SavedAdd your journal

Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 54 Issue: 4

A non-abelian tensor product of algebras with bracket

José Manuel Casas(CITMAga)0000-0002-6556-6131
Emzar Khmaladze(The University of Georgia)0000-0001-9492-982X
Manuel Ladra(Universidad de Santiago de Compostela)0000-0002-0543-4508
Pages
—
Publication Date
2025-08-29
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

We investigate its applications to the universal central extensions and the low-dimensional homology of perfect algebras with bracket.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:We introduce and study a non-abelian tensor product of two algebras with bracket with compatible actions on each other.
📊
Key Findings & Data:We introduce and study a non-abelian tensor product of two algebras with bracket with compatible actions on each other.
💡
Conclusion & Impact:We investigate its applications to the universal central extensions and the low-dimensional homology of perfect algebras with bracket.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:algebras with bracket · associative algebras · non-abelian tensor product

Abstract

We introduce and study a non-abelian tensor product of two algebras with bracket with compatible actions on each other. We investigate its applications to the universal central extensions and the low-dimensional homology of perfect algebras with bracket.

🔖

Cite This Article

Casas, J. M., Khmaladze, E., Ladra, M. (2025). A non-abelian tensor product of algebras with bracket. Hacettepe Journal of Mathematics and Statistics, Volume: 54 Issue: 4. https://doi.org/10.15672/hujms.1535583

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (27)

  1. M. Barr and J. Beck, Homology and standard constructions, Seminar on triples and categorical homology theory (ETH, Zürich, 1966/67), Lecture Notes in Math. 80, 245–335, 1969.
  2. F. Borceux and D. Bourn, Mal’cev, protomodular, homological and semi-abelian categories, Math. Appl. 566, Kluwer Academic Publishers, Dordrecht, 2004.
  3. R. Brown and J.-L. Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (3), 311–335, 1987.
  4. J.M. Casas, Homology with trivial coefficients and universal central extension of algebras with bracket, Comm. Algebra 35 (8), 2431–2449, 2007.
  5. J M. Casas, On solvability and nilpotency of algebras with bracket, J. Korean Math. Soc. 54 (2), 647–662, 2017.
  6. J.M. Casas, E. Khmaladze and M. Ladra, Wells-type exact sequence and crossed extensions of algebras with bracket, Forum Math. 36 (6), 15651584, 2024.
  7. J.M. Casas, E. Khmaladze and N. Pacheco Rego, A non-abelian Hom-Leibniz tensor product and applications, Linear Multilinear Algebra 66 (6), 1133–1152, 2018.
  8. J.M. Casas and T. Pirashvili, Algebras with bracket, Manuscripta Math. 119 (1), 1–15, 2006.
  9. J.M. Casas and T. Van der Linden, Universal central extensions in semi-abelian categories, Appl. Categ. Structures 22 (1), 253–268, 2014.
  10. P. Dedecker and A.S.-T. Lue, A nonabelian two-dimensional cohomology for associative algebras, Bull. Amer. Math. Soc. 72, 1044–1050, 1966.
  11. D. di Micco and T. Van der Linden, An intrinsic approach to the non-abelian tensor product via internal crossed squares, Theory Appl. Categ. 35, 1268–1311, 2020.
  12. G. Donadze, N. Inassaridze, E. Khmaladze and M. Ladra, Cyclic homologies of crossed modules of algebras, J. Noncommut. Geom. 6 (4), 749–771, 2012.
  13. G.J. Ellis, Non-abelian exterior product of Lie algebras and an exact sequence in the homology of Lie algebras, J. Pure Appl. Algebra 46, 111–115, 1987.
  14. G.J. Ellis, A non-abelian tensor product of Lie algebras, Glasgow Math. J. 33 (1), 101–120, 1991.
  15. X. García-Martínez, E. Khmaladze and M. Ladra, Non-abelian tensor product and homology of Lie superalgebras, J. Algebra 440, 464–488, 2015.
  16. A.V. Gnedbaye, A non-abelian tensor product of Leibniz algebras, Ann. Inst. Fourier (Grenoble) 49, 1149–1177, 1999.
  17. D. Guin, Cohomologie et homologie non-abéliennes des groupes, J. Pure Appl. Algebra 50, 109–137, 1988.
  18. D. Guin, Cohomologie des algèbres de Lie croisées et K-théorie de Milnor additive, Ann. Inst. Fourier (Grenoble) 45, 93–118, 1995.
  19. P.J. Higgins, Groups with multiple operators, Proc. London Math. Soc. 3 (3), 366–416, 1956.
  20. H. Inassaridze and N. Inassaridze, Non-abelian homology of groups, K-Theory J. 18, 1–17, 1999.
  21. N. Inassaridze, Nonabelian tensor products and nonabelian homology of groups, J. Pure Appl. Algebra 112 (2), 191–205, 1996.
  22. G. Janelidze, L. Márki and W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra 168 (2-3), 367–386, 2002.
  23. I. V. Kanatchikov, On field theoretic generalizations of a Poisson algebras, Rep. Math. Phys. 40 (2), 225–234, 1997.
  24. J.-L. Loday, Spaces with finitely many non-trivial homotopy groups, J. Pure Appl. Algebra 24 (2), 179–202, 1982.
  25. F.I. Michael,A note on the Five Lemma, Appl. Categ. Structures 21 (5), 441–448, 2013.
  26. D. Quillen, On the (Co-)homology of commutative rings, Proc. Sympos. Pure Math. 17, 65–87, 1970.
  27. Ch.A. Weibel, An introduction to homological algebra, Cambridge Stud. Adv. Math. 38, Cambridge University Press, Cambridge, 1994.