QUADRATIC MODULES OF LIE ALGEBRAS FIBRED OVER NIL(2)-MODULES OF LIE ALGEBRAS

Year 2022, Volume: 5 Issue: 2, 68 - 75, 31.07.2022

Hatice Taşbozan , Aydın Güzelkokar

https://doi.org/10.33773/jum.1089354

Abstract

In this work we illustrade that the forgetful functor mapping a quadratic module of Lie algebra to a nil(2)-module of Lie algebra is a fibration.

Keywords

Crossed module, Lie algebra, nil(2)-module, Quadratic module

References

• [1] J.H.C. Whitehead, Combinatorial Homotopy II , Bull. Amer. Math. Soc. 55, 453496, 1949.
• [2] D.M. Kan, A combinatorial definition of homotopy groups , Annals of Maths. 61, 288312, 1958.
• [3] D. Conduch, Modules croiss gnraliss de longueur 2, J. Pure and Applied Algebra , 34,(1984), 155-178.
• [4] G.J.Ellis,Homotopical aspects of Lie algebras, J. Austral. Math. Soc. (Series A),54, (1993), 393-419.
• [5] A. Grothendieck, Catgories cobress additives et complexe cotangent relatif. In: Lecture Notes in Mathematics, vol. 79. Springer, Berlin (1968).
• [6] D. Guin-Walery and J-L. Loday, Obsructiona l'excision en K-theorie algebrique, in: Algebraic K-Theory (Evanston 1980). Lecture Notes in Math. (1981), 179-216 (1981).
• [7] H.J. Baues, Combinatorial homotopy and 4-dimensional complexes, Walter de Gruyter, 15, 1991.
• [8] I. Aka and Z. Arvasi, Simplicial and crossed Lie algebras, Homology, Homotopy andApplications, Vol. 4 No.(1), (2002) ,43-57.
• [9] Z. Arvasi and E. Ulualan, : Quadratic and 2-crossed modules of algebras. Algebra Colloquium. (14), 669-686 (2007).
• [10] E.Ulualan and E.. Uslu, ,Quadratic modules for Lie algebras, Hacettepe Journal of Mathematics and Statistics, Vol40, (3), (2010), 409-419.
• [11] H. Atik and E. Ulualan,Quadratic modules bibred over nil (2)-modules, Journal of Homotopy and Related Structures, 121, (2017), 83-108.
• [12] K. Ylmaz,aprazlanm Kareler iin Bir Fibrasyon Uygulamas, Cumhuriyet Science Journal 391, (2018): 1-6.[13] K. Ylmaz and E.S. Ylmaz, Baues cobration for quadratic modules of Lie algebras, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, Vol 682, (2019), 1653-1663.
• [14] E.S. Ylmaz and K. Ylmaz On Crossed Squares of Commutative Algebras, Math. Sci. Appl. E-Notes, 82, (2020), 32-41.
• [15] M. Gerstenhaber, A uniform cohomology theory for algebras, Proceedings