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 commutative 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 ∈ 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 ∈ 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 ∈ h∗ that satisﬁes Vw = {v ∈ V : ρ(h)v = w(h)v ∀ h ∈ h} (5) The next lemma should be intuitively reasonable: Lemma 2.1. The set of weights is preserved under equivalence of representations.

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 ∈ 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 ∈ Vλ and h ∈ 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 ∈ 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 ∈ 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) ∈ (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 reasonable, 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 ∈ h, then there is a weight space decomposition M V = Vw (9) w∈Λ(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 = ∞ 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 orthogonal 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 ∈ h∗ such that

gr = {X ∈ g :[h, X] = r(h)X ∀ h ∈ h} (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) r∈R

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. Bigger algebras have more complicated reflection symmetries, which are understood through the Weyl group. Things get compiclated quickly, however – the 27-dimensional representation of SU(3) splits into many SU(2) multiplets, with the Weyl group consisting of 6 elements. Actually, this turns out to be pretty simple – the Weyl group is just S3. But in general, we might hope to avoid getting too specific about the Weyl groups for individual algebras.

4 The Killing form

An alternate way of getting to the reflections that generate the Weyl group is to come up with an inner product on the adjoint of the maximal torus t∗. We can do this by introducing the Killing form: Definition 4.1. The Killing form on a Lie algebra g is defined by

K(X,Y ) = T r(ad(X) ◦ ad(Y )) (14) and is invariant under the adjoint action of G. The definiteness of the Killing form will depend on the Lie algebra. In fact, the Killing form is negative definite on compact groups: Lemma 4.1. If g is the Lie algebra of a compact semi-simple Lie group, the Killing form K is negative definite on g. Proof. The trick we use is to realize that one way to find a G-invariant inner product on g is to average out an arbitrary inner product on g. In this case, Ad is an orthogonal representation, and the elements of its Lie algebra are skew symmetric: ad(X)T = −ad(X). So then

K(X,X) = T r(ad(X)ad(X)) = −T r((ad(X))T ad(X)). (15)

However, we note that for any (nonzero) real matrix A,

T X 2 T r(A A) = Aij > 0 (16) ij and so evidently the Killing form is negative-deﬁnite as claimed. Given that the Weyl group acts on the roots by reﬂection symmetries, we expect that for each root r, there is some wr ∈ W (G, T ) that acts on the torus T and leaves the space Ur invariant. Weyl groups originate from elements in the normalizer N(T ) acting by conjugation, and so the action becomes an isometry when we use the Killing form (since the Killing form gives an adjoint-invariant

5 inner product). Essentially, this inner product allows us to go back and forth between t and t∗. We should probably stop to do an example at this point: let’s choose SU(3). ∗ Let λn denote the Gell-Mann matrices. t is spanned by the two roots

r12(hλ) = λ1 − λ2, r23(hλ) = λ2 − λ3 (17)

The element w12 ∈ W is a reﬂection that swaps λ1 with λ2:

w12(r23) = r13 = r12 + r23 (18) while the element w23 swaps λ2 with λ3:

w23(r12) = r13 = r12 + r23 (19)

Thinking about this a bit, we see that the Cartan matrix for SU(3) should be

2 −1 H = (20) −1 2 which means that the two roots r12 and r23 are separated by angle of θ = 2π/3. The two reﬂections w12 and w23 actually generate the entire Weyl group, which upon closer reﬂection is actually just S3. To be continued...