REPRESENTATIONS OF LIE GROUPS AND LIE ALGEBRAS

Transcription

1 REPRESENTATIONS OF LIE GROUPS AND LIE ALGEBRAS JIAQI JIANG Abstract. This papr studis th rlationship btwn rprsntations of a Li group and rprsntations of its Li algbra. W will mak th corrspondnc in two stps: First w shall prov that a givn rprsntation of a Li group will provid us with a corrsponding rprsntation of its Li algbra. Scond, w shall go backwards and s whthr a givn rprsntation of a Li algbra will hav a corrsponding rprsntation of its Li group. Contnts 1. Introduction to Li groups and Li algbras 1 2. From Rprsntations of Li groups to Li algbras 8 3. From Rprsntations of Li algbras to Li groups 11 Acknowldgmnts 14 Rfrncs Introduction to Li groups and Li algbras In this sction, w shall introduc th notion of Li group and its Li algbra. Sinc a Li group is a smooth manifold, w shall also introduc som basic thory of smooth manifolds hr. Dfinition 1.1. A Li group is a smooth manifold G that also has a group structur, with th proprty that th multiplication map m : G G G and th invrsion map i : G G, givn rspctivly by ar both smooth maps. m(g, h) = gh, i(g) = g 1, On of th most important xampls of Li groups is th group GL(V ). Suppos V is som ral or complx vctor spac. Thn th group GL(V ) dnots th st of all invrtibl linar transformations from V to itslf. Th group multiplication is just composition. If V is finit-dimnsional, thn any basis for V will induc an isomorphism of GL(V ) with GL(n, R) or GL(n, C), with n = dim V. (Hr, GL(n, R) is th gnral linar group consisting of all n n matrics with ral ntris. Analogously, GL(n, C) is th complx gnral linar group). 1 Thrfor, onc w pick a basis for V, w gt a chart φ on GL(V ) which snds an lmnt of GL(V ) to its matrix in th chosn basis (which can b thought as an lmnt Dat: Octobr 1st Both GL(n,R) and GL(n,C) ar quippd with a standard smooth structur. For dtails, plas rfr to chaptr 1 of L s Introduction to Smooth Manifolds. 1

2 2 JIAQI JIANG in R 2n. Obviously, this singl chart covrs th ntirty of GL(V ), thus forming an atlas. This maks GL(V ) a Li group. If w choos anothr basis for V, thn th transition map btwn th two charts is givn by a map of th form A BAB 1 with B th transition matrix btwn th two bass, it follows that th transition maps ar smooth. Thus, this smooth manifold structur on GL(V ) is indpndnt of th choic of basis. This Li group GL(V ) will play an important rol in th latr sctions of this papr. First w rviw th concpt of tangnt vctors on a manifold. Thr ar in fact svral diffrnt ways to dfin tangnt vctors to a smooth manifold. Hr, w will dfin a tangnt vctor at a point p in a smooth manifold M as a drivation at p. Dfinition 1.2. Lt M b a smooth manifold and p b a point of M. A drivation at p is a linar map X : C (M) R that satisfis for any f, g C (M). X(fg) = f(p)xg + g(p)xf Th intuition bhind such dfinition is that w can rgard a drivation as a dirctional drivativ ncountrd in multivariabl calculus thory. Th st of all drivations of C (M) at p is calld th tangnt spac to M at p, and is dnotd by T p M. An lmnt of T p M is calld a tangnt vctor at p. For any X, Y T p M, w can dfin X + Y T p M and cx T p M for all c R by (X + Y )f = Xf + Y f, (cx)f = c(xf) for all f C (M). Undr ths oprations, T p M is clarly a vctor spac. Morovr, it is important to notic that th construction of th tangnt spac at a point is purly a local construction. Proposition 1.3. Suppos M is a smooth manifold, p M, and X T p M. If f, g C (M) and both functons agr on som nighborhood of p, thn Xf = Xg. Th proof of this proposition rquirs th us of smooth bump functions. Th xistnc of such functions is on of th proprtis of a smooth manifold. 2 Proof. Lt h = f g. Thn by linarity of drivations, it is sufficint to show that Xh = 0 whnvr h vanishs in a nighborhood of p. Lt ψ C (M) b a smooth bump function that is qual to 1 on th support of h and is supportd in M \ {p}. Sinc ψ 1 whr h is not zro, thn th product ψh is idntically qual to h. Sinc h(p) = ψ(p) = 0, thn Xh = X(ψh) = 0, which follows from th dfinition of a drivation. Having dfind tangnt vctors on a manifold, w thn nd to xplor how tangnt vctors bhav undr smooth maps. Suppos M and N ar two smooth manifolds and F : M N is a smooth map. For ach p M, thr is a natural map from T p M to T F (p) N, which is calld th push-forward of F : Dfinition 1.4. If M and N ar smooth manifolds and F : M N is a smooth map, for ach p M w dfin a map F : T p M T F (p) N, calld th push-forward of F, by (F X) (f) = X (f F ). for all f C (N). 2 Th proof can b found in chaptr 2 of L s Introduction to Smooth Manifolds.

