MAT 551 FINAL EXAM : PSL(2,R) AND SOo(1, 2) Recall that SL(2,R) := {A ∈M2(R) | det(A) = 1} PSL(2,R) := SL(2,R)/{I2, −I2} . Define I1,2 := 1 0 00 −1 0 0 0 −1 . We define SO(1, 2) := {A ∈M3(R) |AtI1,2A = I1,2 , det(A) = 1} . SOo(1, 2) denotes the identity component (from lecture) . The goal of this project is to show that SOo(1, 2) and PSL(2,R) are homeomorphic / isomorphic ( they are isomorphic as topological groups ) . As you can see, there are many steps along the way towards doing this. The solutions to many of the questions may rely on the solutions to preceding ones. They will be graded independently, for example, say you can’t work out problem n but you use the result of problem n correctly in solving problem n + 1. Then you will be awarded full points for problem n+ 1. There are 100 points in total. This project contains 5 pages. (1) (8pts.) Show that SO(1, 2) and SL(2,R) are topological groups. This means sev- eral things: a) First show that they are groups under matrix multiplication. b) Show that SL(2,R) is a closed subspace of M2(R). c) Show that SO(1, 2) is a closed subspace of M3(R) . d) Show that matrix multiplication and inversion are continuous in the respective subspace topologies. (2) Show that SL(2,R) is connected. Do this as follows. Let ψ be the map from the product of the circle S1 and the disk D = R>0 × R (why is this the disk?) into SL(2,R) given by: ψ : S1 ×D → SL(2,R). ψ((cos θ, sin θ), (y, x)) := ( cos θ sin θ − sin θ cos θ )( y x 0 y−1 ) . 1 2 551 FINAL PROJECT You are to show that this map is a homeomorphism . a) (2pts.) Show that ψ is continuous. You may take for granted that the trig. func- tions are continuous. b) (5pts.) Show that ψ is injective. Next you will show that ψ is surjective. This is harder. Given A ∈ SL(2,R), write A = ( a b c d ) = (v1, v2) . So v1 is the first column vector of A, and v2 is the second. Define a matrix B by B := (||v1||−1 −v1·v2||v1|| 0 ||v1|| ) . c1) (5pts.) Show that AB is an orthogonal matrix with determinant one. c2) (6pts.) Let C be any orthogonal matrix with determinant one. Show that for some angle α we have: C = ( cosα sinα − sinα cosα ) . c3) (2pts.) Show that ψ−1 is given by ψ−1(A) = (AB, B−1) . c4) (2pts.) Now conclude that SL(2,R) is connected (the continuous image of a connected set is connected) . You may assume the circle is connected. c5) (2pts.) Why is PSL(2,R) connected in the quotient topology? (See Thm. 23.5 of the textbook.) Now that we know some things about the topological group PSL(2,R), lets try to see how this could possibly be the same as SO0(1, 2). First of all, SO0(1, 2) is a collection of three by three matrices, but the elements of SL(2,R) are given two by two matrices! We need to represent the elements of SL(2,R) as three by three’s. In other words, given A ∈ SL(2,R) we would like to map A to a 3× 3 matrix, that is, we would like to construct a homomorphism from SL(2,R) into GL3(R). First, some preliminaries. sl2(R) := {u ∈M2(R) | Tr(u) = 0} Tr(u) denotes the trace of the 2× 2 matrix u. Observe that sl2(R) is a three dimensional vector space overR. Given σ ∈ SL(2,R) we define Ad(σ)(u) := σuσ−1 where u ∈ sl2(R) . 