Extensions, crossed modules and pseudo quadratic Lie type superalgebras
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Abstract
Extensions and crossed modules of Lie type superalgebras are introduced and studied. We construct homology and cohomology theories of Lie-type superalgebras. The notion of left super-invariance for a bilinear form is defined and we consider Lie type superalgebras endowed with nondegenerate, supersymmetric and left super-invariant bilinear form. Such Lie type superalgebras are called pseudo quadratic Lie type superalgebras. We show that any pseudo quadratic Lie type superalgebra induces a Jacobi-Jordan superalgebra. By using the method of double extension, we study pseudo quadratic Lie type superalgebras and theirs associated Jacobi-Jordan superalgebras.
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Pouye, M., & Kpamegan, B. (2022). Extensions, crossed modules and pseudo quadratic Lie type superalgebras. Extracta Mathematicae, 37(2), 153-184. https://doi.org/10.17398/2605-5686.37.2.153
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Section
Non-associative Rings and Algebras
References
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[2] I. Bajo, S. Benayadi, M. Bordemann, Generalized double extension and descriptions of quadratic Lie superalgebras, arxiv.org/0712.0228v1, December 2007.
[3] A. Baklouti, S. Benayadi, Symplectic Jacobi-Jordan algebras, Linear Multilinear Algebra 69 (8) (2021), 1557 – 1578.
[4] S. Benayadi, S. Hidri, Leibniz algebras with invariant bilinear form and related Lie algebra, Comm. Algebra 44 (2016), 3538 – 3556.
[5] D. Burde, A. Fialowski, Jacobi-Jordan algebras, Linear Algebra Appl. 459 (2014), 589 – 594.
[6] L.M. Camacho, I. Kaygorodov, V. Lopatkin, M.A. Salim, The variety of dual mock-Lie algebras, Commun. Maths. 28 (2020), 161 – 178.
[7] C. Cuvier, Algèbres de Leibnitz: définitions, propriétés, Ann. Sci. École Norm. Sup. (4) 27 (1) (1994), 1 – 45.
[8] E. Getzler, M. Kapranov, Cyclic operads and cyclic homology, in “ Geometry, Topology & Physics ”, International Press, Cambridge, MA, 1995, 167 – 201.
[9] N. Kamiya, S. Okubo, Jordan-Lie superalgebra and Jordan-Lie triple system, J. Algebra 198 (1997), 388 – 411.
[10] P. Jordan, J. von Neumann, E. Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. of Math. (2) 35 (1934), 29 – 64.
[11] M. Pouye, B. Kpamegan, L. Todjihoundé, Quadratic Lie-type superalgebras, J. Adv. Math. Stud. 14 (3) (2021), 399 – 419.
[12] K.A. Zhevlakov, Solvability and nilpotence of Jordan rings, Algebra i Logika Sem 5 (1966), 37 – 58.