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.