TY - JOUR T1 - A non-abelian tensor product of algebras withbracket AU - Casas, José Manuel AU - Khmaladze, Emzar AU - Ladra, Manuel PY - 2025 DA - August Y2 - 2024 DO - 10.15672/hujms.1535583 JF - Hacettepe Journal of Mathematics and Statistics PB - Hacettepe University WT - DergiPark SN - 2651-477X SP - 1395 EP - 1409 VL - 54 IS - 4 LA - en AB - 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. KW - algebras with bracket KW - associative algebras KW - non-abelian tensor product KW - crossed modules KW - Hochschild homology CR - [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. CR - [2] F. Borceux and D. Bourn, Mal’cev, protomodular, homological and semi-abelian categories, Math. Appl. 566, Kluwer Academic Publishers, Dordrecht, 2004. CR - [3] R. Brown and J.-L. Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (3), 311–335, 1987. CR - [4] J.M. Casas, Homology with trivial coefficients and universal central extension of algebras with bracket, Comm. Algebra 35 (8), 2431–2449, 2007. CR - [5] J M. Casas, On solvability and nilpotency of algebras with bracket, J. Korean Math. Soc. 54 (2), 647–662, 2017. CR - [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. CR - [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. CR - [8] J.M. Casas and T. Pirashvili, Algebras with bracket, Manuscripta Math. 119 (1), 1–15, 2006. CR - [9] J.M. Casas and T. Van der Linden, Universal central extensions in semi-abelian categories, Appl. Categ. Structures 22 (1), 253–268, 2014. CR - [10] P. Dedecker and A.S.-T. Lue, A nonabelian two-dimensional cohomology for associative algebras, Bull. Amer. Math. Soc. 72, 1044–1050, 1966. CR - [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. CR - [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. CR - [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. CR - [14] G.J. Ellis, A non-abelian tensor product of Lie algebras, Glasgow Math. J. 33 (1), 101–120, 1991. CR - [15] X. García-Martínez, E. Khmaladze and M. Ladra, Non-abelian tensor product and homology of Lie superalgebras, J. Algebra 440, 464–488, 2015. CR - [16] A.V. Gnedbaye, A non-abelian tensor product of Leibniz algebras, Ann. Inst. Fourier (Grenoble) 49, 1149–1177, 1999. CR - [17] D. Guin, Cohomologie et homologie non-abéliennes des groupes, J. Pure Appl. Algebra 50, 109–137, 1988. CR - [18] D. Guin, Cohomologie des algèbres de Lie croisées et K-théorie de Milnor additive, Ann. Inst. Fourier (Grenoble) 45, 93–118, 1995. CR - [19] P.J. Higgins, Groups with multiple operators, Proc. London Math. Soc. 3 (3), 366–416, 1956. CR - [20] H. Inassaridze and N. Inassaridze, Non-abelian homology of groups, K-Theory J. 18, 1–17, 1999. CR - [21] N. Inassaridze, Nonabelian tensor products and nonabelian homology of groups, J. Pure Appl. Algebra 112 (2), 191–205, 1996. CR - [22] G. Janelidze, L. Márki and W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra 168 (2-3), 367–386, 2002. CR - [23] I. V. Kanatchikov, On field theoretic generalizations of a Poisson algebras, Rep. Math. Phys. 40 (2), 225–234, 1997. CR - [24] J.-L. Loday, Spaces with finitely many non-trivial homotopy groups, J. Pure Appl. Algebra 24 (2), 179–202, 1982. CR - [25] F.I. Michael,A note on the Five Lemma, Appl. Categ. Structures 21 (5), 441–448, 2013. CR - [26] D. Quillen, On the (Co-)homology of commutative rings, Proc. Sympos. Pure Math. 17, 65–87, 1970. CR - [27] Ch.A. Weibel, An introduction to homological algebra, Cambridge Stud. Adv. Math. 38, Cambridge University Press, Cambridge, 1994. UR - https://doi.org/10.15672/hujms.1535583 L1 - https://dergipark.org.tr/en/download/article-file/4155387 ER -