A Lie group is a special kind of continuous group. The group elements R(a) are labelled by. R(a)R(a) = R(a)R(a) = R(O), R(a) = R- 1 (a).

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1 208 Group Represetatio Theory for Physicists 5.2. Defiitio of a Lie Group; With Examples A Lie group is a special kid of cotiuous group. The group elemets R(a) are labelled by r real parameters a 1, a 2,..., ar, (5-19) The parameters ap may vary over a fiite or a ifiite rage. The space of the r parameters is called the group-parameter space. A group G is called a Lie group of order r if R(a) obeys the followig five postulates: 1. The idetity elemet R(a 0 ) exists, that is, R(ao)R(a) = R(a)R(ao) = R(a), for ay R(a) E G. (5-20) The parameters a 0 of the idetity elemet are usually take as zero, that is, R(a 0 ) = R(O). 2. For ay a we ca fid a such that i.e., for every R(a) a iverse exists: R(a)R(a) = R(a)R(a) = R(O), R(a) = R- 1 (a). (5-21) 3. For give parameters a ad b, we ca fid c i the set of parameters such that R(c) = R(b)R(a), (5-22) where the parameters c are real fuctios of the real parameters a ad b, c=rp(a,b). (5-23) Equatio (5-23) is called the combiatio law of group parameters ad tells us that the group is closed. 4. Associativity. R(a)[R(b)R(c)] = [R(a)R(b)]R(c), rp(rp(c,b),a) = rp(c,rp(b,a)). (5-24) 5. The parameters c i (5-23) are aalytic fuctios of a ad b, ad the a i (5-21) are aalytic fuctios of a. A Lie group is said to be compact if its parameters are bouded. Example 1: The real liear trasformatio group G L(2, R) i two-dimesioal space ( x') =(au a12) (x), R(a) =(au a12). y' a21 a22 y a21 a22 (5-25) The collectio of all 2 x 2 osigular matrices R( a) forms a real liear trasformatio group uder matrix multiplicatio. ts elemets are labelled by four real parameters (a 11, a12, a21, a22) The order of GL(2, R) is therefore four. f we restrict ourselves to the trasformatios with det(r(a)) = 1, we obtai a subgroup of GL(2, R), called the special real liear trasformatio group of dimesio two, ad deoted by SL(2, R). Example 2: The complex liear trasformatio group G L(2, C) i two-dimesioal space. f the parameters a i (5-25) are allowed to be complex, R(a) form a complex liear trasformatio group of dimesio 2. Let akl = bkl + ickt, where bkt ad Ckt are real. Therefore its elemets

2 Lie Groups 209 are characterized by eight real parameters, a 1 = b 11,..., a 8 c22, ad the order of GL(2, C) is therefore eight. Example 3: The group SU2. f the matrices i (5-25) are uitary, that is, u = ( ~ ~) = ( ~~~ ai2), a22 ( 5-26a) utu = 1, (5-27) the the collectio of matrices (5-26a) forms the uitary group U 2 f we further restrict the matrices to those satisfyig det(u) = 1, (5-28) the correspodig group is called the special uitary group SU2. From (5-26a) ad (5-28) we have u-1 = ( d -b) -c a Moreover from (5-27), (5-26a) ad (5-28), we have d = a*, c = -b*, ial 2 + lbl2 1. Therefore the most geeral form of the group elemets of SU2 is ( 5-26b) _ (ei~ cos't/ -ei( si") U - '( c ei si 1/ e-i~ cos 1/ (5-29) t cotais three real parameters ~' 1/ ad (. Thus the order of SU 2 is three. Example 4: The 2-dimesioal rotatio group R 2 A poit P(x, y) i the x-y plae goes to a poit P'(x',y') after a rotatio through agle c.p about the z-axis. From Fig. 5.2, we have j j' ' l.{j P l p Fig Rotatios of poits ad axes. (~'.) (5-30a) Rz(c.p) (c?sc.p -sic.p). (5-30b) sm c.p cos c.p The matrices Rz(c.p)(O ~ c.p < 21!') costitute the 2-dimesioal rotatio group R2. Rz(c.p) i (5-30b) is idetical to the rep D(c.p) of (2-60b) carried by the basis c.p 1 (x) = x ad c.pz(x) = y.