3 REPRESENTATIONS OF LIE GROUPS AND LIE ALGEBRAS 3 It is straightforward to show that th map F : T p M T F (p) N is a linar map btwn th two vctor spacs T p M and T F (p) M. Morovr, it is not hard to vrify th following lmma by simply following th dfinition of push-forward. Lmma 1.5. Lt F : M N and G : N K b smooth maps, and lt p M. (a)f : T p M T F (p) N is linar. (b)(g F ) = G F (c)(id M ) = Id TpM (d)if F is a diffomorphism, thn F : T p M T F (p) N is an isomorphism With th concpt of push-forward, w can now construct a basis for T p M for any p M by th following thorm: 3 Thorm 1.6. Lt M b a smooth n-manifold. For any p M, T p M is an n- dimnsional vctor spac. If (U, φ = (x i )) is any smooth chart containing p, th coordinat vctors ( / x 1 p,..., / x n p ) form a basis for T p M, whr x i ( φ 1) p x i. φ(p) With th concpt of push-forward, w can now xtnd th familiar notion of tangnt vctors to a smooth curv in R n to smooth curvs in manifolds. Dfinition 1.7. If M is a smooth manifold, w dfin a smooth curv in M to b a smooth map γ : J M, whr J R is an intrval. (In most situations, w shall lt J b an opn intrval.) Sinc w can rgard J as an opn submanifold of R with th standard smooth structur, thn it is natural to dfin th tangnt vctor to a smooth curv in M as th push-forward of d/dt to T t0 R. Dfinition 1.8. If γ is a smooth curv in a smooth manifold M, thn th tangnt vctor to γ at t 0 J is th vctor ( ) γ d (t 0 ) = γ T γ(t0)m, dt whr d/dt t0 is th standard coordinat basis for T t0 R. Vctor filds on R n ar familiar objcts to us. Now, w wish to dfin a vctor fild on an abstract smooth manifold M. In ordr to mak a prcis dfinition, w shall first introduc th concpt of th tangnt bundl of M. Dfinition 1.9. Givn a smooth manifold M, th tangnt bundl of M, dnotd by T M, is th disjoint union of th tangnt spacs at all points of M: T M = T p M, t0 p M togthr with th smooth structur dfind in Thorm An lmnt of T M will b writtn as an ordrd pair (p, X). 3 W hav omittd th proof hr. Radrs can find th proof in chaptr 3 of L s Introduction to Smooth Manifolds.

4 4 JIAQI JIANG Th tangnt bundl is quippd naturally with a projction map π : T M M by π(p, X) = p. From our dfinition, th tangnt bundl sms to b mrly a collction of vctor spacs on M. Howvr, with th following thorm, w s that T M has a smooth manifold structur on it such that th projction map π dfind bfor is smooth with rspct to this structur. Thorm For any smooth n-manifold M, th tangnt bundl TM has a topology and smooth structur that mak it into a 2n-dimnsional smooth manifold. With this structur, π : T M M is a smooth map. Howvr, w shall not provid th proof hr. 4 Th smooth charts in th smooth structur dfind by th thorm ar (π 1 (U), φ), whr (U, φ) ar smooth charts for M and th map φ : π 1 (U) R 2n is dfind by 5 ) φ (v i x i = ( x 1 (p),..., x n (p), v 1,..., v n). p Now, w ar rady to dfin a vctor fild on a smooth manifold M. Dfinition If M is a smooth manifold, a vctor fild on M is a continuous map Y : M T M, usually dnotd as p Y p, satisfying (1.12) π Y = Id M A smooth vctor fild on M is thn a smooth map Y : M T M which satisfis (1.12). A rough vctor fild on M is a map (th map dos not nd to b continuous) Y : M T M satisfying (1.10). W will us th notation X(M) to dnot th st of all smooth vctor filds on M. It bcoms a vctor spac undr pointwis addition and scalar multiplication by: (ay + bz) p = ay p + bz p. In additon, if f C (M) and Y X(M), thn fy : M T M dfind by (fy ) p = f(p)y p is anothr smooth vctor fild. W can also rgard Y X(M) as a map C (M) C (M) by f Y f, whr Y f is dfind by Y f(p) = Y p f. From th product rul for tangnt vctors, it follows that for f, g C (M) and Y X(M), (1.13) Y (fg) = fy g + gy f. In gnral, w call a map from C (M) to C (M) which is linar ovr R and satisfis quation (1.13) as a drivation. Actually, th drivations of C (M) can b idntifid with smooth vctor filds on M. W shall mrly quot th statmnt of th following proposition hr. 6 Proposition Lt M b a smooth manifold. A map Y : C (M) C (M) is a drivation if and only if it is of th form Yf = Y f for som Y X(M). 4 For proof, s chaptr 4 of L s Introduction to Smooth Manifolds. 5 W hav usd Einstin summation convntion in th following quation. 6 For proof, s chaptr 4 of L s Introduction to Smooth Manifolds.

