Actor of a crossed module of dialgebras via tetramultipliers

Abstract

We study the representability of actions in the category of crossed modules of dialgebras via tetramultipliers. We deduce a pair of dialgebras in order to construct an object which, under certain circumstances, is the actor (also known as the split extension classifier). Moreover, we give give a full description of actions in terms of equations. Finally, we check that under the aforementioned circumstances, the center coincides with the kernel of the canonical map from a crossed module to its actor.

Keywords

• Dialgebra
• crossed module
• tetramultiplier
• representation
• actor

References

1. [1] J.C. Baez and A.S. Crans, Higher-dimensional algebra. VI. Lie 2-algebras, Theory Appl. Categ. 12, 492–538, 2004.
2. [2] F. Borceux, G. Janelidze and G.M. Kelly, On the representability of actions in a semi-abelian category, Theory Appl. Categ. 14, 244–286, 2005.
3. [3] Y. Boyaci, J. M. Casas, T. Datuashvili and E.Ö. Uslu, Actions in modified categories of interest with application to crossed modules, Theory Appl. Categ. 30, 882–908, 2015.
4. [4] R. Brown, Groupoids and crossed objects in algebraic topology, Homology Homotopy Appl. 1, 1–78, 1999.
5. [5] R. Brown and C.B. Spencer, G-groupoids, crossed modules and the fundamental groupoid of a topological group, Nederl. Akad. Wetensch. Proc. Ser. A. 79, 296–302, 1976.
6. [6] J.M. Casas, T. Datuashvili and M. Ladra, Universal strict general actors and actors in categories of interest, Appl. Categ. Structures 18 (1), 85–114, 2010.
7. [7] J.M. Casas, R. Fernández-Casado, X. García-Martínez and E. Khmaladze, Actor of a crossed module of Leibniz algebras, Theory Appl. Categ. 33, 23–42, 2018.
8. [8] J.M. Casas, R. Fernández-Casado, E. Khmaladze and M. Ladra, Universal enveloping crossed module of a Lie crossed module, Homology Homotopy Appl. 16 (2), 143–158, 2014.

9. [9] J.M. Casas, R. Fernández-Casado, E. Khmaladze and M. Ladra, More on crossed modules in Lie, Leibniz, associative and diassociative algebrs, J. Algebra Appl. 16 (6), 1750107 (17 pp.), 2017.
10. [10] J.M. Casas, N. Inassaridze, E. Khmaladze and M. Ladra, Adjunction between crossed modules of groups and algebras, J. Homotopy Relat. Struct. 9 (1), 223–237, 2014.
11. [11] Sh. Chen, Y. Sheng and Z. Zheng, Non-abelian extensions of Lie 2-algebras, Sci. China Math. 55 (8), 1655–1668, 2012.
12. [12] P. Dedecker and S.-T. Lue, A nonabelian two-dimensional cohomology for associative algebras, Bull. Amer. Math. Soc. 72 1044–1050, 1966.
13. [13] R. Fernández-Casado, Relations between crossed modules of different algebras, Ph.D. thesis, Universidade de Santiago de Compostela, 2015.
14. [14] X. García-Martínez and T. Van der Linden, A characterisation of Lie algebras amongst anti-commutative algebras, J. Pure Appl. Algebra 223(11), 4857–4870, 2019.
15. [15] X. García-Martínez and T. Van der Linden, A characterisation of Lie algebras via algebraic exponentiation, Adv. Math. 341, 92–117, 2019.
16. [16] X. García-Martínez, M. Tsishyn, T. Van der Linden and C. Vienne, Algebras with representable representations, To appear in Proc. Edinburgh Math. Soc., 2021.
17. [17] G. Hochschild, Cohomology and representations of associative algebras, Duke Math. J. 14, 921–948, 1947.
18. [18] S.A. Huq, Commutator, nilpotency, and solvability in categories, Quart. J. Math. Oxford Ser. (2), 19, 363–389, 1968.
19. [19] G. Janelidze, Internal crossed modules, Georgian Math. J. 10 (1), 99–114, 2003.
20. [20] C. Kassel and J.-L. Loday, Extensions centrales dalgèbres de Lie, Ann. Inst. Fourier (Grenoble), 32, 119–142, 1982.
21. [21] E. Khmaladze, On associative and Lie 2-algebras, Proc. A. Razmadze Math. Inst. 159, 57–64, 2012.
22. [22] J.-L. Loday, Dialgebras, Dialgebras and related operads, Lecture Notes in Math., vol. 1763, Springer, Berlin, 2001, pp. 7–66.
23. [23] J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann. 296(1), 139–158, 1993.
24. [24] S. Mac Lane, Extensions and obstructions for rings, Illinois J. Math. 2, 316–345, 1958.
25. [25] N. Martins-Ferreira, A. Montoli and M. Sobral, Semidirect products and crossed modules in monoids with operations, J. Pure Appl. Algebra, 217(2), 334–347, 2013.
26. [26] G. Orzech, Obstruction theory in algebraic categories. I, II, J. Pure Appl. Algebra, 2, 287–314, 1972; ibid. 2 1972, 315–340.
27. [27] A. Patchkoria, Crossed semimodules and Schreier internal categories in the category of monoids, Georgian Math. J. 5 (6), 575–581, 1998.
28. [28] T. Porter, Extensions, crossed modules and internal categories in categories of groups with operations, Proc. Edinburgh Math. Soc. (2) 30 (3), 373–381, 1987.
29. [29] N.M. Shammu, Algebraic and categorical structure of categories of crossed modules of algebras, Ph.D. thesis, University of Bangor, 1992.
30. [30] Y. Sheng and Z. Liu, Leibniz 2-algebras and twisted Courant algebroids, Comm. Algebra, 41 (5), 1929–1953, 2013.
31. [31] R. Tang and Y. Sheng, Cohomological characterizations of non-abelian extensions of strict Lie 2-algebras, J. Geom. Phys. 144, 294–307, 2019.
32. [32] E.Ö. Uslu, S. Çetin and A.F. Arslan, On crossed modules in modified categories of interest, Math. Commun. 22 (1), 103–119, 2017.
33. [33] J.H.C. Whitehead, Combinatorial homotopy II, Bull. Amer. Math. Soc. 55, 453–496, 1949.