Covering Algebras I. Extended Affine Lie Algebras

Abstract

This is the first of what will be a sequence of three papers dealing with a generalization of certain parts of the beautiful work of V. Kac on finiteorder automorphisms of finite-dimensional complex simple Lie algebras. Recall that Kac (see [K2, Chap. 8] and [H, Sect. X.5]) built a Lie algebra from a pair ( σ) composed of a finite-order automorphism σ of a finitedimensional simple Lie algebra over as follows. First from σ he obtains the eigenspaceswhere m is on the order of σ, ζ = e 2πi √ −1/m i ∈ , and i →ī is the natural map of → m (here m denotes the integers modulo m). He

Reflection systems and partial root systems

“…Then (2) holds trivially for α ∈ R im , and it holds for α ∈ R re by 2.3.2. Now suppose we have (2). Then f (x), ξ ∨ = 0 for all ξ ∈ f (R) if and only if x, α ∨ = 0 for all α ∈ R, showing that (3) holds.…”

“…(c) For the statement concerning α, β , it remains in view of (2) and by symmetry to show that 2α + β / ∈ R. Assume to the contrary that ε := 2α + β ∈ R. By (1), we have 2α ∈ R hence 2α ∈ R re by 3.3.1, and therefore ε ∈ R re by (PRS3). Furthermore,…”

“…We claim that R = {0} ∪ ∆ ∪ P (L1) is a partial root system with R re = ∆, which is symmetric if L is not of type P(n), in particular if L is classical. To see this, it suffices by finite dimensionality of L to verify the condition (2). To do so, we will use the description of the root systems in classical Lie algebras given in [23, Prop.…”

“…(d) From the definition of W 1 (R) it is clear that also ϕ: W 1 (R) → W (R) is surjective, so we have (2). An element v ∈ V induces the identity both on Z (by (b)) and onX ∼ = X/Z.…”

“…In the special case A = {α, β}, we write α, β := {α, β} c (2) and call it the closed root interval between α and β. If f : (R, X) → (R , X ) is a morphism of…”