Exploring Triality Explicitly: Convenient bases for $\mathrm{SO}(8)$, $\mathrm{Spin}(1,7)$, and ${\mathrm{G}}_2$

Abstract

The property of triality only appears in one linear simple Lie algebra: $D_4$, a.k.a. $\mathfrak{so}(8,\mathbb{C})$. In this paper several convenient representations and bases are constructed in order to facilitate the exploration of the three fold symmetry known as the triality of representations. In particular the three $8$ dimensional representations for the Euclidean and Lorentzian real forms of $\mathfrak{so}(8,\mathbb{C})$ are constructed, and the maps between representations are given in each case, respectively. Standard results are presented in a modern language, such as how ${\mathfrak{g}}_2$ arises as the intersection of non-conjugate $\mathfrak{spin}(7,\mathbb{R})$ sub-algebras, and as the stabilizer of the outer automorphism group of $\mathfrak{so}(8)$. It is argued that $\mathfrak{spin}(1,7)$ is in some sense the more natural stage for triality to play out upon, as triality can be made to act diagonally in this case, leading to a ${\mathbb{Z}}_3$ grading of the Lie algebra and subsequently a Jordan algebra. Once this is understood a short discussion is had about obstacles to a theory of triality which attempt to explain the three generations of matter via some form of triality.

arXiv comment18 pages, V2 added discussion regarding Jordan Algebra structure of spin(1,7), clarified discussion around the very loose wick rotation/twistor analogy

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