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).
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.