5 REPRESENTATIONS OF LIE GROUPS AND LIE ALGEBRAS 5 Sinc w hav dfind th push-forward of a tangnt vctor, it is natural to xtnd th ida to th vctor filds. Dfinition Suppos F : M N is a smooth map btwn th smooth manifolds M and N. Lt Y X(M) and Z X(N). W say that th vctor filds Y and Z ar F-rlatd if F Y p = Z F (p) for ach p M. Givn two smooth vctor filds V and W, w can gt a third smooth vctor fild, calld Li brackt of V and W, dfind by [V, W ] f = V W f W V f. W shall show that [V, W ] dfind as abov is indd a smooth vctor fild. Lmma Th Li brackt [V, W ] of V, W X(M) is also a smooth vctor fild. Proof. By Proposition 1.14, it is sufficint to show that [V, W ] is a drivation of C (M). For any f, g C (M), w hav [V, W ] (fg) = V (W (fg)) W (V (fg)) = V (fw g + gw f) W (fv g + gv f) = f(v W g) + V fw g + g(v W f) + V gw f f(w V g) W fv g g(w V f) W gv f = f([v, W ]g) + g([v, W ]f) From th dfinition, it is not hard to s that Li brackt has following proprtis. Lmma Th Li brackt satisfis following proprtis for all V, W, Y X(M): (a)bi-linarity: For a, b R, (b)anti-symmtry: (c)jacobi Idntity: [av + bw, Y ] = a[v, Y ] + b[w, Y ], [Y, av + bw ] = a[y, V ] + b[y, W ]. [V, W ] = [W, V ]. [V, [W, Y ]] + [W, [Y, V ]] + [Y, [V, W ]] = 0. Proof. Bi-linarity and anti-symmtry follow simply from our dfinition of th Li brackt. For th proof of Jacobi idntity: ([V, [W, Y ]] + [W, [Y, V ]] + [Y, [V, W ]])f = V [W, Y ]f [W, Y ]V f + W [Y, V ]f [Y, V ]W f + Y [V, W ]f [V, W ]Y f = V W Y f V Y W f W Y V f + Y W V f + W Y V f W V Y f = 0 Y V W f + V Y W f + Y V W f Y W V f V W Y f + W V Y f

