Fine gradings on simple exceptional Jordan pairs and triple systems

Abstract

We give a classification up to equivalence of the fine group gradings by abelian groups on the Jordan pairs and triple systems of types bi-Cayley and Albert, under the assumption that the base field is algebraically closed of characteristic different from 2. The associated Weyl groups are computed. We also determine, for each fine grading on the bi-Cayley and Albert pairs, the induced grading on the exceptional simple Lie algebra given by the Tits-Kantor-Koecher construction.⋆ Supported by the Spanish Ministerio de Economía y Competitividad-Fondo Europeo de Desarrollo Regional (FEDER) MTM2013-45588-C3-2-P and by the Diputación General de Aragón-Fondo Social Europeo (Grupo de Investigación deÁlgebra). 1Proof. Let Γ be a G-grading on T and η Γ : G D → Aut(T) its associated morphism. Consider the projection morphism π : Aut(J)×µ 2 → Aut(J) and the isomorphism f : Aut(T) → Aut(J) × µ 2 of Theorem 2.5. Also, note that the elements of µ 2 (R) FINE GRADINGS ON SIMPLE EXCEPTIONAL JORDAN PAIRS AND TRIPLE SYSTEMS 13are identified with the scalar automorphisms of T R of the form r1 with r ∈ R × and r 2 = 1, which implies that the composition π • f • η Γ : G D → Aut(J) determines the equivalence class of Γ. Then the result follows because the morphisms G D → Aut(J) are in correspondence with the equivalence classes of gradings on J.Remark 2.7. Note that fine gradings on T correspond to maximal quasitori of Aut(T), which are the direct product of a maximal quasitorus of Aut(J) and µ 2 . Corollary 2.8. Let J be a finite-dimensional central simple Jordan F-algebra with associated Jordan triple system T. Let Γ J be a G-grading on J and Γ T the same G-grading on T. Then W(Γ T ) = W(Γ J ).Proof. From Theorem 2.5 we know that Aut(T) ∼ = Aut(J)×{±1}. Hence Aut(Γ T ) ∼ = Aut(Γ J ) × {±1} and the result follows.Proposition 2.9. Let J be a Jordan F-algebra with unity 1, and let T be its associated Jordan triple system. Let Γ be a G-grading on T. If J is central simple, then 1 is homogeneous. Moreover, if G = U(Γ) and 1 is homogeneous, then deg(1) has order 2.