Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 072, 10 pages      arXiv:1305.6946

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

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.

pdf (359 kb)   tex (25 kb)


  1. Baez J., The octonions, Bull. Amer. Math. Soc. 39 (2002), 145-205, math.RA/0105155.
  2. Baez J., Huerta J., The algebra of grand unified theories, Bull. Amer. Math. Soc. 47 (2010), 483-552, arXiv:0904.1556.
  3. Distler J., Garibaldi S., There is no ''theory of everything'' inside $E_8$, Comm. Math. Phys. 298 (2010), 419-436, arXiv:0905.2658.
  4. Douglas A., Kahrobaei D., Repka J., Classification of embeddings of abelian extensions of $D_n$ into $E_{n+1}$, J. Pure Appl. Algebra 217 (2013), 1942-1954.
  5. GAP-Groups, Algorithms, and programming, Version 4.2, 2000,
  6. Humphreys J.E., Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York - Berlin, 1972.
  7. Lisi A.G., An exceptionally simple theory of everything, arXiv:0711.0770.
  8. Lisi A.G., An explicit embedding of gravity and the Standard Model in $E_8$, in Representation Theory and Mathematical Physics, Contemp. Math., Vol. 557, Amer. Math. Soc., Providence, RI, 2011, 231-244, arXiv:1006.4908.
  9. Malcev A.I., Commutative subalgebras of semi-simple Lie algebras, Amer. Math. Soc. Translation 1951 (1951), no. 40, 15 pages.
  10. Minchenko A.N., Semisimple subalgebras of exceptional Lie algebras, Trans. Moscow Math. Soc. 67 (2006), 225-259.
  11. Mkrtchyan R.L., On the map of Vogel's plane, arXiv:1209.5709.
  12. Nesti F., Percacci R., Chirality in unified theories of gravity, Phys. Rev. D 81 (2010), 025010, 7 pages, arXiv:0909.4537.
  13. Vogel P., Algebraic structures on modules of diagrams, J. Pure Appl. Algebra 215 (2011), 1292-1339.

Previous article  Next article   Contents of Volume 10 (2014)