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Actor of a crossed module of dialgebras via tetramultipliers

Yıl 2021, Cilt: 50 Sayı: 4, 1063 - 1078, 06.08.2021
https://doi.org/10.15672/hujms.701217

Öz

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.

Destekleyen Kurum

Ministerio de Economía y Competitividad

Proje Numarası

MTM2016-79661-P

Kaynakça

  • [1] J.C. Baez and A.S. Crans, Higher-dimensional algebra. VI. Lie 2-algebras, Theory Appl. Categ. 12, 492–538, 2004.
  • [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] 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] R. Brown, Groupoids and crossed objects in algebraic topology, Homology Homotopy Appl. 1, 1–78, 1999.
  • [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] 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] 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] 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] 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] 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] Sh. Chen, Y. Sheng and Z. Zheng, Non-abelian extensions of Lie 2-algebras, Sci. China Math. 55 (8), 1655–1668, 2012.
  • [12] P. Dedecker and S.-T. Lue, A nonabelian two-dimensional cohomology for associative algebras, Bull. Amer. Math. Soc. 72 1044–1050, 1966.
  • [13] R. Fernández-Casado, Relations between crossed modules of different algebras, Ph.D. thesis, Universidade de Santiago de Compostela, 2015.
  • [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] X. García-Martínez and T. Van der Linden, A characterisation of Lie algebras via algebraic exponentiation, Adv. Math. 341, 92–117, 2019.
  • [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] G. Hochschild, Cohomology and representations of associative algebras, Duke Math. J. 14, 921–948, 1947.
  • [18] S.A. Huq, Commutator, nilpotency, and solvability in categories, Quart. J. Math. Oxford Ser. (2), 19, 363–389, 1968.
  • [19] G. Janelidze, Internal crossed modules, Georgian Math. J. 10 (1), 99–114, 2003.
  • [20] C. Kassel and J.-L. Loday, Extensions centrales dalgèbres de Lie, Ann. Inst. Fourier (Grenoble), 32, 119–142, 1982.
  • [21] E. Khmaladze, On associative and Lie 2-algebras, Proc. A. Razmadze Math. Inst. 159, 57–64, 2012.
  • [22] J.-L. Loday, Dialgebras, Dialgebras and related operads, Lecture Notes in Math., vol. 1763, Springer, Berlin, 2001, pp. 7–66.
  • [23] J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann. 296(1), 139–158, 1993.
  • [24] S. Mac Lane, Extensions and obstructions for rings, Illinois J. Math. 2, 316–345, 1958.
  • [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] G. Orzech, Obstruction theory in algebraic categories. I, II, J. Pure Appl. Algebra, 2, 287–314, 1972; ibid. 2 1972, 315–340.
  • [27] A. Patchkoria, Crossed semimodules and Schreier internal categories in the category of monoids, Georgian Math. J. 5 (6), 575–581, 1998.
  • [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] N.M. Shammu, Algebraic and categorical structure of categories of crossed modules of algebras, Ph.D. thesis, University of Bangor, 1992.
  • [30] Y. Sheng and Z. Liu, Leibniz 2-algebras and twisted Courant algebroids, Comm. Algebra, 41 (5), 1929–1953, 2013.
  • [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] 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] J.H.C. Whitehead, Combinatorial homotopy II, Bull. Amer. Math. Soc. 55, 453–496, 1949.
Yıl 2021, Cilt: 50 Sayı: 4, 1063 - 1078, 06.08.2021
https://doi.org/10.15672/hujms.701217

