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Maximal Tori, Weyl Groups, and Roots

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Maximal Tori, Weyl Groups, and Roots Maximal tori, Weyl groups, and roots Ethan Lake 9/7/2015 1 Maximal tori Our goal in these notes is to try to understand the most important basic notions of the representation theory of Lie algebras. Of course, this is still very much a work in progress! Throughout, we let G denote a finite group. Definition 1.1. Let g be a finite-dimensional Lie algebra. A torus t is a com- mutative sub-algebra of g. A maximal torus is a torus which is not strictly contained in any other larger torus. Lemma 1.1. If t is a maximal torus of a Lie algebra g, then t is equal to its centralizer. Proof. Since t is commutative by definition, it is contained within its centralizer. Conversely, let Z be in the centralizer of t. Then t + KZ is also a torus, where K is the field we're working over. But t is maximal by assumption, and so Z 2 t. We can also define tori through the group U(1). The representation theory of U(1) is easy { each irreducible representation is labeled by some integer n. The natural way to construct a torus T is to take the product of m circles U(1)×· · ·×U(1), whose irreducible representations will be labeled by m integers. If we take some random G, we can attack the problem of finding the irreducible representations of G by taking the largest possible (the maximal) torus T such that T ⊂ G, and trying to understand the quotient G=T . Rather surprisingly, this turns out to be quite helpful in many cases. Any maximal torus T (now thinking in the group picture) is equipped with a group action, called the Weyl group: Definition 1.2. Given a maximal torus T and a compact, connected Lie group G, the quotient group W (G; T ) = N(T )=T (1) is the Weyl group of G, where N(T ) is the normalizer of T (i.e., the set of all g 2 G such that T is invariant under conjugation by g). 1 The classic (easy) example is for G = SU(2). One choice for the maximal torus is the set of all matrices of the form eiφ 0 (2) 0 e−iφ The normalizer of T then consists of two different types of matrices: eiφ 0 0 e−iφ ; (3) 0 e−iφ eiφ 0 Evidently the Weyl group of SU(2) consists of two elements, which we can write as 1 0 0 −1 ; : (4) 0 1 1 0 We'll come back to the Weyl group later on when we talk about root systems of Lie algebras and their symmetries under reflections. 2 Weights Definition 2.1. Let g be a Lie algebra with (ρ, V ) a representation of g and h any torus in g. A weight of ρ with respect to h is a functional w 2 h∗ that satisfies Vw = fv 2 V : ρ(h)v = w(h)v 8 h 2 hg (5) The next lemma should be intuitively reasonable: Lemma 2.1. The set of weights is preserved under equivalence of representa- tions. Proof. Let (ρ, V ) and (ρ0;V 0) be two representations of g made equivalence through a bijection T : V ! V 0. Furthermore, let Λ; Λ0 denote the set of their weights with respect to some torus h ⊂ g. If λ 2 Λ is a weight of ρ, then Vλ (the weight space of λ) and its image under T must both be non-zero. For a non-zero v 2 Vλ and h 2 h we have ρ0(h)(T v) = T (ρ(h)v) = λ(h)T v 0 0 and so T (Vλ) ⊆ Vλ is also non-zero. So then Λ ⊆ Λ . The reverse inclusion is shown in exactly the same way. This sounds rather opaque at first, until we work some examples to see what it really means. We start with the circle group. Any irreducible representation of S1 is given by a map ρ : S1 ! U(1) whose differential maps from R ! R (think of the tangent space to S(1)!). We can thus label the irreducible representations by an integer index n: 1 ρn : t 2 S ! exp(2πint) (6) 2 When we think about the irreducible representation in terms of the Lie algebras, we expand the exponentials out and see that ρn : t 2 R ! nt. We can extend this way of thinking to the maximal tori quite easily. Since T = (S1)m for some integer m, the irreducible representations are now homo- morphisms ρ :(S1)m ! U(1). Naturally, we have 1 m ρn :(t1; : : : ; tm) 2 (S ) ! exp(2πi(n1t1 + : : : nmtm)): (7) This goes over to the Lie algebra map in the obvious way: ρn :(t1 : : : tm) ! n1t1 + : : : nmtm (8) which can be thought of as living in the the dual space to the Lie algebra of T , t∗. Essentially, a weight means an element of the dual space corresponding to an irreducible representation of T . We now state (without proof!) an important, but perhaps intuitively rea- sonable, result regarding weights: Theorem 2.1. Let (ρ, V ) be a representation of g and h be a torus in g. If ρ(h) is diagonalizable for every h 2 h, then there is a weight space decomposition M V = Vw (9) w2Λ(ρ) where Λ(ρ) is the set of weights of ρ. In the infinite dimensional case, V might not be the direct sum of its weight spaces. However, the sum of V 's weight spaces is still always a direct sum, just as eigenvectors associated with distinct eigenvalues of a linear transform are always linearly independent. Of course, we won't say too much about the case when dim V = 1 anyway. 3 Roots This section begins with a barrage of definitions: Definition 3.1. A sub-algebra h of a Lie algebra g is an ideal if [g; h] ⊂ h. As an example, consider the group U(3). The one-dimensional space I given by multiples of the identity matrix is obviously an ideal. Additionally, the space of all traceless matrices J is also an ideal (the trace of a commutator must be traceless, after all). So then any element in U(3) can be written as a sum of one element in I and one in J: U(3) breaks down to a sum of two ideals. Definition 3.2. A Lie algebra g of dimension greater than one is said to be simple if it has no proper ideals. g is called semi-simple if it has no non-trivial abelian ideals. 3 In particular, we see that U(3) is neither simple or semi-simple. Note that a semi-simple Lie algebra is isomorphic to a product of simple Lie algebras. Algebras like su(n) are simple, while algebras like su(n)⊕su(n) are semi-simple. We need one more definition before we can define the roots of G: Definition 3.3. A Cartan sub-algebra is a Lie sub-algebra whose Lie group is a maximal torus of G. Putting it another way, a Cartan sub-algebra is a maximal abelian sub-algebra. Suppose we know the adjoint representation of G on g. What are the weights of the action of T on G? T definitely acts trivially on the Cartan sub-algebra, and so the trivial weight will come up in the action rank(G) times. The or- thogonal compliment of the Cartan sub-algebra in g will then decompose into orthogonal irreducible representations of T . So finally, we are ready to define the roots of G: Definition 3.4. The roots of G are the (non-trivial) weights of the adjoint representation of G on g. They take integer values on the integer lattice. Putting it another way, let h be a Cartan sub-algebra of g. A root for g with respect to h is a functional r 2 h∗ such that gr = fX 2 g :[h; X] = r(h)X 8 h 2 hg (10) We usually write the set of roots as R(g; h). Looking at our earlier theorem about the weight-space decomposition, we get the following result: Corollary 3.1. Let h be a Cartan sub-algebra of a Lie algebra g. The set of roots R(g; h) is non-empty, and we can perform a root-space decomposition of g with respect to h: ! M g = h ⊕ gr (11) r2R The easy example is for G = SU(2). The Lie algebra is gC = gl(2; C), and the elements of the Cartan sub-algebra are matrices of the form 1 0 h = ασz = α : (12) 0 −1 The two root spaces are given the matrices of the form P (with root 2α) and M (with root −2α), where 1 1 P = (σx + iσy);M = (σx − iσy) (13) 2 2 In general, the strategy of forming things that look like ladder operators is a good way to get the pairs of roots in SU(n) (e.g. in SU(3), we would do the same thing, but use the Gell-Mann matrices instead of the Pauli matrices, giving three raising and three lowering operators). We can now go back at recast our earlier definition of the Weyl group in terms of root systems. 4 Definition 3.5. Let R be a root system in V . The subgroup W (R) ⊂ O(V ) generated by the root reflections is called the Weyl group of the root system. Going back to the SU(2) example, we see that the representations of SU(2) are symmetric with respect to a reflection about their center.
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