In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
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 P ...
, 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
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...
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 \mathbb) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping \omega : V \times V \to F that is
; Bilinear: ...
, 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 ''a, b'' ...
; 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 group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
s", ''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.
*
Lie groups
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