Equivalent formulation: Every compact Lie group has a finite dimensional faithful representation.

Other corollaries of Peter-Weyl (see the proof in Brocker-tom Dieck):

(1) Characters are dense in the space of continuous class functions

(2) If V is a faithful representation, every irreducible representation appears as a summand in the tensor powers V^{\otimes k} \otimes (V^*)^{\otimes l}

(3) If H is a closed subgroup of G, there is a representation V and an element v in V such that the isotropy group G_v is H.

Proof of the Peter-Weyl theorem (following Broecker-tom Dieck).

Definition of maximal torus of a compact Lie group. Existence is guaranteed. These are the maximal connected subgroups. Definition of the Weyl group W.

Prop: W is finite (proof just sketched - see Brocker-tom Dieck).

Examples: U(n) and SO(2n+1)

Theorem: Any element of g of G is in some maximal torus and any two maximal tori are conjugate. 

Corollary: The exponential map \mathfrak g \to G is surjective (as it is surjective on tori). Surjectivity of the 

exponential is in fact equivalent to the fact that every element is contained in a torus since g=exp(X) is contained in the closure of {exp(tX): t \in \R) 

Corollary: There is canonical surjection T/W \to G/conjugacy (in fact this is a homeomorphism). Therefore a character (and hence a representation) of G is completely determined by its restriction to a maximal torus (and the representations of tori that occur in this way are invariant under the Weyl group).

Tori have topological generators. If t is a topological generator of T acting on a manifold M then the fixed point M^T agrees with M^t and also with the set of zeros of the infinitesimal action a(X) for some X \in \mathfrak t with exp(X)=t.

André Weil's proof of the Theorem on tori (using the Lefschetz fixed point theorem). 

If G is a compact Lie group, we've proved its Lie algebra splits as a product \mathfrak g =\mathfrak z \times \mathfrak k with \mathfrak z the center and \mathfrak k semisimple with negative definite Killing form. It's easy to see that if T is a maximal torus then \mathfrak t \otimes \C = \mathfrak z \otimes \C \osum \mathfrak h with \mathfrak h a Cartan subalgebra of \mathfrak k \otimes C. 

The complexified Lie algebra \mathfrak g \otimes \C breaks up in terms of eigenspaces for the adjoint action of T and this precisely corresponds to the root space decomposition with respect to the Cartan subalgebra. Inspecting the action of the Weyl group on the decomposition one sees that it agrees with the Weyl group defined for the root system. 

In order to classify representations of a compact lie group G, recall that it has a finite cover \pi \colon Z \times K \to G with Z a torus integrating \mathfrak z and K a (compact) simply connected Lie group integrating \mathfrak k. The complex representations of K are the same as the representations of the complex semisimple Lie algebra \mathfrak k \otimes \C that we discussed last time and we know what the ones for a torus and a product are so we understand the representations of Z \times K. The representations of G are those which send the finite abelian subgroup \ker \pi to the identity.
Example: The irreducible representations of SO(3) are the representations of SU(2) (or SL(2;\C)) where \pm 1 act as the identity. For each even degree n this can be taken to be the homogeneous polynomials of degree n in two complex variables.

B=\mathfrak h \osum \osum_{\alpha \in \Phi^+} L_\alpha is solvable with [B,B]=\osum_{\alpha \in \Phi^+} L_\alpha (nilpotent) so by Lie's theorem each representation contains an eigenvector v for B with [B,B] acting trivially. These are called highest weight vectors. By the classification of sl(2)-modules, \lambda_i(v)>=0. The set . If V is irreducible then V=U(L)v for any highest weight vector. The set of weights for which this happens are called the dominant weights \Lambda^+=\Z_{\geq 0} \langle \lambda_1, \ldots, \lambda_l\rangle.

Theorem: There is a 1-1 correspondence between \Lambda^+ and the set of isomorphism classes of irreducible finite dimensional representations of L, sending V to V(\lambda) with (unique) highest weight \lambda.

Idea of construction: Induce from a one dimensional representation of the Borel subalgebra B with weight \lambda on the Cartan subalgebra: V = U(L) \otimes_{U(B)} \langle v \rangle. By the PBW theorem this breaks up as a sum of weight spaces lower than \lambda. One can show this has a unique irreducible quotient with highest weight v and this is the irreducible representation V(\lambda). This universal construction shows V(\lambda) is unique.

The set \Pi(\lambda) of weights appearing in V(\lambda) is preserved by the action of the Weyl group and for each \alpha the \alpha-string through \lambda has length \langle \lambda,\alpha \rangle. From this one can actually get the following characterization of \Pi(\lambda): it is the set of weights \mu such that \sigma \mu is less than or equal to \lambda for all \sigma in the Weyl group (for the partial order determined by \Delta).

Example: Some representations of sl(3)

G compact Lie group. For any representation W there is a canonical isomorphism \oplus_{V irred} \Hom^G(V,W) \otimes V \to W  given by evaluation. Can do the usual algebraic constructions with representations - dual tensor, and so on. Over \C there is also the conjugate representation. An invariant Hermitean inner product on V gives an isomorphism between \overline V and V^*. Definition of character and representative function. Example: For G=S^1 the representative functions are the trigonometric polynomials.