6 6 JIAQI JIANG Anothr important proprty of Li brackt is xprssd in th nxt thorm. Thorm Lt F : M N b a smooth map, and lt V 1, V 2 X(M) and W 1, W 2 X(N). If V i is F-rlatd to W i for i = 1, 2, thn [V 1, V 2 ] is F-rlatd to [W 1, W 2 ]. Proof. Fix p M. Thn for any f C (N), w hav F [V 1, V 2 ] p (f) = [V 1, V 2 ] p (f F ) = (V 1 ) p ((V 2 )(f F )) (V 2 ) p ((V 1 )(f F )) = (V 1 ) p ((W 2 f) F ) (V 2 ) p ((W 1 f) F ) = (W 1 ) F (p) (W 2 f) (W 2 ) F (p) (W 1 f) = [W 1, W 2 ] F (p) (f) Th most important application of Li brackts actually occurs in th contxt of Li groups. Suppos G is a Li group. According to th dfinition, for any g G, w hav th map L g : G G, which is calld th lft translation, dfind by L g (h) = gh for all h G, is a smooth map. In fact, it is a diffomorphism of G bcaus th smooth map L g 1 is clarly th invrs for L g. A vctor fild X on G is said to b lft-invariant if it is L g -rlatd to itslf for vry g G. This mans (L g ) X h = X gh, for all g, h G. Sinc L g is also a diffomorphism, thn it follows that (L g ) X = X for vry g G. Suppos X, Y ar both smooth lft-invariant vctor filds on G, thn it follows that (L g ) (ax + by ) = a(l g ) X + b(l g ) Y = ax + by, for all a, b R, so th st of all smooth lft-invariant vctor filds on G is a linar sub-spac of X(G). Mor importantly, this sub-spac is closd undr Li brackts. Thorm Lt G b a Li group. If X and Y ar both smooth lft-invariant vctor filds on G, thn [X, Y ] is also a smooth lft-invariant vctor fild. Proof. Fix som g G. By assumption, w hav (L g ) X = X and (L g ) Y = Y. Thn by Thorm 1.18, w hav [X, Y ] is L g -rlatd to [(L g ) X, (L g ) Y ] = [X, Y ], so w hav (L g ) [X, Y ] = [X, Y ]. Thrfor, [X, Y ] is a smooth lft-invariant vctor fild on G. Now, w ar rady to dfin th Li algbra of a givn Li group. First, w will giv th dfinition for a gnral Li algbra. Dfinition A ral Li algbra is a ral vctor spac g ndowd with a bilinar map calld th brackt [, ] : g g g, which is also anti-symmtric and satisfis th Jacobi idntity for all X, Y, Z g: [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0

7 REPRESENTATIONS OF LIE GROUPS AND LIE ALGEBRAS 7 From Thorm 1.19, it follows that if G is a Li group, th st of all smooth lft-invariant vctor filds on G is a Li algbra undr th Li brackt. W call this Li algbra th Li algbra of G, and dnot it by g. Having dfind th Li algbra of a Li group, w shall first xplor som basic proprtis of g. Thorm Lt G b a Li group. Th valuation map ε : g T G, dfind by ε(x) = X, is a vctor spac isomorphism. Thrfor, dimg = dimt G = dimg. Proof. Clarly by our dfinition of ε, it is a linar map. Thrfor, to prov th thorm, w only nd to find a linar invrsion for ε. For ach V T G, w shall dfin a vctor fild V on G by (1.22) V g = (L g ) V. Clarly, if X is a lft-invariant vctor fild on G such that X = V, thn X has to b givn by th quation Now, w shall chck that V is smooth. To vrify this, it is sufficint to show that V f is smooth for any f C (U) for any opn subst U of G. 7 Lt γ : ( ɛ, ɛ) G b such that γ(0) = and γ (0) = V. 8 Thn for any g U, w hav (1.23) ( V f)(g) = V g f = ((L g ) V )f = V (f L g ) = γ (0)(f L g ) = d dt (f L g γ)(t). t=0 Lt ψ : ( ɛ, ɛ) G R b th map dfind by ψ(t, g) = f L g γ(t). Sinc th multiplication on G, f and γ ar all smooth maps, thn it follows that ψ is also smooth. By th quation (1.23), w can s that ( V f)(g) = ψ/ t (0,g). Sinc ψ is smooth, ψ/ t (0,g) dpnds smoothly on g. So V f is smooth. It is asy to vrify that V is lft-invariant. For all h, g G, by dfinition of V, w hav (L h ) V g = (L h ) (L g ) V = (L hg ) V = V hg, whr th scond quality follows from Lmma 1.5. Now, lt τ : V V b dfind as abov. Th linarity of τ follows from its dfinition. For any V T G, w hav ε(τ(v )) = ε( V ) = ( V ) = (L ) V = V. So ε τ = Id TG. On th othr hand, for any vctor fild X g, w hav (τ(ε(x))) g = (τ(x )) g = X g = (L g ) X = X g. 7 For th dtails, chck Lmma 4.6 which is provd in chaptr 4 of L s Introduction to Smooth Manifolds. 8 Th xistnc of such a smooth curv is not hard to prov. Lt (U, φ) b a smooth coordinat chart cntrd at, and lt V = V i / x i in trms of th coordinat basis. Dfin th map γ : ( ɛ, ɛ) U by γ(t) = (tv 1,..., tv n ). Thn this γ is a smooth curv with γ(0) = and γ (0) = V.

