In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

 E_7½ is a subalgebra of E₈

E8 (mathematics)

In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8...

 containing E₇ defined by Landsberg and Manivel in order
to fill the "hole" in a dimension formula for the exceptional series E_n

En (Lie algebra)

In mathematics, especially in Lie theory, En is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1,2, and k, with k=n-4....

 of simple Lie algebras. This hole was observed by Cvitanovic

Predrag Cvitanovic

Predrag Cvitanović, is currently an endowed Professor of Physics at the Georgia Institute of Technology. He is highly regarded for his work in nonlinear dynamics, particularly his contributions to periodic orbit theory...

, 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.