SIGMA 10 (2014), 072, 10 pages      arXiv:1305.6946      https://doi.org/10.3842/SIGMA.2014.072

The GraviGUT Algebra Is not a Subalgebra of $E_8$, but $E_8$ Does Contain an Extended GraviGUT Algebra

Andrew Douglas ^a and Joe Repka ^b
a) CUNY Graduate Center and New York City College of Technology, City University of New York, USA
^b) Department of Mathematics, University of Toronto, Canada

Received April 04, 2014, in final form July 03, 2014; Published online July 08, 2014

Abstract
The (real) GraviGUT algebra is an extension of the $\mathfrak{spin}(11,3)$ algebra by a $64$-dimensional Lie algebra, but there is some ambiguity in the literature about its definition. Recently, Lisi constructed an embedding of the GraviGUT algebra into the quaternionic real form of $E_8$. We clarify the definition, showing that there is only one possibility, and then prove that the GraviGUT algebra cannot be embedded into any real form of $E_8$. We then modify Lisi's construction to create true Lie algebra embeddings of the extended GraviGUT algebra into $E_8$. We classify these embeddings up to inner automorphism.

Key words: exceptional Lie algebra $E_8$; GraviGUT algebra; extended GraviGUT algebra; Lie algebra embeddings.

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