Lecture 31: One-Parameter Subgroups

Lecture 31: One-Parameter Subgroups 31 One-Parameter Subgroups 31.1 Review Last time, we talked about one-parameter subgroups. Defnition 31.1 A one-parameter group in GLn (C) is a diferentiable homomorphism φ : R −→ GLn (C). For a matrix A ∈ Matn×n (C), the matrix exponential is eA := 1 + A + 1 2 1 A + A3 + · · · , 2! 3! which converges to a matrix in GLn (C).97 For example, φA (t) = etA is a one-parameter group.98 Example  31.2    1 0 1 0 If A = , then An = for all n ≥ 1. Then 0 0 0 0   X   X 1 1 0 1 0 e A n e = A = + = 1 0 0 0 0 n! n≥0 n≥1  0 . 1 Example 31.3  0 Similarly, for A = 0    1 0 0 2 ,A = = A3 = · · · . Then 0 0 0 A e = 31.2  1 0   0 0 + 1 0   1 1 = 0 0  1 . 1 Properties of the Matrix Exponential The matrix exponential fulflls several nice properties. • The product is the exponential of the sum: esA etA = e(s+t)A . In fact, if AB = BA, then eA eB = eA+B , but they must commute.99    λ  λ1 · · · 0 e1 · · · 0  ..  , then eA =  .. . . ..  . .. • If A =  ... . . . .  . 0 ··· λn 0 ··· eλn • If B = P AP −1 , then eB = P eA P −1 . This allows us to easily take the matrix exponential of any diagonalizable matrix. Example  31.4    0 2π 2πi 0 If A = , it has eigenvalues 2πi and −2πi, so diagonalizing gives P AP −1 = . −2π 0 0 2πi   −1 1 0 Then P eA P −1 = eP AP = , since e2πi = 1. Since eA is conjugate to the identity matrix, eA 0 1 itself must be the identity matrix.  In particular, e exponential. 97 With 0 −2π   2π  0  =e  0 0 0 0  , and so the matrix exponential is not injective, unlike the normal the metric ||M || = maxi,j |mij |, every entry converges. is called a one-parameter "subgroup," but it does not have to be injective; it can wrap around. P Ak B ℓ 99 The key fact here is that 1 (A + B)n = k+ℓ=n k! ℓ! when AB = BA; matrix multiplication is not commutative so it is not n! always true. 98 It Lecture 31: One-Parameter Subgroups • Defning the derivative of a matrix to be d dt  a(t) b(t) c(t) d(t)   = a′ (t) c′ (t)  b′ (t) , the derivative is d′ (t)   t2 I + tA + A2 + · · · 2 t2 = 100 0 + A + tA2 + A3 + · · · 2 = AetA , d tA d (e ) = dt dt similarly to the normal exponential. 31.3 One-Parameter Subgroups The matrix exponential is related to one-parameter subgroups in the following manner. Proposition 31.5 Every one-parameter group in GLn (C) is of the form φ(t) = etA for a unique matrix A ∈ Matn×n (C). Proof. We prove uniqueness and existence. • Uniqueness. If φ(t) = etA , then φ′ (t) = AetA , so φ′ (0) = A. So the coefcient A in the one-parameter subgroup is given by taking the derivative and evaluating at 0.101 • Existence. Given φ(t), set A := φ′ (0) ∈ Matn×n . Since φ is a homomorphism, φ(s + t) = φ(s)φ(t) for ∂ all s and t. Taking the derivative ∂s , φ′ (s + t) = φ′ (s)φ(t). Plugging in s = 0, we get φ′ (t) = Aφ(t), and we also have φ(0) = In . Since this is a linear frst-order ordinary diferential equation with an initial condition, there is a unique solution, which is φ(t) = etA . Defnition 31.6 For G ≤ GLn (C), a one-parameter group in G is a one-parameter group φ(t) in GLn (C) such that φ(t) ∈ G for all t ∈ R. For a one-parameter group in G, φ(t) = etA for some A ∈ Matn×n (C) as well. Guiding Question Given a group G, what are the one-parameter groups in G? What is the corresponding set of matrices A for which etA ∈ G for all t? Let’s see an example. Example 31.7 (Diagonal Matrices) Let    λ1  G =  ...   0 ··· .. . ···  0   ..  ≤ GL (C)  n .   λn where λi ̸= 0. The one-parameter groups in G are determined by the matrices A such that etA ∈ G for all t ∈ R. Here, etA ∈ G for all t ∈ R if and only if A is diagonal. 101 Thinking of φ as a trajectory, A is essentially the velocity of the particle when it is passing through the identity. Lecture 31: One-Parameter Subgroups Proof. If  φ(t) = etA λ′1 (t) · · ·  .. .. ′ then φ (t) =  . . 0 ···  λ1 (t) · · ·  .. .. = . . 0 ···  0 ...  , λn (t)  0 ...  . Then ′ λn (t) λ′1 (0)  A = φ′ (0) =  ...  0 ··· .. . ···  0 ...   λ′n (0) must be diagonal.  a1 · · ·  .. . . If A =  . .   ta 0 e 1  . . tA ..  is diagonal, then tA is diagonal, and so e =  .. 0 · · · an 0 diagonal matrix A does correspond to a one-parameter subgroup in G. ··· .. . ···  0 .  ..  ∈ G. So every etan We can also do the same with upper triangular invertible matrices. Example31.8 (Upper Triangular Matrices)   c · · · c   11 1n    .. ...  Let G =  ...  ≤ GLn (C), where cii ̸= 0 for all i. Then etA ∈ G for all t ∈ R if and only if .     0 · · · cnn   a11 · · · ⋆  .. .  . .. A= . .. . 0 ··· ann c′11 (0)  Proof. If φ(t) is upper triangular, then A = φ′ (0) =  ... 0  ··· .. . ···  c′1n (0) ...   must also be upper triangular. c′nn (0) Also, if A is upper triangular, so is An for all n, and thus so is etA . So the image of φ is in G. Problem 31.9 For  1  .. G = . 0 what are the corresponding matrices A? 0 a The answer is that A is of the form  .. . 0  ··· .. . ···  ⋆ . ..  ≤ GLn (C), 1 a ··· .. . ···  ⋆  ... . 0 We can also look at the one-parameter groups for unitary matrices. Example 31.10 (Unitary Matrices) For Un = {M ∗ = M −1 } ≤ GLn (C), etA ∈ Un if and only if A∗ = −A is skew-Hermitian for some matrix A ∈ Matn×n (C). Lecture 31: One-Parameter Subgroups Proof. We have  ∗ ∗ A2 (A∗ )2 (e ) = I + A + + ··· = I ∗ + A∗ + + · · · = e(A ) . 2! 2! A ∗ ∗ ∗ If etA is unitary, then (etA )∗ = (etA )−1 , so etA = e−tA . Diferentiating gives A∗ etA = −Ae−tA , and taking t = 0 gives A∗ = −A. ∗ Conversely, if A∗ = −A, then (etA )∗ = etA = e−tA = (etA )−1 , and so etA ∈ Un for all t.