Again, students who have no formal coursework in this area can follow most of the text if they are willing to accept a few facts. Although the construction of the spin groups in §1.3 is of great importance for later representation theory and mathematical physics, this material can easily be omitted on a first reading.
Compact Lie Groups
Basic Notions
- Manifolds
- Lie Groups
- Lie Subgroups and Homomorphisms
- Compact Classical Lie Groups
- Exercises
A closed subgroup of a Lie group is a separate Lie group with respect to the relative topology. As usual, this is a closed subgroup of U(n), and thus SU(n) is also a compact Lie group.
Basic Topology
- Connectedness
- Simply Connected Cover
- Exercises
If G is a Lie group and H is a connected Lie subgroup such that G/H is also connected, then G is connected. If H is a discrete normal subgroup of a connected Lie group G, then H is contained in the center of G. Since Chi is the continuous image of a connected set G, Ch is connected.
The Double Cover of SO(n)
- Clifford Algebras
- Exercises
From the multilinear algebra it is directly evident (Exercise 1.26) that ι(x) is adjoint of(x) with respect to the natural form of na. A maps Pinn(R) to O(n): It is well known (Exercise 1.32) that every rectangular matrix is a product of reflections.
Integration
- Volume Forms
- Fubini’s Theorem
- Exercises
More generally, the corresponding measure dg is the unique left invariant normalized Borel measure on G. Regress the relationωg = lg∗−1ωe to show that the left invariant measure dg can be scaled this way.
Representations
Basic Notions
- Definitions
- Examples
- Exercises
Although this method works (e.g. Exercise 2.12), we take an equivalent path that realizes Cn(C) as a certain endomorphism. Show that there exists an analogous Spinn(R)-interleaved isomorphism with respect to the spin action on.
Operations on Representations
- Constructing New Representations
- Irreducibility and Schur’s Lemma
- Exercises
It also says that a representation(π,V) of a compact Lie group is better than a homomorphismπ:G→G L(V); it is a homomorphism to the unitary group of V with respect to the G-invariant inner product (Exercise 2.20). See Theorem 3.19 for the generalization to the infinite dimensional setting of unitary representations of compact groups.
Examples of Irreducibility
- SU(2) and V n (C 2 )
- Spin and Half-Spin Representations The spin representation S =
- Exercises
A relatively small dose of functional analysis (Exercise 3.14) can be used to further show that L2(Sn−1). Finally, the operators ι(ej)(−1)deg can then be used to show that W =. c) Show that·,·isO(n)-invariant over Vm(Rn).
Harmonic Analysis
Matrix Coefficients
- Schur Orthogonality
The following statement calculates the L2inner product of the matrix coefficients corresponding to irreducible representations. Let U,V be irreducible finite-dimensional unitary representations of a compact Lie group G with G-invariant inner products(·,·)Uen(·,·)V.
Infinite-Dimensional Representations
- Basic Definitions and Schur’s Lemma
- Canonical Decomposition
- Exercises
Using the definition of the adjoint map of T, T∗ : W → V, it immediately follows that T∗ ∈ HomG(W,V) and that T∗ is nonzero, injective and has close range (Problem 3.11 ). Using the same argument as above applied to the self-adjoint interlaced operators S+S∗and S−i S∗, it follows that S is a multiple of the identity. As in the finite-dimensional case, the above definition of the isotypic component V[π] is well defined, and V[π] is the closure of the sum of all submodules of V-equivalent.
Show that the orthogonal projection of P: V →. If Vγ ⊆V is any submodule equivalent to Eπ, use irreducibility and maximality to show that P Vγ = {0}. c) Show that the definition of the isotype component V[π]in.
The Peter–Weyl Theorem
- The Left and Right Regular Representation
- Main Result
- Applications
- Exercises
As a result, write C(G)G-unambiguously for the set of G-finite vectors associated with each operation. We first show that C(G)G-fin, with respect to the left operation, is the set of matrix coefficients. In particular, f = ffV,v0 ∈ MC(G), so the set of right G-finite vectors is included in the set of left G-finite vectors.
A function f ∈C(G) is called a continuous class function if f(ghg−1)= f(h)for allg,h∈G. 1) The span ofχ is equal to the set of continuous class functions in C(G)G-n.
Fourier Theory
- Convolution
- Plancherel Theorem
- Projection Operators and More General Spaces
- Exercises
Since we recognized 2πθ as an invariant measure on S1 and Zas parametrizes S1 with n corresponding to the (one-dimensional) representation eiθ → ei nθ, it seems likely that this result can be generalized to any compact Lie groupG. Since these are simple exercises, we leave them to the reader (Exercise 3.26). 1) Let G be a compact Lie group. For [π]∈G, it will turn out that the operator (dimEπ) γ (χEπ) is an orthogonal G-intertwining projection of V onto V[π].
