Lie $3-$algebra and super-affinization of split-octonions

Abstract

The purpose of this study is to extend the concept of a generalized Lie $3-$ algebra, known to the divisional algebra of the octonions $\mathbb{O}$, to split-octonions $\mathbb{SO}$, which is non-divisional. This is achieved through the unification of the product of both of the algebras in a single operation. Accordingly, a notational device is introduced to unify the product of both algebras. We verify that $\mathbb{SO}$ is a Malcev algebra and we recalculate known relations for the structure constants in terms of the introduced structure tensor. Finally we construct the manifestly super-symmetric $\mathcal{N}=1\mathbb{SO}$ affine super-algebra. An application of the split Lie $3-$algebra for a Bagger and Lambert gauge theory is also discussed.

17 pages without figures

References in corpus (6)

• Gauge Symmetry and Supersymmetry of Multiple M2-Branes
• Algebraic structures on parallel M2-branes
• Modeling Multiple M2's
• Multiple M2-branes and Generalized 3-Lie algebras
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Cited by in corpus (2)

• A non-associative quaternion scalar field theory
• Dirac and Maxwell systems in split octonions