Gradings on the real forms of the Albert algebra, of g2, and of f4

Abstract

We find all the fine group gradings on the real forms of the Albert algebra and of the exceptional Lie algebras g 2 and f 4 . ͑pO s ͒, J ͑0,0,e͒ = R 3 , J ͑0,0,g͒ = 0, ͑g e͒ J ͑0,1,g͒ = 0 ͑͑pO s ͒ g ͒, 053516-19 Gradings on g 2 and f 4 J. Math. Phys. 51, 053516 ͑2010͒ This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.12.232.124 On: Wed, 26 Nov 2014 20:30:56͑2͒ If J is compact, ͑i͒ the Z 2 5 -grading on JªA ͑1,1,1͒ ͑pO͒, J ͑0,0,e͒ = R 3 , J ͑0,0,g͒ = 0, ͑g e͒ J ͑0,1,g͒ = 0 ͑͑pO͒ g ͒, J ͑1,0,g͒ = 1 ͑͑pO͒ g ͒, J ͑1,1,g͒ = 2 ͑͑pO͒ g ͒. ͑3͒ If J is nonsplit and noncompact, ͑i͒ the Z 2 5 -grading on JªA ͑−1,1,−1͒ ͑pO͒, J ͑0,0,e͒ = R 3 , J ͑0,0,g͒ = 0, ͑g e͒ J ͑0,1,g͒ = 0 ͑͑pO͒ g ͒, J ͑1,0,g͒ = 1 ͑͑pO͒ g ͒, J ͑1,1,g͒ = 2 ͑͑pO͒ g ͒; ͑ii͒ the Z 2 3 ϫ Z-grading on JªA ͑−1,1,−1͒ ͑pO͒, J ͑−2,e͒ = ͗2͑e 0 − e 2 ͒ − 1 ͑1͒͘, J ͑−2,g͒ = 0, ͑g e͒ J ͑−1,g͒ = ͗ 0 ͑x͒ − 2 ͑x͒: x ͑pO͒ g ͘, J ͑0,e͒ = ͗e 1 ,e 0 + e 2 ͘, J ͑0,g͒ = ͗ 1 ͑x͒: x ͑pO͒ g ͘, ͑g e͒ J ͑1,g͒ = ͗ 0 ͑x͒ + 2 ͑x͒: x ͑pO͒ g ͘, J ͑2,e͒ = ͗2͑e 0 − e 2 ͒ + 1 ͑1͒͘, J ͑2,g͒ = 0, ͑g e͒.Also, in the same way, we can have a detailed description of the fine gradings on the real forms of f 4 .

Weyl groups of some fine gradings one6

“…Note, if ϕ = diag(α 1 , α 2 , α 3 , α 4 , α 5 , α 6 ) ∈ GL 6 (F), that ϕ commutes with every inner automorphism of P 5 , although it does not commute necessarily with θ. More precisely, ϕθ ϕ −1 θ −1 acts in L 0 as Ad(ϕϕ t ) and it acts in e σ(1) ∧ e σ(2) ∧ e σ(3) ∈ L 1 with eigenvalue α σ(1) α σ(2) α σ (3) α σ(4) α σ(5) α σ (6) . Thus, the condition φθφ −1 = θ ψ, for ψ = diag(β 1 , β 2 , β 3 , β 4 , β 5 , β 6 ) with β i ∈ {±1}, is equivalent to the conditions α i 2 = β i for all i and Π 6 i=1 α i = 1.…”

“…Since the action by conjugation of Aut(Γ 5 ) on P 5 takes inner automorphisms to inner automorphisms, any element in W(Γ 5 ) must have zero blocks in the positions (2, 1) and (3,1). Also, the zero block in the position (3, 2) is a consequence of the fact that the torsion group of G 5 is preserved.…”

“…. , 4} of W f 4 that we have chosen are, respectively, the ones related to the automorphisms ψ (1,2,3) , ψ (2,3) , ψ c and the extension of τ described in [12,Theorem 4.2] and with the notations used there.…”

“…The knowledge of the corresponding Weyl groups could be applied to the classification up to isomorphism of the gradings on e 6 [14, Corollaries 1.26 and 1.27], which, until now, remains unknown. Also, they would help the classification of the gradings on the five real forms of e 6 , by means of the technique developed in [3]. Note that the algebra e 6 has the striking quantity of 14 fine gradings up to equivalence, whose universal groups and types will be brought in Section 4.…”

“…That work can be considered as the beginning of the theory of gradings on Lie algebras over arbitrary groups. Until that moment, gradings usually were considered over cyclic groups, with the obvious exception of the Cartan decomposition (root space decomposition) and a collection of examples compiled in [21,Chapter 3,§3]. They proposed the search of the remaining fine gradings as a way of looking for an alternative approach to the structure theory and the representation theory.…”