3 210 Group Represetatio Theory for Physicists There is oly oe real parameter, thus r = 1. The matrix Rz('P) of (5-30b) is orthogoal, that is, (5-31) so R 2 is also called the special orthogoal group of dimesio 2, or S02. Comparig (5-27) with (5-31) it is see that if the uitary trasformatios are restricted to be real, the uitary group degeerates to the orthogoal group. For example, (5-29) goes over to (5-30b) whe ~ ( 0, ad T/ = 'P From Fig. 5.2 it is see that if the poit Pis kept fixed ad the coordiates axes are rotated through the agle -ip, the the same relatio (5-30) holds betwee the coordiates x', y' of the same poit P i the ew axes i' ad j', ad its old coordiates x, y. S t is easy to see that the hierarchies of the groups so far metioed are GL(2, C) :J )\ (2, R) :J R2, ad GL(2, C) :J U2 :J SU2 :J R2. Example 5: The 3-dimesioal rotatio group R 3 aalogy with (5-31), the trasformatio matrices for rotatios through agles a, (3, about the x, y, z axes, respectively, are R,(a) = G 0 0 ) ('oofi 0 sifi) c?sa - si a, Ry((J) = O 1 0 ' sm a cos a - si (3 0 cos (3 Rz(r) = ('"'7 si~/ -si/ COS/ 0 D (5-32) Alteratively, we ca first rotate the poit P through agle about the z axis, the rotate through agle (3 about the y axis, ad fially rotate through agle a about the z axis. The set of agles (a, (3, 1) are the Euler agles (Rose 1957). As a result of these three rotatios, the poit P goes to the poit P'. The relatio betwee the coordiates of P ad P' is (5-33) D(a,(3,1) = R(a,(3,1) = Rz(a)Ry((J)Rz(r) cos a cos (3cos1- si a si/, - cos a cos (3 si si a cos/, cos a si (3) = si a cos (3 cos + cos a si/, - si a cos (3 si + cos a cos/, si a si (3 ( - si (3 cos /, si (3 si /, cos (3 (5-34) Dis a orthogoal matrix v- 1 = i5. The Lie group R 3 is also called the special orthogoal group S0 3 of dimesio three. Chapter 6 we discuss this rotatio group i more detail. Let us ow cosider a ew coordiate system xyz which is obtaied from rotatig successively the origial coordiate system through agles /, (3 ad a about the axes z, y, ad z, respectively (see the diagram i Bohr 1969, p. 76). From the discussio i Example 4 we kow that the relatio betwee the old ad ew coordiates is give by (5-35) 5.3. Lie Algebras Oe great cotributio of Sophus Lie to the theory of Lie groups was to cosider those elemets which differ ifiitesimally from the idetity, ad to show that from them oe ca obtai most of the properties of the Lie group.