8 8 JIAQI JIANG Thus, τ ε = Id g. From th proof of Thorm 1.21, w s that if V is a lft-invariant rough vctor fild on th Li group G, w thn hav V = V. This shows that vry lft-invariant vctor fild on a Li group is a smooth vctor fild. Bfor finishing this sction, w shall prov an important proprty of connctd Li groups, as statd in th following thorm. Thorm Suppos G is a connctd Li group. Lt U G b any opn nighborhood of th idntity. Thn vry lmnt of G can b writtn as a finit product of lmnts in U, that is, U gnrats G. Proof. Lt U b an opn nighborhood of. For vry n N, w lt U n = {u 1 u 2 u n : u 1, u 2,..., u n U}. Fix som g U n. Thn thr xist som u U and v U n 1 such that g = vu = L v u. Sinc L v is a diffomorphism, thus a homomorphism, it thn follows that g L v (U) U n whr L v (U) is opn. Thrfor, U n is an opn st. Lt W = n NU n. Thn W is also an opn st. Now, w want to show that W is also closd. Lt g W, th closur of W. Sinc th invrsion map i : G G is smooth and clarly i i = Id, thn th invrsion map i is a diffomorphism of G, thus a homomorphism. So U 1 = i(u) is also a opn nighborhood of. It follows that gu 1 = L g (U 1 ) is an opn nighborhood of g. Sinc g W, thn gu 1 W. Lt h gu 1 W. Thn, thr xists som u U such that h = gu 1. So g = hu. Sinc h W, thn h U k for som k N. Thn it follows that g U k+1 W. So W = W, which implis that W is closd. Thrfor, w hav shown that W is both closd and opn. Sinc G is connctd and clarly W, thn w hav G = W = U n. This implis that U gnrats n N G. 2. From Rprsntations of Li groups to Li algbras Having introducd th concpt of a Li group and its Li algbra, w shall now xplor th connction btwn th two. In both mathmatics and physics, th study of th rprsntations of a Li group plays a major rol in th undrstanding of continuous symmtry. On of th basic tools in studying th rprsntations of Li groups is th us of th corrsponding rprsntations of Li algbras. In this sction, w shall focus on how to gt a corrsponding rprsntation of th Li algbra, givn a rprsntation of its Li group. First, w will giv th dfinition of a rprsntation of a Li group and that of a rprsntation of a Li algbra. Dfinition 2.1. A rprsntation of a Li group G on a finit-dimnsional vctor spac V is a smooth group homomorphism ρ : G GL(V ). Namly, for any h, g G and v V, w hav ρ(hg)(v) = (ρ(h) ρ(g))(v).

9 REPRESENTATIONS OF LIE GROUPS AND LIE ALGEBRAS 9 Dfinition 2.2. A rprsntation of a Li algbra g on a finit-dimnsional vctor spac V is a Li algbra homomorphism ρ : g gl(v ) = End(V ). Namly, for any X, Y g and v V, w hav ρ([x, Y ])(v) = (ρ(x) ρ(y ))(v) (ρ(y ) ρ(x))(v). Th important aspct of th Li algbra of a Li group ariss from th fact that ach Li group homomorphism inducs a Li algbra homomorphism, as statd in th nxt thorm. Thorm 2.3. Suppos G and H ar Li groups and lt g, h b thir Li algbras. If F : G H is a Li group homomorphism, thn for vry X g, thr is a uniqu vctor fild Y h that is F -rlatd to X. Lt this vctor fild b dnotd by F X, thn th map F : g h dfind in this way is a Li algbra homomorphism. Proof. If thr xists a vctor fild Y h that is F -rlatd to X, thn w must hav Y = F X. Thn according to th rmark w hav mad aftr Thorm 1.21, Y is uniquly dtrmind by Y = Y = F X. Now, w nd to show that th vctor fild Y dtrmind this way is indd F - rlatd to X. Sinc F is a Li group homomorphism, w hav So by Lmma 1.5, w gt It thn follows that (F L g )(g ) = F (gg ) = F (g)f (g ) = (L F (g) F )(g ). F (L g ) = (F L g ) = (L F (g) F ) = (L F (g) ) F. F X g = F (L g ) X = (L F (g) ) F X = (L F (g) ) Y = Y F (g). This provs that X and Y ar F -rlatd. Thus, w dfin a map F : g h by X F X = Y, whr X g and Y is th uniqu vctor fild in h that is F -rlatd to X as w hav shown abov. Now what rmains to show is that F is a Li algbra homomorphism. This is a immdiat consqunc of Thorm 1.18, which stats prcisly that F [X, Y ] = [F X, F Y ]. Th map F : g h dfind in th prvious thorm will b calld th inducd Li algbra homomorphism of F. In som litratur, F is calld th diffrntial of map F. Now considr a rprsntation ρ : G GL(V ) of a Li group G on a finitdimnsional (ral or complx) vctor spac V. By dfinition, th map ρ is actually a Li group homomorphism btwn th Li group G and th Li group GL(V ). Thus, by applying Thorm 2.3, w gt a inducd Li algbra homomorphism ρ : g Li(GL(V )), whr Li(GL(V )) dnots th Li algbra of th Li group GL(V ). In fact, th map ρ is indd a rprsntation of th Li algbra g on th finit-dimnsional vctor

