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 (Vogel 1999), and is related by some observations made by (Deligne 1996). (Landsberg Manivel) 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

• E7½

References

• Deligne, Pierre (1996), "La série exceptionnelle de groupes de Lie", Comptes Rendus de l'Académie des Sciences, Série I 322 (4): 321–326, ISSN 0764-4442 
• Deligne, Pierre; Gross, Benedict H. (2002), "On the exceptional series, and its descendants", Comptes Rendus Mathématique 335 (11): 877–881, doi:10.1016/S1631-073X(02)02590-6, ISSN 1631-073X, http://www.numdam.org/item/10.1016/S1631-073X(02)02590-6.pdf 
• Landsberg, J. M.; Manivel, L. (2006), "A universal dimension formula for complex simple Lie algebras", Advances in Mathematics 201 (2): 379–407, doi:10.1016/j.aim.2005.02.007, ISSN 0001-8708 
• Vogel, Pierre (1999), The universal Lie algebra, Preprint, http://www.math.jussieu.fr/~vogel/A299.ps.gz