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| h such that [T (H) (H)]n | = 0, for all H h and some n Then | is called a weight vector if n = 1, that is, T (H)| = (H)| , H h In particular, | is called a highest (or lowest) weight vector if T (E + )| = 0 , + + , (or T (E )| = 0 , ) , (B10)

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where + , or , is the set of the positive, or negative, roots The interest of this notion is the fundamental result obtained by Cartan which says that the irreducible, nite-dimensional, Hermitian representations of simple Lie algebras are in one-toone correspondence with highest-weight vectors for the geometrical construction of representations in those terms) We illustrate this with the simplest example, namely, su(2), with the familiar angular momentum basis { J o , J + , J } and = { } In the unitary spin-j irreducible representation D j of SU (2), of dimension 2 j + 1, with standard basis {| j, m | m = j, , j}, the highest (or lowest) weight vector | j , j (or | j, j ) satis es the relation J + | j, j = 0 (or J | j, j = 0) and has an isotropy subalgebra u(1) (by this, we mean the subalgebra that annihilates the given vector) Notice that here C 2 = J 2 + J + J + J J + is indeed diagonal in D j , with eigenvalue j ( j + 1) o

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A Lie group G may be de ned in several ways, for instance, as a smooth manifold with a group structure such that the group operations ( g 1 , g 2 ) g 1 g 2 , g g 1 are C k , for some k v 2 This actually implies that all group operations are in fact (real) analytical A Lie group is said to be simple, or semisimple, if it has no nontrivial invariant subgroup, or Abelian invariant subgroup Let G be a Lie group For g G consider the map Lg : G G with Lg ( g ) = g g The derivative of this map at g G, denoted T g (Lg ), sets up an isomorphism T g (Lg ) : T g G T g g G between the tangent spaces at g and g g For any vector X T e G (the tangent space at the identity), let us de ne a vector eld X on G by X g = T e (Lg )X Such a vector eld is said to be left-invariant It can be demonstrated that the usual Lie bracket [X , Y ] of two left-invariant vector elds is again a leftinvariant vector eld Using this fact, we see that if Y g = T e (Lg )Y , Y T e G, the relation [X , Y ] = [X , Y ] de nes a Lie bracket on the tangent space, T e G, at the identity of the Lie group G Thus, equipped with this bracket relation, T e G becomes a Lie algebra which we denote by g Next, it can be shown that for any X g, there exists a unique analytical homomorphism, t X (t) of R into G such that d X (0) = X (t)|t=0 = X , dt and for each X g we then write exp X = X (1) The map X exp X is called the exponential map and it de nes a homeomorphism between an open neighborhood.

1 See J Moore, The Death of Competition: Leadership and Strategy in the Age of Business Ecosystems (Chichester, UK: John Wiley & Sons, 1996); M Zeleny, R J Cornet, and J A F Stoner, Moving from the Age of Specialization to the Era of Integration, Human Systems Management 9 (1999): 153 170

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