10 10 JIAQI JIANG spac V. This is a consqunc of th important fact that Li(GL(V )) is isomorphic to gl(v ) in th Li algbra sns, which is statd in th following thorm. Thorm 2.4. Th composition of th natural maps Li(GL(V )) T Id GL(V ) gl(v ) givs a Li algbra isomorphism btwn Li(GL(V)) and gl(v). Proof. Without loss of gnrality, w will assum that V is a finit-dimnsional complx vctor spac. Fix a basis for V. Thn according to th rmark mad in sction on, GL(V ) can b thn idntifid with GL(n, C). Analogously, gl(v ) can b idntifid with gl(n, C). This provids us with globally dfind coordinats on GL(V ) by considring th corrsponding matrix ntris Xj i of X GL(V ) rsprct to th basis w hav prviously chosn. Thn w can gt a natural isomorphism T Id GL(V ) gl(v ) which taks th form A i j ( A i ) j, Id X i j whr (A i j ) dnots th lmnt A gl(v ) with rspct to th basis w hav chosn. It thn follows that any A = (A i j ) gl(v ) dtrmins a lft-invariant vctor fild A Li(GL(V )) as dfind by quation 1.22, which in this cas bcoms ) A X = (L X ) (A i j, Xj i Id for any X GL(V ). Sinc L X is th linar map Y XY for Y GL(V ), thn it follows that th coordinat rprsntation of th push-forward (L X ) is givn by ) (2.5) (L X ) (A i j = XjA i j k, X Xj i Id X i k whr (Xk i ) is th matrix rprsntation of X undr th chosn basis. Not that Einstin summation convntion is usd in quation 2.5. It follows that for any A gl(v ), th lft-invariant vctor fild A Li(GL(V )) dtrmind by A is givn by A = XjA i j k. X Clarly, th map and th composition map X i k A gl(v ) A Li(GL(V )) X Li(GL(V )) X Id T Id GL(V ) (X i j) gl(v ) ar linar invrsions of ach othr. Thus, w hav a natural isomorphism btwn Li(GL(V )) and gl(v ) in th sns of vctor spacs. What rmains to b shown is that this natural isomorphism is indd a Li algbra isomorphism. For any