For [π]∈G, show that V[0π]is the largest subspace of V that is a direct sum of irreducible submodules equivalent to Eπ.
Lie Algebras
Basic Definitions
- Lie Algebras of Linear Lie Groups
- Lie Algebras for the Compact Classical Lie Groups
- Exercises
The statements concerning the basic properties of the Lie bracket in part (b) are elementary and are left as an exercise (Exercise 4.1). It is therefore clear that Definition 4.1 generalizes in the obvious way to realize the Lie algebra of Sp(n) inside gl(n,H). Show that the Lie bracket is linear in each variable, skew symmetric and satisfies the Jacobi identity.
Show that it is possible to linearly reparametrize ϕ, i.e. replace ϕ(t) with ϕ(st) for some nonzero ∈ R such that ϕ([−1,1])⊆expU. 4) Show that ϕ (t)=et Xfort ∈R and conclude that ϕ is indeed real analytic and above all smooth.
Further Constructions
- Lie Algebra Homomorphisms
- Lie Subgroups and Subalgebras
- Exercises
Let A be a connected Lie subgroup H×G with a Lie algebra (Theorem 4.14) and let πH and πG be Lie group homomorphisms projecting A onto H and G, respectively. Then AndπH is a Lie algebra isomorphism for and h, so Theorem 4.15 implies, that πH is a covering map from AtoH. ZG(H)= {g∈G|gh=hg,h ∈H}, is a Lie subgroup of G with a Lie algebra, the centralizer of H in g,. 2) If h⊆g, show that the centralizer is hin G,.
Show that kerϕ is a closed Lie subgroup of the Lie-algebra kerdϕ. 2)Show that the Lie subgroupϕ(H)ofGhas Lie-algebradϕh.
Abelian Lie Subgroups and Structure
Abelian Subgroups and Subalgebras
- Maximal Tori and Cartan Subalgebras
- Examples
- Conjugacy of Cartan Subalgebras
- Exercises
The most general compact Abelian Lie group is isomorphic to Tk ×F, where F is a finite Abelian group. Since it is easy to see a maximal abelian subalgebra ofu(n) (Exercise 5.2), it follows that T is a maximal torus and its corresponding Cartan subalgebra. Rewriting kushtingg0 =g0gascg0g=g, the usual argument using Theorem 4.6 shows that the Lie algebra of Aisa (Exercise 4.22).
For every compact classical Lie group in §5.1.2, show that the given subgroupT is a maximal torus and that the given subalgebratis is its corresponding Cartan subalgebra.
Structure
- Exponential Map Revisited
- Lie Algebra Structure
- Commutator Theorem
- Compact Lie Group Structure
- Exercises
Assume that is a connected Abelian Lie subgroup of G. 2) Show that ZG(S) is the union of all maximal tori containing S and is therefore connected. Since it spends N from Theorem 1.15, this shows that. a) Let be the Lie algebra of a Lie subgroup of a linear group. ThusKi =expki = Si and in particular, Si is a normal closed Lie subgroup with Lie algebra.
Show that any closed normal Lie subgroup of G is a product of some of these with a central subgroup.
Roots and Associated Structures
Root Theory
- Representations of Lie Algebras
- Weights
- Roots
- Compact Classical Lie Group Examples
- Exercises
It is easy to check (Exercise 6.1) that the differentials of the representations of Lie groups given in Definition 2.10 give the following representations of Lie algebras. It follows that if gi is a Lie algebra of a compact Lie group, Realized by G ⊆ U(n), then gC can be identified with g⊕ige equipped with the standard Lie bracket inherited from gl(n,C) (Exercise 6.3). Since it is easy to see that z(gC)=z(g)C (Exercise 6.5), it follows from semisimplicity and Theorem 5.18 that H =0.
Show that expX =(cosλ)I +sinλλX. 2) Assume the Lie algebra of a compact Lie group G with Ga Lie subgroup of U(n).
The Standard sl(2, C ) Triple
- Cartan Involution
For the next theorem, recall that the nonsingular Bis ongα×g−αfrom Theorem 6.16 and that−B(X, θY) is an Ad-invariant inner product ongCforX,Y ∈gC. Since these are the bracket relations for the standard basis of sl(2,C), part (a) is completed (cf. Exercise 4.21). Continuing the notation from Theorem 6.20, the set{Eα,Hα,Fα} is called astandardsl(2,C)-triple connected to α and the group{Iα,Jα,Kα} is called standardsu(2)-connected to α. a) The only multiple of αin(gC) is ±α.
Use the maximal torus theorem, Lemma 6.14, and Theorem 6.9 to show that det Adg=1 ongC and therefore a gas well.
