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Odd-quadratic Leibniz superalgebras
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AbstractAn odd-quadratic Leibniz superalgebra is a (left or right) Leibniz superalgebra with an odd, supersymmetric, non-degenerate and invariant bilinear form. In this paper, we prove that a left (resp. right) Leibniz superalgebra that carries this structure is symmetric (meaning that it is simultaneously a left and a right Leibniz superalgebra). Moreover, we show that any non-abelian (left or right) Leibniz superalgebra
does not possess simultaneously a quadratic and an odd-quadratic structure. Further, we obtain an inductive description of odd-quadratic Leibniz superalgebras using the procedure of generalized odd double extension and we reduce the study of this class of Leibniz superalgebras to that of odd-quadratic Lie superalgebras. Finally, several non-trivial examples of odd-quadratic Leibniz superalgebras are included.
Title: Odd-quadratic Leibniz superalgebras
Description:
AbstractAn odd-quadratic Leibniz superalgebra is a (left or right) Leibniz superalgebra with an odd, supersymmetric, non-degenerate and invariant bilinear form.
In this paper, we prove that a left (resp.
right) Leibniz superalgebra that carries this structure is symmetric (meaning that it is simultaneously a left and a right Leibniz superalgebra).
Moreover, we show that any non-abelian (left or right) Leibniz superalgebra
does not possess simultaneously a quadratic and an odd-quadratic structure.
Further, we obtain an inductive description of odd-quadratic Leibniz superalgebras using the procedure of generalized odd double extension and we reduce the study of this class of Leibniz superalgebras to that of odd-quadratic Lie superalgebras.
Finally, several non-trivial examples of odd-quadratic Leibniz superalgebras are included.
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