E7½ Explained

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, where (56) is the 56-dimensional irreducible representation of E₇. This representation has an invariant symplectic form, and this symplectic form equips 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", Comptes rendus de l'Académie des Sciences, Série I 322 (1996) 427–432.
• P. Deligne, "La série exceptionnelle de groupes de Lie", Comptes rendus de l'Académie des Sciences, Série I 322 (1996) 321–326.
• P. Deligne, R. de Man, "La série exceptionnelle de groupes de Lie II", Comptes rendus de l'Académie des Sciences, Série I 323 (1996) 577–582.