Lattices
- Definitions
- Relations
- Center and Fundamental Group
- Exercises
While the proof of part (b) of Theorem 6.30 is deferred to §7.3.6, in this section we prove at least that the simply connected cover of a compact semisimple Lie group is still compact. The proof of this lemma is a straightforward generalization of the proof of the Maximal Torus Theorem, Theorem 5.12 (Exercise 6.26). Use the fact that π is a local diffeomorphism to show that ϕ is a local diffeomorphism if and only if ϕ is a local diffeomorphism.
Conclude that G/Z(G)∼=G/Z(G) and use this to show that expG(g) is compact and therefore closed. 8) It remains to show that expG(g) is open.
Weyl Group
- Group Picture
- Classical Examples
- Simple Roots and Weyl Chambers
- The Weyl Group as a Reflection Group
- Exercises
For the sake of symmetry, note that the above conditionα(H) >0 is equivalent to the condition B(H,hα) >0. Similarly, rhα acts trivially on z(g)∩t and acts on it as a reflection across a hyperplane perpendicular to hα (Exercise 6.28). Show that there is a one-to-one correspondence between continuous class functions on G and continuous W-invariant functions on T (cf. Exercise 7.10).
For each compact classical Lie group in §6.4.3, check whether the given system of simple roots and the associated Weyl room is correct.
Highest Weight Theory
Highest Weights
- Exercises
4)Conclude using the root system ofso(n,C) and Theorem 2.33 that the weight vector(x1−i x2)p of weight p1 must be the highest weight vector ofHp(Rn) for n ≥3. For p ≤ m, show that there is only one vector with the highest weight and conclude that pCn is irreducible with the highest weight p. 2)Forn=2m, examine the wedge product of elements of the form2j−1±i e2j to find a basis of weight vectors. Forp 1) If V(λ) and V(λ) are irreducible representations of G, show that the weight of V(λ)⊗V(λ) is of the form µ+µ, where µ is a weight of V(λ) and µ is a weight of V(λ ). 2) By looking at vectors with highest weight, showV(λ+λ) appears exactly once as a summan iV(λ)⊗V(λ). Thus, by construction, the bounded differential atTe(UG)ate is clearly invertible, so it is a local differential from UGate. Thus, perhaps reducing UgandUG, we can assume that UG/T is an open neighborhood of eT inG/T and that mapsUg exp. Recall that the invariant measures above are given by integration over unique (up to ± 1) normalized left-volume invariantωG forms. 2)Show that the map f → |W(G)|−1d f|T defines a norm-preserving isomorphism between the L1 class functions on G and the W-invariant L1 functions on T. 3)Show that the map f → |W(G)|−12 D f|Tdefines a unit isomorphism between the L2 class functions on G to the W-invariant L2 functions on T, where D(eH)=. Let G be a compact Lie group, T a maximal torus and +(gC) a system of positive roots with a corresponding simple system (gC)= {α1,. In particular, it shows that when G is a simply connected compact Lie group with a semisimple Lie algebra, then the irreducible representations are parameterized by a set of dominant algebraic weights, P. An affine Weyl group is a group generated by transformations of the form H →wH+Z zaw∈W((gC)∨)inZ ∈2πi R∨. Lemma 7.40. Let G be a compact connected Lie group with maximal torus T. a) The affine Weyl group is generated by reflections over the hyperplanes α−1(2πi n) for α∈(gC) and n∈Z. b) An affine Weyl group acts simply transitively on the set of niches. Using part (1) of Exercise 7.21, compare dominant terms with showmµis given by the expression. Using the definitions, it is easy to see that these maps are well-defined, inverse, and G-interleaved (Exercise 7.28). By Theorem 4.14, there exists a unique connected Lie subgroup of G L(n,C) with Lie algebragC. Write the GC for this subgroup and call it the complexization of G. b) Fix+(gC) system of positive roots and recalln+=. The corresponding Borel subalgebra is b=tC⊕n+. c) Let N,B and Abe be uniquely connected Lie subgroups of the group G L(n,C) with Lie algebrasn+,b and a=it. It is easy to check that these maps are well-defined, G-interleaved, and inverse of each other (Exercise 7.31). This condition, written in local coordinates, gives rise to the standard Cauchy-Riemann equations (Exercise 7.32). Br¨ocker, T., en tom Dieck, T., Representations of Compact Lie Groups, vertaal uit die Duitse manuskrip, gekorrigeerde herdruk van die 1985-vertaling, Graduate Texts in Mathematics, 98, Springer-Verlag, New York, 1995 Goodman, R., en Wallach, N., Representations and Invariants of the Classical Groups, Encyclopedia of Mathematics and its Applications, 68, Cambridge University Press, Cambridge, 1998. Harish-Chandra, On some appliations of the universal enveloping algebra of 'n semi-eenvoudige Lie-algebra, Trans. McKay, W., and Patera, J., Tables of Dimensions, Indices, and Branching Rules for Representations of Simple Lie Algebras, Marcel Dekker, New York, 1981.Weyl Integration Formula
Weyl Character Formula
Borel–Weil Theorem
