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 P2/S3, the projective plane P2 divided out by the symmetric group S3 of permutations of coordinates. It was introduced by {{harvtxt|Vogel|1999}}, and is related by some observations made by {{harvtxt|Deligne|1996}}. {{harvtxt|Landsberg|Manivel|2006}} 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
References
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- {{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | last2=Gross | first2=Benedict H. | title=On the exceptional series, and its descendants | doi=10.1016/S1631-073X(02)02590-6 | mr=1952563 | year=2002 | journal=Comptes Rendus Mathématique | issn=1631-073X | volume=335 | issue=11 | pages=877–881| url=http://www.numdam.org/item/10.1016/S1631-073X(02)02590-6.pdf }}
- {{Citation | last1=Landsberg | first1=J. M. | last2=Manivel | first2=L. | title=A universal dimension formula for complex simple Lie algebras | arxiv=math/0401296 | doi=10.1016/j.aim.2005.02.007 | doi-access=free | mr=2211533 | year=2006 | journal=Advances in Mathematics | issn=0001-8708 | volume=201 | issue=2 | pages=379–407}}
- {{citation|first=Pierre|last=Vogel|url=http://www.math.jussieu.fr/~vogel/A299.ps.gz|title=The universal Lie algebra|year=1999|series=Preprint}}