ar X iv : m at h - ph / 0 41 10 37 v 1 1 0 N ov 2 00 4 The fine gradings of sl ( 3 , C ) and their symmetries 1

We describe the normalizers for all non-conjugate maximal Abelian subgroups of diagonalizable automorphisms of sl(3, C) and show their relation to the symmetries of equations related to the graded contraction. Introduction — Admissible gradings of a simple Lie algebra L over the complex or real number field are basic structural properties of each L. Examples of exploitation of coarse gradings like Z2 are easy to find in the physics literature. Here we are interested in the opposite extreme: the fine gradings of L which only recently were described [8,9]. Our aim is to point out the interesting symmetries of the fine gradings on the example of sl(3, C). Such symmetries can be used in the study of graded contractions of L. Indeed, they are the symmetries of the system of quadratic equations for the contraction parameters. Graded contractions are a systematic way of forming from L a family of equidimensional Lie algebras which are not isomorphic to L. An insight into such parameter–dependent families of Lie algebras provides a group theoretical tool for investigating relations between different physical theories through their symmetries. In this contribution we consider finite dimensional L and announce specific results for sl(3, C) only. The decomposition Γ : L = ⊕ i∈I Li is called a grading if, for any pair of indices i, j ∈ I, there exists an index k ∈ I such that 0 6= [Li, Lj] ⊆ Lk. There are infinitely many gradings of a given Lie algebra. We do not need to distinguish those gradings which can be transformed into each other. More precisely, if Γ : L = ⊕ i∈I Li is a grading and g is an automorphism of L, then Γ̃ : L = ⊕ i∈I g(Li) is also a grading. Two such gradings are called equivalent. A grading Γ : L = ⊕ i∈I Li is a refinement of the grading Γ : L = ⊕ i∈J Lj if for any i ∈ I there exists j ∈ J such that Li ⊆ Lj. A grading which cannot be properly refined is called fine. For construction of a grading of L, one can use any diagonalizable automorphism g from AutL. It is easy to see that decomposition of L into eigenspaces of g is a In: Proceedings of XXIII International Colloquium on Group Theoretical Methods in Physics (A.N. Sissakian, G.S. Pogosyan and L.G. Mardoyan, eds.), JINR Dubna 2002, Vol. 1, pp. 57–61.