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An Introduction to the Theory of Topological Groups and Their Representations

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An Introduction to the Theory of Topological Groups and Their Representations AN INTRODUCTION TO THE THEORY OF TOPOLOGICAL GROUPS AND THEIR REPRESENTATIONS VERN PAULSEN Abstract. These notes are intended to give an introduction to the representation theory of finite and topological groups. We assume that the reader is only familar with the basics of group theory, linear algebra, topology and analysis. We begin with an introduction to the theory of groups acting on sets and the representation theory of finite groups, especially focusing on representations that are induced by actions. We then proceed to introduce the theory of topological groups, especially compact and amenable groups and show how the "averaging" technique allows many of the results for finite groups to extend to these larger families of groups. We then finish with an introduction to the Peter- Weyl theorems for compact groups. 1. Review of Groups We will begin this course by looking at finite groups acting on finite sets, and representations of groups as linear transformations on vector spaces. Following this we will introduce topological groups, Haar measures, amenable groups and the Peter-Weyl theorems. We begin by reviewing the basic concepts of groups. Definition 1.1. A group is a non-empty set G equipped with a map p : G×G ! G; generally, denoted p(g; h) = g·h, called the product satisfying: • (associativity) (g · h) · k = g · (h · k); for every g; h; k 2 G; • (existence of identity) there is an element, denoted e 2 G such that g · e = e · g = g for every g 2 G; • (existence of inverses) for every g 2 G, there exists a unique element, denoted g−1; such that g · g−1 = g−1 · g = e: Often we will write a group as, (G; ·); to denote the set and the specific product. Recall that the identity of a group is unique. Definition 1.2. If G is a group, then a non-empty subset, H, of G is called a subgroup provided that: • e 2 H; • g; h 2 H; then g · h 2 H; Date: January 31, 2011. 1 2 VERN PAULSEN • g 2 H, then g−1 2 H: A subgroup H ⊆ G is called normal provided that the set g · H · g−1 = fg · h · g−1 : h 2 Hg is always a subset of H, for every g 2 G: A group, G, is called abelian, or commutative, provided, g · h = h · g for every, g; h 2 G: When G is abelian, then every subgroup is normal. When N ⊆ G is a normal subgroup of G then we may define a quotient group, G/N whose elements are left cosets, i.e., subsets of G of the form g · N = fg · h : h 2 Ng; and with the product defined by (g1 · N) · (g2 · N) = (g1 · g2) · N: Following are a few examples of groups, subgroups and quotient groups, that we assume that the reader is familiar with: • (Z; +){the additive group of integers, with identity, e=0, • (nZ; +){the normal, subgroup of Z, consisting of all multiples of n; • (Z=(nZ); +){the quotient group, usually denoted, (Zn; +) and often called the cyclic group of order n. • (Q; +){the additive group of rational numbers, with identity, e=0, • (R; +){the additive group of real numbers, with identity, e=0, • (C; +){the additive group of complex numbers, with identity, e=0, ∗ • (Q ; ·){the multiplicative group of non-zero rationals, with identity, e=1, ∗ • (R ; ·){the multiplicative group of non-zero reals, with identity, e=1, + ∗ • (R ; ·){the multiplicative subgroup of (R ; ·), consisting of positive reals, ∗ • (C ; ·){the multiplicative group of non-zero complex numbers, with identity, e=1, • (T; ·){the multiplicative subgroup of complex numbers of modulus 1. Definition 1.3. Given two groups, G1;G2, a map, π : G1 ! G2 is called a homomorphism, provided that π(g · h) = π(g) · π(h); for every g; h 2 G1: A homomorphism that is one-to-one and onto, is called an isomorphism. Problem 1.4. Prove that if G is a group and g 2 G satisfies, g ·g = g, then g = e-the identity of G. Problem 1.5. Prove that if π : G1 ! G2 is a homomorphism, and ei 2 Gi; i = 1; 2 denotes the respective identities, then • π(e1) = e2; • π(g−1) = π(g)−1; • N = fg 2 G1 : π(g) = e2g ⊆ G1 is a normal subgroup. The set N is called the kernel of the homomorphism and is denoted ker(π): • Prove that there is a well-defined homomorphism, π~ : G1=N ! G2 given by π~(g · N) = π(g): We call π~ the induced quotient map. • Prove that if π(G1) = G2; then