On Quadratic Left Leibniz Algebras and Related Lie-Yamaguti Structures

Abstract:

      A left Leibniz algebra equipped with an invariant nondegenerate skew-symmetric bilinear form (i.e., a skew-symmetric quadratic Leibniz algebra) is constructed. The notion of $T^*$-extension of Lie-Yamaguti algebras is introduced and it is observed that the trivial extension of a Lie-Yamaguti algebra is a quadratic Lie-Yamaguti algebra. It is proved that every symmetric (resp., skew-symmetric) quadratic Leibniz algebra induces a quadratic (resp., symplectic) Lie-Yamaguti algebra.