4 Lie Groups 211 We begi with the Taylor expasio of the group elemets R(a), R(a) = R(O) + ap Xp +..., (5-36) ( ar(:)) ' Oa a=o (5-37) are called ifiitesimal geerators or simply geerators of the Lie group. For a Lie group of order r, there are r liearly idepedet geerators. To explore the eighborhood of the idetity, we oly eed retai terms liear i a i (5-36), that is, The iverse elemet is R(a) 1 + ap Xp. (5-38) (5-39) Suppose that there are two ifiitesimal elemets ad each has oly oe o-vaishig parameter, (5-40) Accordig to the defiitio of the Lie group, R(a) = 1 + c:xp, R(b) = 1 + c:x 11 O the other had from (5-40) we have R(a)R(b) = R(c) = 1 +er Xr, R(b)R(a) = R(c') = 1 + c'r Xr, [R(a), R(b)] = c: 2 c; 11 xr, c; 11 (Cr_ c'r)jc:2. (5-41) (5-42) Comparig (5-41) with (5-42), we get a importat relatio: (5-43) amely, the commutator of two geerators is a liear combiatio of the r geerators. The coefficiets c; 11 are called the structure costats of the Lie group. They have the followig two properties. 1. They are ati-symmetric with respect to the subscripts. 2. Accordig to the Jacobi idetity we have (5-44) (5-45a) (5-45b) The r geerators spa a real r-dimesioal vector space Cr. Ay vector i the space ca be expressed as ap Xp. The product of two basis vectors i the space is defied by their commutator (5-43). The set {Xp} is thus closed uder liear combiatios ad multiplicatios defied by (5-43), that is, {Xp} costitutes a algebra ad is called the Lie algebra correspodig to the give Lie group. f ap are real, it is called a real algebra, otherwise it is a complex Lie algebra.

5 212 Group Represetatio Theory for Physicists Results obtaied by Lie reduce the searchig for irreps of the Lie group with a ifiite umber of elemets, to a search for irreps of the Lie algebra with a fiite umber of elemets. Havig foud irreps of the Lie algebra, the irreps of the Lie group are also kow. Therefore the Lie algebra plays a crucial role i the theory of Lie groups. For a give Lie group, we always first fid its correspodig Lie algebra. physical problems, it ofte occurs that a certai kid of Lie algebra emerges aturally; evertheless, the correspodig Lie group does ot have a simple physical meaig. such cases, we oly deal with the Lie algebra ad do ot bother about the related Lie group at all. the space Lr = { Xp : p = 1, 2,... r }, ay vector ca be expressed as (5-46) where ap ca be thought of as the coordiates of a abstract vector X. Accordig to (5-2), the basis vectors ad the coordiates trasform i the followig ways: X' = B 11 X a'p = AP a 11 A = tr 1 p p 11' 11 ' (5-47a,b,c) the ew coordiate system with the basis { X~}, the structure costats are C~~, [X~,x~J = c~~x~. (5-48) From ( 5-9b), the relatio betwee the ew ad old structure costats is B µb VAT c>- C, p11 T - p 11 >. µv (5-49) Equatio (5-48) shows that the Lie algebra of the same Lie group may take differet forms due to the differet choices of the group parameters. This poit merits special attetio whe we are dealig with the classificatio of Lie algebras. Example 1: The group GL(2, R). Usig (5-25) ad (5-37) we get the four geerators Xi = e11 = 0 ~), X2 = ei2 = ( ~ ~), X3 = e21 = ( ~ ~), X4 = e22 = ( ~ (5-50) t ca be show that those geerators obey the followig commutatio relatios: (5-51) Example 2: The group S0 2 From (5-30b) ad (5-37) we obtai ( 0-1) X'+' = 1 0. Example 3: The group S0 3. From (5-32) ad (5-37) we have Xi=O H), X,=U ~, Xs=(. (5-52) (5-53a) They obey the commutatio relatios (5-53b)

