Abstract
This paper introduces the category of gb-triple systems and studies some of their algebraic properties. Also provided is a functor from this category to the category of Leibniz algebras.
Highlights
A Triple system is a vector space g over a field K together with a K-trilinear map T : g⊗3 → g
A gb-triple system is a K-vector space g equipped with a trilinear operation
For a gb-triple system g and a subalgebra S of g, we define the left normalizer of S in g by
Summary
A Triple system is a vector space g over a field K together with a K-trilinear map T : g⊗3 → g. The following example provides a Leibniz 3algebra that is not a gb-triple system. The two-dimensional complex Leibniz 3-algebra Ł (see [6, Theorem 2.14]) with basis {a1, a2}, dim([Ł, Ł, Ł]Ł) = 1, and brackets A subalgebra I of a gb-triple system g is called ideal (resp., left ideal, resp., right ideal) of g if it satisfies the condition [g, I, g]g ⊆ I (resp., [g, g, I]g ⊆ I, resp., [I, g, g]g ⊆ I).
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