11 REPRESENTATIONS OF LIE GROUPS AND LIE ALGEBRAS 11 A, B gl(v ), w hav [ [ A, B] X = XjA i j k (2.6) = X i ja j k = X i ja j k Bk r = X i k X i k ], Xq p Br q Xr p X ( X p q B q ) r X Xr p Xq p Br q Xq p BrA q r k X X p k X ), X X i r ( X i ja j k Bk r X i jb j k Ak r X i r X p r ) (X ja i j k X whr w hav usd th fact that Xq p / Xk i is qual to 1 if p = i and q = k, and 0 othrwis, as wll as th fact that th mixd partial drivativs of a smooth function can b takn in any ordr. Evaluating quation 2.6 at th idntity map Id GL(V ) rsults [ A, B] Id = ( A i kbr k BkA i k ) r. Id X i r X i k Sinc th matrix rprsntation of [A, B] gl(v ) is just (A i k Bk r B i k Ak r), thus according to th natural isomorphism w gt [ A, B] = [A, B]. This provs that th natural isomorphism btwn Li(GL(V ) and gl(v ) is a Li algbra isomorphism. Thus w can simply idntify Li(GL(V )) with gl(v ). Combining Thorm 2.3 and Thorm 2.4, w gt th following corollary. Corollary 2.7. If ρ : G GL(V ) is a rprsntation of th Li group G on a finit-dimnsional vctor spac V, thn th inducd homormophism ρ : g gl(v ) is a rprsntation of th Li algbra g of th Li group G on th vctor spac V. 3. From Rprsntations of Li algbras to Li groups In th prvious sction, w saw that givn a rprsntation ρ : G GL(V ) of th Li group G on a finit-dimnsional vctor spac V, w obtaind an corrsponding rprsntation ρ : g gl(v ) of its Li algbra on th sam vctor spac V. Howvr, givn a rprsntation of th Li algbra g of th Li group G on a finitdimnsional vctor spac V, can w thn find an corrsponding rprsntation of th Li group G on th vctor spac V? Th answr is ys if w impos th additional condition that th Li group G is simply-connctd. In ordr to rach our ultimat goal, w shall us an important fact that vry Li subalgbra corrsponds to som Li subgroup. 9 Th proof of this fact involvs a lot of nw machinry which w hav not introducd in this papr. Thrfor, w will only stat th thorm hr without offring th proof A Li subalgbra of th Li algbra g is a linar sub-spac h g which is also closd undr brackts. A Li subgroup of a Li group G is a subgroup H of G which also has a Li group structur and is an immrsd submanifold of G. 10 For dtails of th proof, plas rfr to ithr chaptr 20 of L s Introduction to Smooth Manifolds or Lctur 8 of Rprsntation Thory:A First Cours by Fulton and Harris.

12 12 JIAQI JIANG Thorm 3.1. Suppos G is a Li group with Li algbra g. If h is any Li subalgbra of g, thn thr xists a uniqu connctd Li subgroup of G whos Li algbra is h. Th most important application of Thorm 3.1 is to prov th following thorm. Thorm 3.2. Suppos G and H ar Li groups with G simply connctd, and lt g and h b thir Li algbras. For any Li algbra homomorphism φ : g h, thr xists a Li group homomorphism Φ : G H such that Φ = φ. In ordr to prov Thorm 3.2, w shall first prov th following lmmas. Lmma 3.3. Suppos G and H ar Li groups with Li algbras g and h. Thn g h is a Li algbra with th brackt dfind by [(X, Y ), (X, Y )] = ([X, X ], [Y, Y ]). Th Li algbra g h dfind abov is indd isomorphic to Li(G H), th Li algbra of th Li group G H. Proof. Th vrification for th vctor spac g h bing a Li algbra with th brackt dfind in th lmma is an immdiat consqunc of th proprtis satisfid by th brackts for g and h. What rmains to show is that Li(G H) is isomorphic to g h. Lt (U, x i ) and (V, y j ) b charts which rspctivly contain G and H. Thn, it follows that (U V, x i y j ) is a chart which contains G H. So for any tangnt vctor Z T (G H), w hav Z = z xi x i + z yj y j. By Thorm 1.21, w know that vry lft-invariant vctor fild W on a Li group is uniquly dtrmind by its valu at, namly W. Thus, w gt a vctor spac isomorphism ψ : Li(G H) g h by considring th composition of maps Z Z = z xi x i + z ( yj y j z xi x i, z ) ( ) yj y j ZG, ZH, whr Z H and Z G ar th uniqu lft-invariant vctor filds dtrmind rspctivly by z xi / x i T G and z yj / y j T H. To complt th proof, w nd to show that this isomorphism is indd a Li algbra isomorphism. For any W, Ṽ Li(G H), ) ) [ W, Ṽ ] vxk xi = (w x i v x i w xk vyl yj x i x k + (w y j j w yl vy y j y l, whr w hav usd th fact that w xi / y j = 0 and w yi / x j = 0 (analogously for v xi and v yj ). 11 So w hav [ W, Ṽ ] = ([ W G, V G ], [ W H, VH ] ). Thn it follows that ψ : Li(G H) g h is a Li algbra isomorphism. ( ) 11 This is bcaus for Z = x i, Zg h = (L g h ) Z = (L g h ) x i = yl L x i L xj x i x j g h + y l g h, whr L is th coordinat rprsntation of L g h. Sinc L g h = L g L h, thn yj L clarly x i = 0. From this w can s that w xi and w yj can only dpnd on x i and y j rspctivly.

