Extensions, crossed modules and pseudo quadratic Lie type superalgebras
DOI:
https://doi.org/10.17398/2605-5686.37.2.153Keywords:
Lie type superalgebras, Jacobi-Jordan superalgebras, extension, crossed module, homology, cohomology, double extension, pseudo quadratic Lie type superalgebrasAbstract
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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