Fine gradings and gradings by root systems on simple Lie algebras

Abstract

Abstract. Given a fine abelian group grading Γ : L = g∈G Lg of a finite dimensional simple Lie algebra over an algebraically closed field of characteristic zero, with G being the universal grading group, it is shown that the induced grading by the free group G/ tor(G) on L is a grading by a (not necessarily reduced) root system. Some consequences for the classification of fine gradings on the exceptional simple Lie algebras are drawn.

Weyl groups of some fine gradings one6

“…In fact, such grading is a root grading for the root system of F 4 , taking into account the interesting relationship between fine gradings and root gradings which has been recently stated in [11].…”

“…Also note that, for an element with this particular shape, B necessarily belongs to the group τ 1 , τ 2 . This follows, for example, from the fact that B represents the action on the coarsening of Γ 1 induced by taking G 1 modulo its torsion subgroup, and this coarsening is a grading by the root system of type A 2 (according to [11]). …”

“…A Lie algebra L over F is Φ-graded, with Φ being a reduced root system, if L contains as a subalgebra a finite-dimensional simple Lie algebra g = h ⊕ ( α∈Φ g α ) whose root system is Φ relative to a Cartan subalgebra h = g 0 and L = α∈Φ∪{0} L(α) where L(α) = {x ∈ L | [h, x] = α(h)x for all h ∈ h} (and L(0) = α∈Φ [L(α), L(−α)]). According to [11], if Γ is a fine grading on a simple finitedimensional Lie algebra L with universal group G, then L is graded by an irreducible (possibly nonreduced) root system Φ of rank equal to the free rank of G. Moreover, the fine grading Γ comes from mixing the Z rank(Φ) -grading given by the root grading with certain type of grading over a finite group in the related coordinate algebra.…”

Multiplicity-Free Gradings on Semisimple Lie and Jordan Algebras and Skew Root Systems

“…The classification for E 7 and E 8 can be extracted from the recent work of Yu [12]. Moreover, recently many fine gradings on simple Lie algebras and gradings by root systems were related by Elduque in [3] .…”

Maximal finite abelian subgroups of E8

“…If T = 0 or, equivalently, if the free rank of the universal group of Γ is infinite, then Γ induces a grading by a not necessarily reduced root system [Eld13] and it is determined by a fine grading on the coordinate algebra of the grading by the root system. Associative, alternative, Jordan or structurable algebras appear as coordinate algebras.…”