13 REPRESENTATIONS OF LIE GROUPS AND LIE ALGEBRAS 13 Lmma 3.4. Suppos G and H ar connctd Li groups with Li algbras g and h rspctivly. Lt F : G H b a Li group homomorphism. Thn F is a smooth covring map if and only if th inducd homormorphism F : g h is an isomorphism. Th proof for Lmma 3.4 rquirs som xtra machinry which w hav not introducd in this papr. So for this rason, w shall not provid th proof hr. 12 Now,w ar rady to prov Thorm 3.2. Proof. By Lmma 3.3, th Li algbra of G H is isomorphic to g h. Lt r g h b th graph of φ, namly r = {(X, φx) : X g}. Clarly, r is a vctor sub-spac of g h. Bcaus φ is a Li algbra homomorphism, w thn hav [(X, φx), (Y, φy )] = ([X, Y ], [φx, φy ]) = ([X, Y ], φ[x, Y ]) r. So r is in fact a Li subalgbra of g h. Thn by Thorm 3.1, thr xists a uniqu connctd Li subgroup R G H whos Li algbra is r. Lt π 1 : G H G and π 2 : G H H b th projction maps to G and H rspctivly. Obviously, π 1 and π 2 ar Li group homomorphisms, so it follows that Π = π 1 R : R G is also a Li group homomorphism. W shall show that Π is a smooth covring map in th nxt stp. Sinc G is simply connctd, thn it will follow that Π is bijctiv, and thrfor is a Li group isomorphism. 13 In ordr to show that Π is a smooth covring map, it is sufficint to show that its inducd Li algbra homomorphism Π is an isomorphism according to Lmma 3.4. Considr th following squnc of maps R G H π1 G. Th composition of ths maps is xactly Π. Thus by Lmma 1.5, it follows that th inducd Li algbra homomorphism Π is just th composition of th maps r g h ϖ1 g, whr ϖ 1 dnots th projction g h g. This implis that Π is just th projction ϖ 1 : g h g rstrictd to r. Sinc r is th graph of φ, thn r h=0. So Π = ϖ 1 r : r g is an isomorphism. Thrfor, by Lmma 3.4, Π is a smooth covring map of th simply connctd Li group G and thus a Li group isomorphism. Now dfin a Li group homomorphism Φ : G H by Φ = π 2 R Π 1. Sinc Π = π 1 R, thn according to th dfinition of Φ, w hav π 2 R = Φ π 1 R. Sinc th Li algbra homomorphism inducd by th projction π 2 : G H H is just th projction ϖ 2 : g h h, thn by Lmma 1.5, w hav ϖ 2 r = Φ ϖ 1 r : r h. 12 For dtails of th proof, plas rfr to chaptr 9 of L s Introduction to Smooth Manifolds. 13 This follows from th proprty of covring maps of simply connctd spacs: If X is a simply connctd spac, thn any covring map π : X X is a homomorphism. For proofs, plas rfr to L s Introduction To Topological Manifolds.

14 14 JIAQI JIANG Thrfor, for any X g, w hav This shows that Φ = φ. φx = ϖ 2 r (X, φx) = Φ ϖ 1 r (X, φx) = Φ X. By invoking Thorm 3.2, w shall finally rach th ultimat goal of this sction. Corollary 3.5. Lt G b a simply connctd Li group and g b its Li algbra. If φ : g gl(v ) is a rprsntation of th Li algbra g on a finit-dimnsional vctor spac V, thn thr xists a rprsntation Φ : G GL(V ) of th Li group G on th sam vctor spac V such that th corrsponding rprsntation Φ of G on V is xactly φ. Proof. By Thorm 2.4, w know that th Li algbra of GL(V ) is xactly gl(v ). Sinc th rprsntation φ : g gl(v ) is a Li algbra homomorphism, thn by Thorm 3.2, thr xists a Li group homomorphism Φ : G GL(V ) such that Φ = φ. Thn th map Φ is th dsird rprsntation of Li group G on th vctor spac V. Acknowldgmnts. I shall thank my mntor Jonathan Glason, without whos gnrous hlp and support, I would not hav possibly pickd this intrsting topic and finishd this papr. Also I shall thank profssor Ptr May for organizing this amazing REU, from which I hav truly larnd a lot of fun mathmatics. Rfrncs [1] John M. L. Introduction to Smooth Manifolds. Springr-Vrlag Nw York, Inc [2] William Fulton and Jo Harris. Rprsntation Thory - A First Cours. Springr-Vrlag Nw York, Inc [3] Anthony W. Knapp. Li groups Byond An Introduction. Birkhäusr Boston