6 Lie Groups Fiite Trasformatios Equatio (5-38) is the expressio for ifiitesimal trasformatios. Now let us fid the expressio for fiite trasformatios. Cosider first the sigle parameter group S0 2 The couterparts of (5-36) ad (5-37) are ( 0-1) R(8c.p) = l+8c.px1p, Xcp = l O. (5-54) Let the ifiitesimal agle 8c.p = c.p / N, where N is a arbitrarily large umber. Therefore Applyig R( 8c.p) N times, we obtai the fiite rotatio R(~) ~ (1+ ~X, t = t, (:) (~x, r c.p 2 2 c.p 3 3 (5 55) c.pxcp + - 2, x'/! + - 3, x'/! N~oo.. = cosc.p o + sic.p (~ -D = G~:; ~~~:). This is the familiar result (5-30b). Equatio (5-55) ca be writte formally as R(c.p) = ecpx''. (5-56) the above discussio we igored the uchaged z-compoet. f the z-compoet is icluded, the the geerator Xcp i (5-52) goes over to X 3 i (5-53a). Lettig X 3 = -ijz, we get the represetative matrix of the operator Jz i the Cartesia coordiate system as show i (5-58b). The group elemets of S0 2 thus take the well-kow form Aalogously, we itroduce for S0 3 From (5-53) we have J'.'. = (~ ~ -~) ' Jy = ( ~ ~ ~) ' Jz = (~ -~ ~o). 0 i 0 -i (5-57) (5-58a) (5-58b) (5-59) J'.l'.,y,z are the three compoets of agular mometum. Equatio (5-58b) is their matrix represetatio i the 3-dimesioal Cartesia basis. The rotatio operators correspodig to (5-32) are (5-60) The operator for a rotatio through agle c.p about a axis with orietatio agle ( (}', 'P') ca be expressed as R(c.p) = e-iip J =exp [-i'{l( Jx si(}' cos '{! 1 + Jy si(}' si c.p 1 + Jz cos B')]. (5-61)

7 214 Group Represetatio Theory for Physicists Such a rotatio ca be writte as a product of three rotatios R( 'P) = R( r.p 1, e 1, O)R( r.p, 0, O)R(O, -e 1, -r.p 1 ) = R(r.p', e', O)R(r.p, -e', -r.p'), (5-62) amely, first rotate the -axis oto the z-axis, the rotate through agle 'P about the z-axis, ad fially brig the z-axis back to the -axis. Usig (5-62) ad (5-34) we ca get the matrix form of the rotatio R ( 'P) i the 3-dimesioal space x, y ad z. The trasitio from the ifiitesimal trasformatio (5-54) to the fiite trasformatio ca be exteded to the more geeral case (5-63a, b) t should be metioed that it is ot always possible to write the fiite trasformatio i the form ( 5-63b). f the trasformatio ca be put i this form, the the group parameters ap are said to be caoical. For example i ( 5-61) ax = 'P sie' cos tp 1, ay = 'P sie' si rp', az = 'P cos O' are caoical parameters. f we choose the Euler agles o:, /3 ad 'Y as the group parameters of 803, from (5-34) ad (5-60), we have Sice Jy ad Jz do ot commute, (5-64) Therefore the Euler agle o:, /3 ad 'Y are ot caoical parameters Correspodece betwee Lie Groups ad Lie Algebras The classificatios of Lie groups ad Lie algebras are i oe-to-oe correspodece. This correspodece is based o the two relatios (5-65) ad (5-66) which follow. Let Rp, R 11 be two ifiitesimal elemets. Makig a expasio of ( 5-63b) ad retaiig terms up to e, 2, we obtai Therefore 2 2 Rp ~ 1 + cxp + ~! x;, R11 ~ 1 + e,x11 + ~! x;. [Rp,R 11 ] = c 2 [Xp,X11] = c 2 C;11Xr. (5-65) RpR 11 R; 1 R; 1 =1+c 2 [Xp,X11]=1 + c 2 C; 11 Xr. Accordig to the above two relatios it is easy to establish the followig correspodeces: (5-66) Lie groups la. Abelia Lie groups Lie algebras lb. Abelia Lie algebras p,g'=l,2,...,r. (5-67a) [Xp,X11] = 0, p, G' = 1, 2,..., r. (5-67b) 2a. Subgroups Gs of a Lie group G. Let Xi, XJ,..., Xk be the geerators of Gs. Let Ri = l+e,xi,rj = l+cxj. Therefore 2b. Subalgebras As of a Lie algebra A. By (5-68a), [Ri, Rj] is a elemet of the group algebra of G 8 Usig (5-65) we kow that Xi, Xi,..., Xk form a subalgebra A 8 of A, that is,

