In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
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, 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,W ...
of E
7. 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 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 that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
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
{{algebra-stub