E7½

In mathematics, the Lie algebra E_7½ 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 E_n of simple Lie algebras. This hole was observed by Cvitanovic, Deligne, Cohen and de Man. E_7½ 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 E_7½.

See also

• Vogel plane

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 E_7½", Advances in Mathematics, 201 (1): 143–179, arXiv:math.RT/0402157, doi:10.1016/j.aim.2005.02.001, MR 2204753