551 FINAL PROJECT 3 (3) (2pts.) Show that Ad(σ)(u) ∈ sl2(R) if and only if u ∈ sl2(R). (4) (2pts.) Show that Ad(σ) acts by linear transformations on sl2(R), in other words, for any u, v ∈ sl2(R) and scalars α, β Ad(σ)(αu+ βv) = αAd(σ)(u) + βAd(σ)(v) . (5) (2pts.) Show that Ad(σ · τ) = Ad(σ) ◦ Ad(τ) for all σ, τ ∈ SL(2,R). (6) (2pts.) Show that Ad(I2) = I . That is, Ad takes the identity to the identity. (7) (2pts.) Conclude that Ad is a homomorphism of SL(2,R) into invertible linear transformations of sl2(R): Ad : SL(2,R)→ GL(sl2(R)) . (8) (6pts.) Show that the kernel of Ad is {I2,−I2} . The next series of questions is devoted to characterizing the image of Ad. (9) (2pts.) Show that E1 := ( 0 1 −1 0 ) , E2 := ( 1 0 0 −1 ) , E3 := ( 0 1 1 0 ) is a basis of sl2(R) . (10) (2pts.) Show that( x y z −x ) = y − z 2 E1 + xE2 + y + z 2 E3 . (11) (8pts.) Let σ ∈ SL(2,R) be given by σ = ( a b c d ) . Show that the matrix presentation ofAd(σ)with respect to the ordered basisE1, E2, E3 is given by [Ad(σ)] = 12(a2 + b2 + c2 + d2) −(ab+ cd) 12(a2 − b2 + c2 − d2)−(ac+ bd) ad+ bc bd− ac 1 2 (b2 + a2 − (c2 + d2)) cd− ab 1 2 (a2 + d2 − (c2 + b2)) . (12) (2pts.) Use the preceding problem to show that Ad is a continuous map. 4 551 FINAL PROJECT Next we define a quadratic form Q on sl2(R) by the following rule : Q(u, v) := −1 2 Tr(uv) u , v ∈ sl2(R) . (13) Show that Q has signature (1, 2) by direct calculation. Do this by checking that a) (1pt.) Q(E1, E1) = 1 b) (1pt.) Q(E2, E2) = −1 c) (1pt.) Q(E3, E3) = −1 d) (1pt.) Q(Ei, Ej) = 0 for i 6= j. (14) (2pts.) Show that Q(Ad(σ)(u), Ad(σ)(v)) = Q(u, v) for all σ ∈ SL(2,R) and u, v ∈ sl2(R) . (15) (2pts.) Explain why this implies that Ad(SL(2,R)) ⊂ O(1, 2) . (16) (3pts.) Use problems c4), (6), (12) and our two connectedness Theorems from lecture to show that, in fact Ad(SL(2,R)) ⊆ SO0(1, 2) . (17) (2pts.) We have the commutative diagram, where pi is the quotient map, and ι is the induced map. SL(2,R) pi Ad // SOo(1, 2) PSL(2,R) ι 77 Explain why ι is injective. (18) (6pts.) lemma. Let G and H be groups (not necessarily topological groups) . Let ρ be a homomorphism from H to G. Let G act on a set S, and let o ∈ S be any point. Assume that ρ(H) acts transitively on S, and contains the stabilizer of o in G, i.e. Go ⊂ ρ(H). Show that ρ(H) = G, in other words ρ is surjective. You are to apply this lemma in the situation where ρ = Ad, G = SO0(1, 2), 551 FINAL PROJECT 5 H = SL(2,R) and S is defined to be a sheet of the hyperbolic paraboloid S = {u = xE1 + yE2 + zE3 ∈ sl2(R) |Q(u, u) = 1 , x > 0} . (19) a) (2pts.) Use problem 11 to show that [Ad(σ)] = 12(a2 + a−2) 0 12(a2 − a−2)0 1 0 1 2 (a2 − a−2) 0 1 2 (a2 + a−2) σ = (a 0 0 a−1 ) . b) (2pts.) Let t := ln(a2). Write the previous matrix down in terms of cosh(t) and sinh(t). c) (3pts.) Next show that [Ad(σ)] = 1 0 00 cos 2θ sin 2θ 0 − sin 2θ cos 2θ σ = ( cos θ sin θ− sin θ cos θ ) . (20) (6pts.) Compute the stabilizer of E1 ∼ (1, 0, 0) in SO0(1, 2). Compare this with part c) of problem (19) and conclude that the image of Ad contains SO0(1, 2)E1 . (21) (6pts.) Finally, show that the image of Ad acts transitively on S. Conclude by the lemma that Ad is surjective . (Hint: Draw pictures of the orbits on the paraboloid under the action of the preceding matrices)
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