Homotopy of Lie-Rinehart Crossed Module Morphisms
Year 2019,
Volume: 9 Issue: 1, 202 - 212, 28.06.2019
Ayşe Çobankaya
Selim Çetin
Abstract
In this paper our aim is to give the concept homotopy of morphisms of Lie-Rinehart crossed modules. We show that the homotopy relation gives rise to an equivalence relation. Additionally a groupoid structure of Lie- Rinehart crossed module morphisms and their homotopies.
References
- [1] Akça, İ.İ., Sidal, Y., Homotopies of Lie crossed module morphisms, arXiv: 1609.09297v1[math.CT], 2016.[2] Brown, R., and Higgings, P. J., Tensor Products and Homotopies for w− groupoids andcrossed complexes, Journal of Pure and Applied Algebra 47, 1-33, 1987.[3] Cabello, J.G. and Garzon, A.R., Closed model structures foralgebraic models for n−types, Journal of Pure and Applied Algebra 103 (3), 287-302, 1995.[4] Casas, J.M., Ladra, M., Pirashvili, T., Crossed modules for Lie-Rinehart algebras,Journal of Algebra, 274, 192-201, 2004.[5] Casas, J.M., Ladra, M., Pirashvili, T., Triple cohomology of Lie-Rinehart algebras andthe canonical class of associative algebras, Journal of Algebra, 291, 144-163, 2005.[6] Herz, J., Pseudo-algbres de Lie, C. R. Acad. Sci. Paris, 236, 1935-1937, 1953.[7] Huebschmann, J., Poisson cohomology and quantization , J. Reine Angew Math.408,57-113, 1990.[8] Mackenzie, K., Lie Groupoids and Lie algebroids in differential geometry, London math.soc. lecture note ser., vol. 124, Cambridge Univ. Press, 1987.[9] Whitehead, J.H.C., Note on a previous paper entitled On adding relations to homotopygroups, Ann. of Math. (2) , 47, 806-810, 1946.[10] Whitehead, J.H.C., Combinatorial Homotopy I and II, Bull. Amer. Math. Soc., 55,231-245 and 453-456, 1949.[11] Whitehead, J.H.C., On adding relations to homotopy groups, Ann. of Math. (2), 42, 409-428, 1941.
Lie-Rinehart Çapraz Modül Morfizimlerinin Homotopisi
Year 2019,
Volume: 9 Issue: 1, 202 - 212, 28.06.2019
Ayşe Çobankaya
Selim Çetin
Abstract
Bu makalede amacımız Lie-Rinehart çapraz modüllerin morfizmlerinin homotopi kavramını vermektir. Homotopi bağıntısı bir denklik bağıntısı oluşturduğunu, buna ek olarak Lie- Rinehart çapraz modül morfizmleri ve homotopileri bir groupoid yapısı olduğunu gösterdik.
References
- [1] Akça, İ.İ., Sidal, Y., Homotopies of Lie crossed module morphisms, arXiv: 1609.09297v1[math.CT], 2016.[2] Brown, R., and Higgings, P. J., Tensor Products and Homotopies for w− groupoids andcrossed complexes, Journal of Pure and Applied Algebra 47, 1-33, 1987.[3] Cabello, J.G. and Garzon, A.R., Closed model structures foralgebraic models for n−types, Journal of Pure and Applied Algebra 103 (3), 287-302, 1995.[4] Casas, J.M., Ladra, M., Pirashvili, T., Crossed modules for Lie-Rinehart algebras,Journal of Algebra, 274, 192-201, 2004.[5] Casas, J.M., Ladra, M., Pirashvili, T., Triple cohomology of Lie-Rinehart algebras andthe canonical class of associative algebras, Journal of Algebra, 291, 144-163, 2005.[6] Herz, J., Pseudo-algbres de Lie, C. R. Acad. Sci. Paris, 236, 1935-1937, 1953.[7] Huebschmann, J., Poisson cohomology and quantization , J. Reine Angew Math.408,57-113, 1990.[8] Mackenzie, K., Lie Groupoids and Lie algebroids in differential geometry, London math.soc. lecture note ser., vol. 124, Cambridge Univ. Press, 1987.[9] Whitehead, J.H.C., Note on a previous paper entitled On adding relations to homotopygroups, Ann. of Math. (2) , 47, 806-810, 1946.[10] Whitehead, J.H.C., Combinatorial Homotopy I and II, Bull. Amer. Math. Soc., 55,231-245 and 453-456, 1949.[11] Whitehead, J.H.C., On adding relations to homotopy groups, Ann. of Math. (2), 42, 409-428, 1941.