builds upon their work by utilizing computer algebra software (Mathematica) to analyze the equations that define the structure constants. Furthermore, we explore the concept of α-inner-derivations and α-centroids of complex Hom-trialgebras. The findings reveal that for 2- and 3-dimensional algebras, there is only one trivial α-inner-derivation. However, there exist 23 non-isomorphic α-inner-derivations for 2- and 3-dimensional algebras. Regarding α-centroids, we identify trivial isomorphism classes for 2- and 3-dimensional Hom-trialgebras. Additionally, there are 11 non-isomorphic classes for 2-dimensional Hom-trialgebras and 19 for 3-dimensional algebras. The range of dimensions for both α-inner-derivations and α-centroids spans from 0 to 3.
    },
     year = {2023}
    }
    

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    T1  - Classification, α-Inner Derivations and α-Centroids of Finite-Dimensional Complex Hom-Trialgebras
    AU  - Bouzid Mosbahi
    AU  - Ahmed Zahari
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    JF  - Pure and Applied Mathematics Journal
    JO  - Pure and Applied Mathematics Journal
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    PB  - Science Publishing Group
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    AB  - This article focuses on the classification, α-inner-derivations and α-centroids of complex Hom-trialgebras up to dimension three. The initial research on these algebras was conducted by Loday and Ronco, and this paper builds upon their work by utilizing computer algebra software (Mathematica) to analyze the equations that define the structure constants. Furthermore, we explore the concept of α-inner-derivations and α-centroids of complex Hom-trialgebras. The findings reveal that for 2- and 3-dimensional algebras, there is only one trivial α-inner-derivation. However, there exist 23 non-isomorphic α-inner-derivations for 2- and 3-dimensional algebras. Regarding α-centroids, we identify trivial isomorphism classes for 2- and 3-dimensional Hom-trialgebras. Additionally, there are 11 non-isomorphic classes for 2-dimensional Hom-trialgebras and 19 for 3-dimensional algebras. The range of dimensions for both α-inner-derivations and α-centroids spans from 0 to 3.
    
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Author Information
  • Department of Mathematics, Faculty of Sciences, University of Sfax, Sfax, Tunisia

  • IRIMAS-Department of Mathematics, Faculty of Sciences, University of Haute Alsace, Mulhouse, France

  • Department of Mathematics, Faculty of Sciences, University of Gafsa, Gafsa, Tunisia

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