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

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

2010-04-23

arxiv.org/abs/1004.4228v3

Modern Physics Letters A, Volume: 26, Issue: 35(2011) pp. 2663-2675

Physics

Mathematical Physics

17 pages without figures

Scientific paper

10.1142/S0217732311037005

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.

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