8 Lie Groups 215 (5-68a) 3a. variat subgroups. f the elemets Ri, Rj,..., Rk belog to a ivariat subgroup G 8, oe has from (1-28) RpRiR; 1 E Gs, Ri E Gs, p 1,2,...,r (5-68b) 3b. variat subalgebras. From (5-69a) ad (5-66) it is kow that [Xa, Xp] = C~pXb, a,b=i,j,...,k, p= 1,2,...,r. (5-69b) Thus (5-69a) The algebra Xi,..., Xk is called the ivariat subalgebra of A. 4a. Simple Lie group. A Lie group which has o ivariat subgroups is a simple Lie group. 5a. Semi-simple Lie group. The Lie group which has o Abelia ivariat subgroups is a semi-simple Lie group. 4b. Simple Lie algebra A Lie algebra which has o ivariat subalgebra is a simple Lie algebra. Sb. Semi-simple Lie algebra. The Lie algebra which has o Abelia ivariat subalgebras is a semi-simple Lie algebra. 6a. Theorem 5.1: A semi-simple Lie group is a direct product of a set of simple Lie groups, where Gi are simple ad [Gi, Gil= 0. ( 5-70a) 6b. Theorem 5.1': A semi-simple Lie algebra is a direct sum of a set of simple Lie algebras, where Ai are simple, [Ai,AJ] = 0 ad the itersectios betwee ay Ai ad Ai are zeroes. 7. A compact Lie algebra is oe correspodig to a compact Lie group. (5-70b) t is importat to distiguish betwee the semi-simple ad o-semi-simple Lie groups, sice Abelia ivariat subgroups, though apparetly the easiest to deal with, ca actually be the most troublesome from the poit of view of represetatios. Fortuately, i most physical applicatios we deal oly with semi-simple Lie groups. Below we maily cocer ourselves with semi-simple Lie groups. (The criteria for semi-simple Lie groups is give i Sec ) a semi-simple Lie algebra the maximum umber of liearly idepedet geerators, deoted H1,..., H1, which commute with oe aother, is called the rak of the Lie algebra or the rak of the correspodig Lie group, desigated by l. (A equivalet defiitio of rak is give i Sec ). The set of operators H 1, H 1 form a subalgebra, called the Carta subalgebra. Naturally, ay Lie group must be of at least rak 1. Example 1: For S02, there is oly oe geerator lz. Naturally ]z commutes with itself. Therefore S0 2 is a Abelia group with rak l = 1. Example 2: For S03, there are three geerators l., ly ad lz. Each of them oly commutes with itself. S0 3 is a o-abelia group of rak 1. S02 ad S03 are both simple.

9 216 Group Represetatio Theory for Physicists 5.6 Liear Trasformatio Groups Secs. 5.2 ad 5.3, we gave the geeral defiitios of Lie groups ad Lie algebras. Sec. 5.2 we also gave some simple examples. We will ow exted these examples to the geeral liear trasformatio groups. These groups are the most useful oes i physics. Assume R(a) :::: R(a 1, a 2,, ar) is a -dimesioal liear trasformatio, R(a) x ~ x' = R(a)x, (5-71a) or equivaletly x":x:::: Raf3(a)xf3, a= 1,2,...,. (5-71b) Here x may be real or complex. The set of all x matrices R(a) forms a liear trasformatio group i -dimesioal space. t ca be further classified ito the followig categories: 1. GL(, C) = GL(), the geeral complex liear trasformatio group. The matrix elemets Raf3(a) are complex umbers. The group cotais 2 2 real parameters; therefore the order is r= GL(, R), the geeral real liear trasformatio group. The matrix elemets are restricted to real umbers. There are 2 real parameters. The order is r = 2 3. SL(, C), SL(, R), the special liear trasformatio groups. These two groups are obtaied from GL(, C) ad GL(, R) by requirig that the determiats of the trasformatios be uity. Their orders are equal to ad 2-1, respectively. Obviously we have GL(, C) :J SL(, C) :J SL(, R), GL(, R) :J SL(, R). 