# Gradings on simple Lie algebras by Alberto Elduque

By Alberto Elduque

Gradings are ubiquitous within the idea of Lie algebras, from the basis area decomposition of a fancy semisimple Lie algebra relative to a Cartan subalgebra to the attractive Dempwolff decomposition of as a right away sum of thirty-one Cartan subalgebras. This monograph is a self-contained exposition of the type of gradings through arbitrary teams on classical basic Lie algebras over algebraically closed fields of attribute no longer equivalent to two in addition to on a few nonclassical basic Lie algebras in optimistic attribute. different vital algebras additionally input the level: matrix algebras, the octonions, and the Albert algebra. many of the offered effects are contemporary and feature no longer but seemed in publication shape. This paintings can be utilized as a textbook for graduate scholars or as a reference for researchers in Lie thought and neighboring components. This booklet is released in cooperation with Atlantic organization for examine within the Mathematical Sciences (AARMS)

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Suppose that Γ admits a realization as a G0 -grading for some group G0 . We will say that G0 is a universal group of Γ if for any other realization of Γ as a G-grading, there exists a unique homomorphism G0 → G that restricts to identity on Supp Γ. Note that, by deﬁnition, G0 is a group with a distinguished generating set, Supp Γ. A standard argument shows that, if a universal group exists, it is unique up to an isomorphism over Supp Γ. We will denote it by U (Γ). The following proposition shows that U (Γ) exists and depends only on the equivalence class of Γ.

The induction hypothesis implies that W = v2 D ⊕ · · · ⊕ vn−1 D. In particular, v1 and vn are not in W and hence Iv1 = Ivn = V . Now if there exists r ∈ I such that rv1 = 0 and rvn = 0, we are done. Otherwise the map d : Ivn → Iv1 , rvn → rv1 , is well-deﬁned. Clearly, d is a homomorphism of R-modules and a homogeneous map of degree (deg vn )−1 deg v1 . Hence d ∈ D. Finally, by deﬁnition of d, we have r(vn d − v1 ) = 0 for all r ∈ I. But this means vn d − v1 ∈ W , which is a contradiction. A structure theorem.

Let FG be the vector space that has G as a basis. Then FG is a coalgebra with comultiplication and counit deﬁned, respectively, by Δ(g) = g ⊗ g and ε(g) = 1, for all g ∈ G. It is wellknown that a G-grading is equivalent to the structure of an FG-comodule. (For the background on coalgebras, comodules, etc. ) The equivalence is set up as follows. 2) ρΓ (v) := v ⊗ g for all v ∈ Vg , g ∈ G. Conversely, given a coaction ρ : V → V ⊗ FG, we can deﬁne a grading Γ on V by setting Vg := {v ∈ V | ρ(v) = v ⊗ g} for all g ∈ G.