E7½

In mathematics, the Lie algebra E_7½ is a subalgebra of E₈ containing E₇ defined by Landsberg and Manivel in orderto 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.

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. [http://arxiv.org/abs/math.RT/0402157" The sextonions and E_7½"] . Adv. Math. 201 (2006), no. 1, 143--179.