E7½
In mathematics, the Lie algebra E7½ is a subalgebra of E8 containing E7 defined by Landsberg and Manivel in order to fill the "hole" in a dimension formula for the exceptional series En of simple Lie algebras. This hole was observed by Cvitanovic, Deligne, Cohen and de Man. E7½ has dimension 190, and is not simple: as a representation of its subalgebra E7, it splits as E7 ⊕ (56) ⊕ R, where (56) is the 56-dimensional irreducible representation of E7. This representation has an invariant symplectic form, and this symplectic form equips (56) ⊕ R with the structure of a Heisenberg algebra; this Heisenberg algebra is the nilradical in E7½.
See also
References
- A.M. Cohen, R. de Man, Computational evidence for Deligne's conjecture regarding exceptional Lie groups, C. R. Acad. Sci. Paris, Série I 322 (1996) 427—432.
- P. Deligne, La série exceptionnelle de groupes de Lie, C. R. Acad. Sci. Paris, Série I 322 (1996) 321—326.
- P. Deligne, R. de Man, La série exceptionnelle de groupes de Lie II, C. R. Acad. Sci. Paris, Série I 323 (1996) 577—582.
- Landsberg, J. M.; Manivel, L. (2006), "The sextonions and E7½", Advances in Mathematics, 201 (1): 143–179, arXiv:math.RT/0402157
, doi:10.1016/j.aim.2005.02.001, MR 2204753
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