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Representations of SL(2, R) Representations of SL(2; R) Mateusz S. Szczygie l Faculty of Physics University of Warsaw June 8, 2018 Plan of the presentation 1. Lie algebra sl(2; C) and sl(2; R). 2. Complex representations of sl(2; C). 3. Representations of sl(2; R). 4. Lie group SL(2; R). 5. Unitary representations of SL(2; R). First we will start from the complex representations of sl(2; C), because it is a complexification of sl(2; R). Also we first want to outline the theory on the level of Lie algebras in order to obtain the Lie group representations by the exponential map. Lie algebra sl(2; C) It consists of 2 by 2 traceless matrices equipped with the scalar product hXjY i = Tr(XY ); X; Y 2 sl(2; C): (1) We have the orthogonal basis of sl(2; C) consisting of Pauli matrices: 0 1 0 −i 1 0 σ = ; σ = ; σ = ; (2) 1 1 0 2 i 0 3 0 −1 and the commutation relation hiσ iσ i iσ i ; j = − k : (3) 2 2 ijk 2 Lie algebra sl(2; C) It is convenient to introduce a triplet of operators A−;A+;N, which we will call the standard triplet, satisfying commutation relations [N; A±] = ±A±; [A+;A−] = 2N: (4) Example of such a triplet: 1 0 1 0 0 2 0 A+ = ;A− = ;N = 1 : (5) 0 0 1 0 0 − 2 It can be constructed as another basis of sl(2; C) by transformation 1 1 A = (σ ± iσ );N = σ (6) ± 2 1 2 2 3 Representations of sl(2; C) Let us introduce the Casimir operator 1 C = (σ2 + σ2 + σ2); (7) 4 1 2 3 in terms of the triplet A−;A+;N 1 C = (A A + A A ) + N 2: (8) 2 + − − + The Casimir operator commutes with the representation. Let us define a specific representation l −1 l 2 l A− = @w + lw ;A+ = −w @w + lw; N = w@w; (9) where l 2 C. It satisfies the sl(2; C) commutation relations. The Casimir operator reads Cl = l(l + 1): (10) Representations of sl(2; C) Let us define the space of monomials η k W := fw j k 2 Z + ηg: (11) We have the representation l,η η sl(2; C) 3 X 7! X 2 L(W ); (12) l,η parametrized by l; η 2 C. The action of X : l k k−1 A−w = (k + l)w ; l k k+1 A+w = (−k + l)w ; (13) N lwk = kwk: l,η l,η+n We have X = X for n 2 Z. Representations of sl(2; C) 1. The representation Xl,η is irreducible if and only if l; −l2 = Z + η. 2. Consider Xl;−l. Then it has an invariant subspace fwk j k = −l; −l + 1;:::g. Let us define Xl;lw as Xl;−l restricted to this subspace. It is 1 irreducible if l 6= 0; 2 ; 1;::: 3. Consider Xl;l. Then it has an invariant subspace fwk j k = : : : ; l − 1; lg. Let us define Xl;hw as Xl;l restricted to this subspace. It is 1 irreducible if l 6= 0; 2 ; 1;::: 1 l;−l l;l 4. If l = 0; 2 ; 1;:::, then X = X and we have an invariant subspace fwk j k = −l; : : : ; lg. Let us define Xfin as Xl;−l = Xl;l restricted to this subspace. Representations of sl(2; C) l; −l2 = Z + η l − 2 l − 1 l l + 1 l + 2 lowest weight −l − 2 −l − 1 −l −l + 1 −l + 2 highest weight l − 2 l − 1 l l + 1 l + 2 finite −l − 1 −l l l + 1 Equivalence of representations sl(2; C) For l; −l2 = Z + η let us introduce diagonal linear map l,η l,η Wl : X ! X . Γ(−l + k) W wk := wk: (14) l Γ(l + 1 + k) By direct calculation we can check that −1 l −1−l Wl X Wl = X ;X 2 sl(2; C): (15) This shows the equivalence of representations Xl,η and X−l−1,η for l; −l2 = Z + η. This condition is given by the behavior of the Γ function. Consistently the Casimir operator is invariant with respect to transformation l 7! −1 − l: Cl = l(l + 1) = C−1−l: (16) Alternative forms of representation Let us compute Xl;− := wlXlw−l. We obtain l;− A− = @w; l;− N = w@w − l; (17) l;− 2 A+ = −w @w + 2lw: With this representation in the case of Xl;lw, the representation space is fwkjk = 0; 1;:::g. For the representation Xl;fin, the representation space is fwkjk = 0; 1;:::; 2lg. Alternative forms of representation Another form of representations is Xl;+ := w−lXlwl. We obtain l;+ A− = @w + 2l; l;+ N = w@w + l; (18) l;+ 2 A+ = −w @w: With this representation