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 (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

• 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, MR 1378507
• 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, MR 1952563
• Landsberg, J. M.; Manivel, L. (2006), "A universal dimension formula for complex simple Lie algebras", Advances in Mathematics, 201 (2): 379–407, arXiv:math/0401296, doi:10.1016/j.aim.2005.02.007, ISSN 0001-8708, MR 2211533
• Vogel, Pierre (1999), The universal Lie algebra, Preprint