Theorem: Representative functions coming from non-isomorphic irreducible representations are orthogonal in L^2(G)

Elementary properties of characters: (1) If V and W are isomorphic, \chi_V=\chi_W (2) \chi_V(h)=\chi_V(ghg^{-1}) (3) \chi_{V \oplus W} = \chi_V + \chi_W (4) chi_{V\otimes W) = \chi_V \chi_W (5) \chi_{V^*}(g)=\chi_{\overline V}(g)=\overline{\chi_V(g)} = \chi_V(g^{-1}) (6) \chi_V(e)=\dim V

Proposition: (7) \int_G \chi_V(g) = \dim V^G

(8) \langle \chi_V, \chi_W \rangle = \dim \Hom^G(V,W). Hence if V and W are irreducible this is 1 if V and W are isomorphic and 0 otherwise.

Corollary: V is isomorphic to W iff \chi_V=\chi_W; If V=\osum V_j^{n_j} with V_j irreducible (and distinct) then \|\chi_V\|^2 = \sum n_j^2. In particular V is irreducible if and only if its character is unital in L^2(G)

Proposition: Let G and H be compact Lie groups. Irreducible representations of G\times H are exactly the tensor products of irreducibles of G with irreducibles of H (with the action (g,h) v\otimes w = (gv) \otimes (gw)).

Example: Representations of compact abelian groups.

Remark: A real Lie subalgebra M of L such that M \otimes_\R \C is called a real form of L. A Chevalley basis gives a standard real form called the split real form (just take the real lie subalgebra spanned by the Chevalley basis). All the real forms of L can be obtained from U. They are conjugate to p+in with p and n the +1, -1 eigenspaces of an involution of U.  Example: L=sl(l+1;\C). The split real form is sl(l+1;\R), the compact form is su(l+1) and there are "intermediate" real forms su(p,q) with p+q=l+1 (Lie algebra of automorphisms of a hermitean form on \C^{l+1} with signature (p,q)).

Proposition. The function x \mapsto \Delta(x) induces a bijective correspondence between Weyl chambers and bases of \Phi. 

Examples. The Weyl group acts compatibly on Weyl chambers and bases.

Theorem: 1. The Weyl group acts simply transitively on bases and Weyl chambers

2. If \alpha is in \Phi then there is \sigma in \mathcal W sending \alpha to \Delta (i.e. every root is part of a base)

3. \mathcal W is generated by simple reflections (reflections determined by simple roots

4. The closure of a Weyl chamber is a fundamental domain for the action of \mathcal W on E.

Definition of Cartan matrix associated to \Delta. Examples.

Proposition: The Cartan matrix determined \Phi up to isomorphism.

The Coxeter graph and the Dynkin diagram. The Dynkin diagram is just a visual depiction of the Cartan matrix. Examples. 

A root system is irreducible if it can not be partitioned into non-empty subsets. This is equivalent to saying that \Delta can not be so partitioned or that the Coxeter graph is connected. Any root system can be written uniquely as a cartesian product of irreducible root systems. If L is semisimple with simple factors L_i then the root system of L is the cartesian product of the root systems of the L_i

Prop: Let (E,\Phi) be an irreducible root system with base \Delta. Then

1. There's a unique maximal root and all its coefficients are positive.

2. The Weyl group orbit of any root spans E

3. At most two root lengths occur in \Phi (these are called long and short roots). Moreover roots of the same length form a single orbit of \mathcal W

4. The maximal root is long.

Theorem: If \Phi is an irreducible root system then it is either A_l for l>=1, B_l for l>=2, C_l for l>=3, D_l for l>=4 or G_2, F_4, E_6, E_7, or E_8.

Flavor of the proof: proof that the Coxeter graph of an irreducible root system can not contain a loop.

The root systems above all arise from semi-simple Lie algebras. A_l from sl(l+1), B_l from so(2l+1), C_l form Sp(2l) and D_l from so(2l). The other come from more complicated Lie algebras called exceptional. G_2 is the (complexification of the) Lie algebra of derivations of the octonions and F_4 the complexification of the Lie algebra of the group of isometries of the octonionic projective plane.

Proposition: If \Phi is a root system, Aut(\Phi) is the semi-direct product of the group \Gamma = \{ f \in Aut(\Phi) \colon f(\Delta)=\Delta} of graph automorphisms with the Weyl group.

If L is semisimple over a field of char 0, Int(L) denotes the subgroup of GL(L) generated by \{ exp(ad(x)): x \in L nilpotent\}. This is a group of automorphisms of L because the binomial formula for the iterated application of a derivation shows more generally that if \delta is a nilpotent derivation of an algebra then exp(\delta) is an automorphism. Over the complex numbers, exp is the matrix exponential so exp(ad x) is just Ad(exp x).

Theorem: All Cartan subalgebras of a semisimple Lie algebra over an algebraically closed field are conjugate by Int(L) (and hence the root system associated to such an algebra is well defined up to isomorphism).