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
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Pr ...
, Cohen and de Man. E
7½ has dimension 190, and is not simple: as a representation of its subalgebra E
7, it splits as , where (56) is the 56-dimensional
irreducible representation of E
7. This representation has an invariant
symplectic form In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping that is
; Bilinear: Linear in each argument ...
, and this symplectic form equips with the structure of a
Heisenberg algebra
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form
::\begin
1 & a & c\\
0 & 1 & b\\
0 & 0 & 1\\
\end
under the operation of matrix multiplication. Elements ...
; this Heisenberg algebra is the
nilradical in E
7½.
See also
*
Vogel plane In mathematics, the Vogel plane is a method of parameterizing simple Lie algebras by eigenvalues α, β, γ of the Casimir operator on the symmetric square of the Lie algebra, which gives a point (α: β: γ) of ''P''2/''S''3, the projective 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.
*
Lie groups
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