Universal central extensions of braided crossed modules of Lie algebras

Yıl 2022, Cilt: 51 Sayı: 4, 1013 - 1028, 01.08.2022

Alejandro Fernández-fariña Manuel Ladra

https://doi.org/10.15672/hujms.901199

Cited By: 1

Öz

In this paper, we give a natural braiding on the universal central extension of a Lie crossed module with a given braiding in the category of Lie crossed modules. We also construct the universal central extension of a braided Lie crossed module in the category of braided Lie crossed modules, showing that, when one of these constructions exists, both of them exist and coincide.

Anahtar Kelimeler

Lie algebra, Lie crossed module, braided, universal central extension

Destekleyen Kurum

Agencia Estatal de Investigación, Xunta de Galicia

Proje Numarası

MTM2016-79661-P, ED431C 2019/10, ED481A-2017/064

Kaynakça

• [1] R. Brown and J.L. Loday, Van Kampen theorems for diagrams of spaces, Topology, 26 (3), 311–335, 1987.
• [2] J.M. Casas and M. Ladra, Perfect crossed modules in Lie algebras, Comm. Algebra, 23 (5), 1625–1644, 1995.
• [3] J.M. Casas and T. Van der Linden, Universal central extensions in semi-abelian categories, Appl. Categ. Struct. 22 (1), 253–268, 2014.
• [4] D. Conduché, Modules croisés généralisés de longueur 2, J. Pure Appl. Algebra, 34 (2- 3), 155–178, 1984.
• [5] G.J. Ellis, A nonabelian tensor product of Lie algebras, Glasgow Math. J. 33 (1), 101–120, 1991.
• [6] A. Fernández-Fariña and M. Ladra, Braiding for categorical algebras and crossed modules of algebras I: Associative and Lie algebras, J. Algebra Appl. 19 (9), 2050176, 30 pp., 2020.
• [7] A. Fernández-Fariña and M. Ladra, Braiding for categorical algebras and crossed modules of algebras II: Leibniz algebras, Filomat, 34 (5), 1443–1469, 2020.
• [8] T. Fukushi, Perfect braided crossed modules and their $mod-q$ analogues, Hokkaido Math. J. 27 (1), 135–146, 1998.
• [9] S.A. Huq, Commutator, nilpotency and solvability in categories, Quart. J. Math. Oxford, 2 (19), 363–389, 1968.
• [10] G. Janelidze and G.M. Kelly, Galois theory and a general notion of central extension, J. Pure Appl. Algebra, 97 (2), 135–161, 1994.
• [11] G. Janelidze, L. Márki and W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra, 168 (2-3), 367–386, 2002.
• [12] A. Joyal and R. Street, Braided monoidal categories, Macquarie Math. Reports No. 860081, 1986.
• [13] K.J. Norrie, Crossed modules and analogues of group theorems, Ph.D. thesis, King’s College, University of London, 1987.
• [14] E. Ulualan, Braiding for categorical and crossed Lie algebras and simplicial Lie algebras, Turkish J. Math. 31 (3), 239–255, 2007.