In mathematics, the Vogel plane is a method of parameterizing
simple Lie algebra
In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras is one of the major achievements of Wilhelm Killing and Élie Cartan.
A direct sum of s ...
s by eigenvalues α, β, γ of the
Casimir operator
In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operato ...
on the symmetric square of the Lie algebra, which gives a point (α: β: γ) of ''P''
2/''S''
3, the
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...
''P''
2 divided out by the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
''S''
3 of
permutations of coordinates. It was introduced by , and is related by some observations made by . generalized Vogel's work to higher symmetric powers.
The point of the projective plane (modulo permutations) corresponding to a simple complex Lie algebra is given by three eigenvalues α, β, γ of the Casimir operator acting on spaces ''A'', ''B'', ''C'', where the
symmetric square of the Lie algebra (usually) decomposes as a sum of the complex numbers and 3 irreducible spaces ''A'', ''B'', ''C''.
See also
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E7½
References
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*{{citation, first=Pierre, last=Vogel, url=http://www.math.jussieu.fr/~vogel/A299.ps.gz, title=The universal Lie algebra, year=1999, series=Preprint
Lie groups
Lie algebras