Finite coproducts in the category of quadratic modules of Lie algebras

Year 2022, Volume: 11 Issue: 3, 210 - 221, 31.12.2022

Emre Özel Ummahan Ege Arslan İbrahim İlker Akça

https://doi.org/10.54187/jnrs.1184962

Abstract

In this study, we will construct finite coproduct objects in the category of quadratic modules of Lie algebras with a new approach using the idea of quasi-quadratic modules.

Keywords

Quadratic modules, Quasi-quadratic modules, Coproduct objects

References

• J. H. C. Whitehead, Combinatorial homotopy. II, Bulletin of the American Mathematical Society, 55(5), (1949) 453-496.
• M. Gerstenhaber, On the deformation of rings and algebras, Annals of Mathematics Second Series, 79(1) (1964) 59-103.
• S. Lichtenbaum, M. Schlessinger, The cotangent complex of a morphism, Transactions of the American Mathematical Society, 128(1), (1967) 41-70.
• T. Porter, Some categorical results in the theory of crossed modules in commutative algebras, Journal of Algebra, 109(2), (1987) 415-429.
• P. Dedecker, A. S-T. Lue, A nonabelian two-dimensional cohomology for associative algebras, Bulletin of the American Mathematical Society, 72(6), (1966) 1044-1050.
• Z. Arvasi, U. E. Arslan, Annihilators, multipliers and crossed modules, Applied Categorical Structures, 11(6), (2003) 487-506.
• U. E. Arslan, İ. İ. Akça, G. I. Onarlı, O. Avcıoğlu, Fibrations of 2-crossed modules, Mathematical Methods in the Applied Sciences, 42(16), (2019) 5293-5304.
• U. E. Arslan, Serdar Hürmetli, Bimultiplications and annihilators of crossed modules in associative algebras, Journal of New Theory, (35), (2021) 72-90.
• A. Aytekin, Categorical structures of Lie-Rinehart crossed module, Turkish Journal of Mathematics, 43(1), (2019) 511-522.
• C. Kassel, J. L. Loday, Extensions centrales d'alg\'ebres de Lie, Annales de l'Institut Fourier (Grenoble), 32(4), (1982) 119-142.
• J. M. Casas, M. Ladra, Colimits in the crossed modules category in Lie algebras, Georgian Mathematical Journal, 7(3), (2000) 461-474.
• G. J. Ellis, Homotopical aspects of Lie algebras, Journal of the Australian Mathematical Society (Series A), 54(3), (1993) 393-419.
• D. Conduch\'e, Modules Crois\'es G\'en\'eralis\'es de Longueur 2, Journal of Pure and Applied Algebra, 34(2-3), (1984) 155-178.
• J. F. Martins, The fundamental 2-crossed complex of a reduced CW-complex, Homology, Homotopy and Applications, 13(2), (2011) 129-157.
• B. Gohla, J. F. Martins, Pointed homotopy and pointed lax homotopy of 2-crossed module maps, Advances in Mathematics, 248, (2013) 986-1049.
• İ. Akça, K. Emir, J. F. Martins, Pointed homotopy of 2-crossed module maps on commutative algebras, Homology, Homotopy and Applications, 18(1), (2016) 99-128.
• P. Carrasco, T. Porter, Coproduct of 2-crossed modules: Applications to a definition of a tensor product for 2-crossed complexes, Collectanea Mathematica, 3(67), (2016) 485-517.
• R. Brown, N. D. Gilbert, Algebraic models of 3-types and automorphism structures for crossed modules, Proceedings of the London Mathematical Society, (3)59, (1989) 51-73.
• J. L. Loday, Space with finitely many non-trivial homotopy groups, Journal of Pure and Applied Algebra, 24(2), (1982) 179-202.
• Z. Arvasi, Crossed squares and 2-crossed modules of commutative algebras, Theory and Applications of Categories, 3(7), (1997) 160-181.
• Z. Arvasi, T. Porter, Freeness conditions for 2-crossed modules of commutative algebras, Applied Categorical Structure, 6(4), (1998) 455-471.
• H. J. Baues, Combinatorial homotopy and 4-dimensional complexes, Walter de Gruyter, Berlin, 1991.
• E. Ulualan, E. Uslu, Quadratic modules for Lie algebras, Hacettepe Journal of Mathematics and Statistics, 40(3), (2011) 409-419.
• A. Odabaş, E Ulualan, On free quadratic modules of commutative algebras, Bulletin of the Malaysian Mathematical Sciences Society, 39(3), (2016) 1059-1074.
• K. Yılmaz, E. S. Yılmaz, Baues cofibration for quadratic modules of Lie algebras, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(2), (2019) 1653-1663.
• E. Özel, Pointed homotopy theory of quadratic modules of Lie algebras, Master's Thesis, Eskişehir Osmangazi University (2017) Eskişehir, Türkiye (in Turkish).
• U. E. Arslan, E. Özel, On homotopy theory of quadratic modules of Lie algebras, Konuralp Journal of Mathematics, 10(1), (2022) 159-165.
• R. Brown, P. J. Higgins, R. Sivera, Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids, Tracts in Mathematics, European Mathematical Society, 2010.
• E. Özel, U. E. Arslan, On quasi quadratic modules of Lie algebras, Journal of New Theory(41), (2022) 62-69.