We start with some basic properties of semisimple Lie algebras over C, and establish some notation to be used throughout. The assertions not proved here are proved in the standard books on Lie Algebras, e.g., those of Dynkin, Jacobson or Sophus Lie (Séminaire). Let L be a semisimple Lie algebra over C, and H a Cartan subalgebra of L. Then H is necessarily abelian and L = H ⊕ α =0 L α where α ∈ H * and L α = {X ∈ L : [H, X] = α(H)X, ∀ H ∈ H} Note that H = L 0. The α's are linear functions on H, called roots. We adapt the convention that L γ = 0 if γ is not a root. Then [L α , L β ] ⊆ L α+β. The rank of L equals dim C H = l, say. The roots generate H * as a vector space over C. Write V for H * Q , the vector space over Q generated by the roots. Then dim Q V = l. Let γ ∈ V. Since the Killing form is nondegenerate there exists an H γ ∈ H such that (H, H γ) = γ(H) for all H ∈ H. Define (γ, δ) = (H γ , H δ) for all γ, δ ∈ V. This is a symmetric, nondegenerate, positive bilinear form on V. Denote the collection of all roots by Σ. Then Σ is a subset of the nonzero elements of V satisfying: 0. Σ generates V as a vector space over Q. 1. α ∈ Σ ⇒ −α ∈ Σ and kα / ∈ Σ for k an integer = ±1. 2. 2(α, β)/(β, β) ∈ Z for all α, β ∈ Σ. (Write α, β = 2(α, β)/(β, β). These are called Cartan integers.) 3. Σ is invariant under all reflections w α (α ∈ Σ) (where w α is the reflection in the hyperplane orthogonal to α, i.e., w α v = v − 2(v, α)/(α, α)α). which has a Z-basis which is a C-basis for V , we say M is a lattice in V. We can now state the following corollaries to Theorem 2. Corollary 1. (a) Every finite dimensional L-module V contains a lattice M invariant under all X m α /m! (α ∈ Σ, m ∈ Z +); i.e., M is invariant under U Z. (b) Every such lattice is the direct sum of its weight components; in fact, every such additive group is. Proof. (a) By the theorem of complete reducibility of representations of semisimple Lie algebras over a field of characteristic 0 (See Jacobson, Lie Algebras, p. 79), we may assume that V is irreducible. Using Theorem 3, we find v + and set M = U − Z v +. M is finitely generated over Z since only finitely many monomials in U − Z fail to annhilate v +. Since U − v + = V and since U − Z spans U − over C, we see that M spans V over C. Before completing the proof of (a), we will first show that if c i v i = 0 with c i ∈ C, v i ∈ M and v 1 = 0, then there exist n i ∈ Z, n 1 = 0, such that c i n i = 0. To see this, let u ∈ U Z be such that the component of uv 1 in Cv + is nonzero. Then c i uv i = 0 implies c i n i = 0 where n i v + is the component of uv i in Cv +. We have n i ∈ Z by Lemma 12 and n 1 = 0 by choice of u. Finally, suppose a basis for M is not a basis for V. Let l be minimal such that there exist v 1 ,. .. , v l ∈ M linearly independent over Z but linearly dependent over C. Suppose l i=1 c i v i = 0. Then there exist n i ∈ Z, n 1 = 0 such that l i=1 c i n i = 0. We see that 0 = n 1 l i=1 c i v i = l i=2 c i (n 1 v i − n i v 1). Since n 1 v i − n i v 1 , i = 2, 3,. .. , 4 are linearly independent ...
This article explains how consensus decision making has operated in practice in the General Agreement on Tariffs and Trade/World Trade Organization (GATT/WTO). When GATT/WTO bargaining is law-based, consensus outcomes are Pareto-improving and roughly symmetrical. When bargaining is power-based, states bring to bear instruments of power that are extrinsic to rules, invisibly weighting the process and generating consensus outcomes that are asymmetrical and may not be Pareto-improving. Empirical analysis shows that although trade rounds have been launched through law-based bargaining, hard law is generated when a round is closed, and rounds have been closed through power-based bargaining. Agenda setting has taken place in the shadow of that power and has been dominated by the European Community and the United States. The decision making rules have been maintained because they help generate information used by powerful states in the agenda-setting process. Consensus decision making at the GATT/WTO is organized hypocrisy, allowing adherence to the instrumental reality of asymmetrical power and the sovereign equality principle upon which consensus decision making is purportedly based.
on the occasion of his 60th birthday § 1. Introduction Our purpose here is to study the irreducible representations of semisimple algebraic groups of characteristic p # 0, in particular the rational representations, and to determine all of the representations of corresponding finite simple groups. (Each algebraic group is assumed to be defined over a universal field which is algebraically closed and of infinite degree of transcendence over the prime field, and all of its representations are assumed to take place on vector spaces over this field.)To state our first principal result, we observe that relative to a Cartan decomposition of a semisimple algebraic group, there is described in §5 below (in a somewhat more general context) a standard way of converting an isomorphism on the universal field into one on the group, and that relative to a choice of a set S of simple roots, an irreducible rational projective representation of the group is characterized by a function from S to the nonnegative integers, to be called, together with the corresponding function on the Cartan subgroup of the decomposition, the high weight of the representation [13, Exp. 14 and 15 J
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