in the case of Xl;lw, the representation space is fwkjk = 0; −1;:::g. For the representation Xl;fin, the representation space is fwkjk = 0; −1;:::; −2lg. SL(2; R) group It consists of 2 by 2 matrices over R whose determinant is equal ∼ to 1. Its center is f1; −1g = Z2. We have the group P SL(2; R) := SL(2; R)=Z2 with the neutral element as a center. This group is isomorphic to homographies preserving C+ := fz 2 C : Im z > 0g. The important decomposition (valid also for SL(2; C)) h h 1 0 1 t 1 0 11 12 = ; (19) h21 h22 s2 1 0 1 s1 1 where h11 − 1 h22 − 1 t = h12; s1 = ; s2 = : (20) h12 h12 If h12 = 0 the decomposition is invalid. We will treat this decomposition as a tool to heuristically construct group representations. Representations of SL(2; R) Alternative form of decomposition 1 0 1 t 1 0 = es2A− etA+ es1A− : (21) s2 1 0 1 s1 1 To find representations of the group we need to exponentiate l l l A+;N ;A−, obtaining tAl −1 l e − f(w) = (1 + tw ) f(w + t); l etN f(w) = f(etw); (22) tAl l w e + f(w) = (1 + tw) f : 1 + wt Thus we have the representation by homographies l −1 l l h11w + h12 h f(w) = (h11 + h12w ) (h21w + h22) f (23) h12w + h22 Representations of SL(2; R) l;− l;− l;− By using operators A+ ;N ;A− we have tAl;− e − f(w) = f(w + t); l;− etN f(w) = e−lt f(etw); (24) tAl;− l w e + f(w) = (1 + tw) f : 1 + wt The representation l;− 2l h11w + h12 h f(w) = (h21w + h22) f (25) h12w + h22 These formulas can suffer from possible multivaluedness of power functions. This is solved by going to the representations of the universal cover SLf (2; R) or by other tricks showed in the following slides. Lie algebra sl(2; R) Lie algebra sl(2; R) consists of matrices of the form X X X = 11 12 : (26) X21 −X11 It has a scalar product Tr(XY ) = 2X11Y11 + X12Y21 + X21Y12: (27) • Let us consider representation X of sl(2; R). It can be extended by complex linearity to representation of sl(2; C). Therefore representations of sl(2; R) are the same as complex representations of sl(2; C). Representations of sl(2; R) The main parameter describing representations of sl(2; C) was l and η. This parametrization is standard for the theory of su(2). In the case of sl(2; R) the popular in the literature parametrization is given by parameter m, defined by m + 1 m = −2l − 1; l = − : (28) 2 The representation Xl with l expressed by m, will be denoted Xm. In our new notation we have m + 1 Amwk = k − wk−1; − 2 N mwk = kwk; (29) m + 1 Amwk = − k − wk+1: + 2 m m2 1 The Casimir operator is given by C = 4 − 4 . Representations of sl(2; R) 1. The representation preserving Wη is dented Xm,η. It is irreducible if and only if m; −m2 = 2Z + 1 + 2η. m; m+1 2. Consider X 2 . Then it has an invariant k m+1 m+3 subspace fw j k = 2 ; 2 ;:::g. m;lw m; m+1 Let us define X as X 2 restricted to this subspace. It is irreducible if m 6= −1; −2; −3;:::. m;− m+1 3. Consider X 2 . Then it has an invariant subspace k m+3 m+1 fw j k = :::; − 2 ; − 2 g. m;hw m;− m+1 Let us define X as X 2 restricted to this subspace. It is irreducible if m 6= −1; −2; −3;:::. m; m+1 m;− m+1 4. If m = −1; −2; −3;:::, then X 2 = X 2 and we k m+1 m+1 have invariant subspace fw j k = 2 ;:::; − 2 g. m;lw m;hw X and X give representations of P SL(2; R) for m 2 2Z + 1 and of SL(2; R) for m 2 Z. Unitarity of representations of SL(2; R) Let η 2 R. 1. Suppose that we equip the space Wη with the sesquilinear scalar product (fjg) where f; g 2 W, defined on the canonical basis by k k0 (w jw ) = δk;k0 : (30) Then (X−m,ηfjg) + (fjXm,ηg) = 0: (31) 2. Suppose we have another sesquilinear product, (fjg)m, where f; g 2 W, defined on the canonical basis by m 1 k k0 Γ( 2 + 2 + k) (w jw )m = δk;k0 m 1 : (32) Γ(− 2 + 2 + k) Then m,η m,η (X fjg)m + (fjX g)m = 0: (33) Unitary representations of SL(2; R) We have positive scalar product in the following cases: 1.
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