Öz

Proje Numarası

MTM2016-79661-P

Kaynakça

  • [1] J.C. Baez and A.S. Crans, Higher-dimensional algebra. VI. Lie 2-algebras, Theory Appl. Categ. 12, 492–538, 2004.
  • [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] 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] R. Brown, Groupoids and crossed objects in algebraic topology, Homology Homotopy Appl. 1, 1–78, 1999.
  • [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] 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] 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] 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] 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] 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] Sh. Chen, Y. Sheng and Z. Zheng, Non-abelian extensions of Lie 2-algebras, Sci. China Math. 55 (8), 1655–1668, 2012.
  • [12] P. Dedecker and S.-T. Lue, A nonabelian two-dimensional cohomology for associative algebras, Bull. Amer. Math. Soc. 72 1044–1050, 1966.
  • [13] R. Fernández-Casado, Relations between crossed modules of different algebras, Ph.D. thesis, Universidade de Santiago de Compostela, 2015.
  • [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] X. García-Martínez and T. Van der Linden, A characterisation of Lie algebras via algebraic exponentiation, Adv. Math. 341, 92–117, 2019.
  • [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] G. Hochschild, Cohomology and representations of associative algebras, Duke Math. J. 14, 921–948, 1947.
  • [18] S.A. Huq, Commutator, nilpotency, and solvability in categories, Quart. J. Math. Oxford Ser. (2), 19, 363–389, 1968.
  • [19] G. Janelidze, Internal crossed modules, Georgian Math. J. 10 (1), 99–114, 2003.
  • [20] C. Kassel and J.-L. Loday, Extensions centrales dalgèbres de Lie, Ann. Inst. Fourier (Grenoble), 32, 119–142, 1982.
  • [21] E. Khmaladze, On associative and Lie 2-algebras, Proc. A. Razmadze Math. Inst. 159, 57–64, 2012.
  • [22] J.-L. Loday, Dialgebras, Dialgebras and related operads, Lecture Notes in Math., vol. 1763, Springer, Berlin, 2001, pp. 7–66.
  • [23] J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann. 296(1), 139–158, 1993.
  • [24] S. Mac Lane, Extensions and obstructions for rings, Illinois J. Math. 2, 316–345, 1958.
  • [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] G. Orzech, Obstruction theory in algebraic categories. I, II, J. Pure Appl. Algebra, 2, 287–314, 1972; ibid. 2 1972, 315–340.
  • [27] A. Patchkoria, Crossed semimodules and Schreier internal categories in the category of monoids, Georgian Math. J. 5 (6), 575–581, 1998.
  • [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] N.M. Shammu, Algebraic and categorical structure of categories of crossed modules of algebras, Ph.D. thesis, University of Bangor, 1992.
  • [30] Y. Sheng and Z. Liu, Leibniz 2-algebras and twisted Courant algebroids, Comm. Algebra, 41 (5), 1929–1953, 2013.
  • [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] 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] J.H.C. Whitehead, Combinatorial homotopy II, Bull. Amer. Math. Soc. 55, 453–496, 1949.
Toplam 33 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Matematik
Yazarlar

José Manuel Casas Bu kişi benim 0000-0002-6556-6131

Rafael Fernandez-casado 0000-0003-1347-6879

Xabier Garcia Martinez 0000-0003-1679-4047

Emzar Khmaladze Bu kişi benim 0000-0001-9492-982X

Proje Numarası MTM2016-79661-P
Yayımlanma Tarihi 6 Ağustos 2021
Yayımlandığı Sayı Yıl 2021 Cilt: 50 Sayı: 4

Kaynak Göster

APA Casas, J. M., Fernandez-casado, R., Garcia Martinez, X., Khmaladze, E. (2021). Actor of a crossed module of dialgebras via tetramultipliers. Hacettepe Journal of Mathematics and Statistics, 50(4), 1063-1078. https://doi.org/10.15672/hujms.701217
AMA Casas JM, Fernandez-casado R, Garcia Martinez X, Khmaladze E. Actor of a crossed module of dialgebras via tetramultipliers. Hacettepe Journal of Mathematics and Statistics. Ağustos 2021;50(4):1063-1078. doi:10.15672/hujms.701217
Chicago Casas, José Manuel, Rafael Fernandez-casado, Xabier Garcia Martinez, ve Emzar Khmaladze. “Actor of a Crossed Module of Dialgebras via Tetramultipliers”. Hacettepe Journal of Mathematics and Statistics 50, sy. 4 (Ağustos 2021): 1063-78. https://doi.org/10.15672/hujms.701217.
EndNote Casas JM, Fernandez-casado R, Garcia Martinez X, Khmaladze E (01 Ağustos 2021) Actor of a crossed module of dialgebras via tetramultipliers. Hacettepe Journal of Mathematics and Statistics 50 4 1063–1078.
IEEE J. M. Casas, R. Fernandez-casado, X. Garcia Martinez, ve E. Khmaladze, “Actor of a crossed module of dialgebras via tetramultipliers”, Hacettepe Journal of Mathematics and Statistics, c. 50, sy. 4, ss. 1063–1078, 2021, doi: 10.15672/hujms.701217.
ISNAD Casas, José Manuel vd. “Actor of a Crossed Module of Dialgebras via Tetramultipliers”. Hacettepe Journal of Mathematics and Statistics 50/4 (Ağustos 2021), 1063-1078. https://doi.org/10.15672/hujms.701217.
JAMA Casas JM, Fernandez-casado R, Garcia Martinez X, Khmaladze E. Actor of a crossed module of dialgebras via tetramultipliers. Hacettepe Journal of Mathematics and Statistics. 2021;50:1063–1078.
MLA Casas, José Manuel vd. “Actor of a Crossed Module of Dialgebras via Tetramultipliers”. Hacettepe Journal of Mathematics and Statistics, c. 50, sy. 4, 2021, ss. 1063-78, doi:10.15672/hujms.701217.
Vancouver Casas JM, Fernandez-casado R, Garcia Martinez X, Khmaladze E. Actor of a crossed module of dialgebras via tetramultipliers. Hacettepe Journal of Mathematics and Statistics. 2021;50(4):1063-78.