π~ is an isomorphism. ∗ + Problem 1.6. Prove that R =R and Z2 are isomorphic. GROUP REPRESENTATION THEORY 3 + t Problem 1.7. Prove that the map, π :(R; +) ! (R ; ·) given by π(t) = e ; is an isomorphism. Problem 1.8. Prove that the map π :(R; +) ! (T; ·) given by π(t) = 2πit e = cos(2πt)+isin(2πt) is an onto homomorphism with kernel, Z: Deduce that R=Z and T are isomorphic. ∗ + Problemp 1.9. Prove that the map π :(C ; ·) ! (R ; ·) defined by π(z) = jzj = a2 + b2; where z = a + ib is a onto, homomorphism with kernel, T: ∗ + Deduce that C =T and R are isomorphic. Problem 1.10. Let G1;G2 be groups and let G1 × G2 = f(g1; g2): g1 2 G1; g2 2 G2g denote their Cartesian product. Show that G1 × G2 is a group with product, (g1; g2)·(h1; h2) = (g1h1; g2h2) and identity, (e1; e2): Show that N1 = f(g1; e2): g1 2 G1g is a normal subgroup and that (G1 × G2)=N1 is isomorphic to G2: Prove a similar result for the other variable. We now examine some other ways to get groups. The Matrix Groups n n We let R and C denote the vector spaces of real and complex n-tuples. n Recall that, using the canonical basis for R , we may identify the (real) linear n n n maps from R to R ; L(R ) with the real n × n matrices, which we denote, n Mn(R): Similarly, the (complex) linear maps, L(C ) can be identified with n Mn(C ): Under these identifications, composition of linear maps becomes matrix multiplication. Recall also, that the key properties of the determinant map, are that det(A) 6= 0 if and only if the matrix A is invertible and that det(AB) = det(A)det(B): We let GL(n; R) = fA 2 Mn(R): det(A) 6= 0g; which by the above remarks is a group under matrix multiplication, with identity the identity matrix, I. This is called the general linear group. Using the fact that ∗ det : GL(n; R) ! R is a homomorphism, we see that the kernel of this map is the normal subgroup, denoted SL(n; R) = fA 2 GL(n; R): det(A) = 1g and called the special linear group. The groups, GL(n; C); SL(n; C) are defined similarly. t Given a matrix, A = (ai;j), we let A = (aj;i) denote the transpose and ∗ let A = (a ¯j;i) denote the conjugate, transpose, also called the adjoint. The orthogonal matrices, O(n) are defined by O(n) = fA 2 Mn(R): t A A = Ig; which is easily seen to be a subgroup of GL(n; R) and the special orthogonal matrices by SO(n) = fA 2 O(n): det(A) = 1g; which can be easily seen to be a normal subgroup of O(n): Similarly, the unitary matrices, U(n) are defined by U(n) = fA 2 ∗ Mn(C): A A = Ig and the special unitary matrices, SU(n) are defined by SU(n) = fA 2 U(n): det(A) = 1g: Problem 1.11. Let SL(n; Z) denote the set of n × n matrices with inte- ger entries whose determinant is 1. Prove that SL(n; Z) is a subgroup of 4 VERN PAULSEN SL(n; R); but is not a normal subgroup.(Hint: Cramer's Rule.) Exhibit in- finitely many matrices in SL(2; Z). Problem 1.12. Prove that SO(n) is a subgroup of SL(n; R). Is it a normal subgroup? Problem 1.13. Let Hn = fA 2 GL(n; C): jdet(A)j = 1g: Prove that Hn is a normal subgroup of GL(n; C) and that SL(n; C) is a normal subgroup of Hn. Identify the quotient groups, GL(n; C)=Hn and Hn=SL(n; C), up to isomorphism. The Permutation Groups Let X be any non-empty set. Any one-to-one, onto function, p : X ! X; is called a permutation. Note that the composition of any two permutations is again a permutation and that every permutation function has a function inverse that is also a permutation. Also if idX : X ! X denotes the identity map, then p ◦ idX = idX ◦ p = p: Thus, the set of permutations of X, with product defined by composition forms a group with identity, e = idX : This group is denoted, Per(X). Note that this group, up to isomorphism, only depends on the cardinality of X. Indeed, if Y is another set of the same cardinality as X and φ : X ! Y is a one-to-one, onto map, then there is a group isomorphism, π : P er(X) ! P er(Y ) givien by π(p) = φ ◦ p ◦ φ−1: Thus, when X is a set with n elements, Per(X) can be identified with the set of permutations of the set f1; : : : ; ng and this group is called the symmetric group on n elements and is denoted Sn: Free Groups with Generators and Relations The free group F2 on two generators, say a,b, consists of all expres- sions of the form ai1 bj1 ··· aim bjm ; where m is an arbitrary, non-negative integer and i1; j1; : : : ; im; jm are arbitrary integers.
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