4. U ad SU, the uitary group ad uimodular uitary group i dimesios. Restrictig matrices R(a) to be uitary, that is, (5-72a) we get the uitary group U of order r = 2 The uitary group is compact, sice by (5-72a) the matrix elemets Raf3(a) :5 1. The coditio (5-72a) also stipulates that det R(a) =exp (irp). (5-72b) Demadig that the determiats of R(a) equal uity, we obtai the uimodular uitary group SU of order r = 2-1. The uitarity (5-72a) esures that the quatity ::=l lx 'l 2 is a ivariat uder the uitary trasformatio, L lx 'l 2 = L lx' 'l 2 (5-73) a=l a=l The fudametal role of uitary groups i quatum mechaics is easily uderstood whe oe realizes that the probabilistic ature of quatum theory requires a preservatio of squares of absolute values of various ier product of wave fuctios. 5. The group U(, m). All the liear trasformatios which keep the quatity +m L lx 'l 2 - L lxf31 2 (5-74) a=l f3=+l ivariat form the group U(, m) with order r = ( + m) 2 U(, m) is a ocompact group. Obviously, U = U(, 0) = U(O, ), GL(, m) :J U(, m). Similarly we ca defie the group SU(, m), with order r = ( + m) 2-1.

10 Lie Groups The complex orthogoal group O(, C). All the complex liear trasformatios which leave :::=l (xa) 2 ivariat form the complex orthogoal group. From we have ~)x'a) 2 = L Raf3Raf3'Xf3xf3' = L (x 13 ) 2, ( 5-75) a=l af3f3' f3=1 (5-76a) Thus R(a) are orthogoal matrices, R(a)R(a) = 1. (5-76b) O(, C) has (-1)/2 complex parameters (see Sec. 5.8), therefore it is of order r = (-1). From (5-76b) we have det(r(a))det (R(a)) = 1, det (R(a)) = ±1. (5-76c) The trasformatio matrices of O(, C) ca be divided ito two sets, oe is associated with det (R(a)) = +1, ad the other with det (R(a)) = -1. The set with determiat +1 forms a subgroup respresetig proper rotatios, the uimodular complex orthogoal group SO(, C). We ca decompose the group O(, C) ito cosets with respect to the subgroup SO(, C), that is, O(, C) = SO(, C) EB SO(, C) x, (5-77) where is the space iversio operator. The quotiet group O(,C)/SO(,C) is a group of order 2. The set with determiat -1 represets rotatio-reflectios. Ay elemet of SO(, C) ca be reached from the idetity via cotiuous paths i parameter space, while the elemets with det(r(a)) = -1 caot. other words, the group O(, C) cosists of two discoected parts ad we caot go from oe part to the other cotiuously. 7. The real orthogoal group O. Restrictig the matrices of O(, C) to be real leads to the real orthogoal group, deoted by O or O(), which is of order r = ~( - 1). By further requirig det ( R( a)) = 1, we get the uimodular orthogoal group S O. t is still of order ~( - 1). Similarly, the group O also cosists of two discoected parts. Obviously we have O(, C) :::> SO(, C) :::> SO, O :::> SO. 8. The group O(, m). All the real liear trasformatio which leave the quatity +m L (xa)2 - L (xf3)2 (5-78) a=l f3=+l ivariat form the group O(, m) with order r = t[( - 1) + m(m - 1)] +m. O(, m) is a ocompact group. The Loretz group 0(3, 1) is a special case of O(, m). Obviously, we have O = O(, 0) = 0(0, ). 9. Complex symplectic, real symplectic ad uitary symplectic groups Sp(2, C), Sp(2, R), ad Spz Suppose x = (x1,...,x;x- 1,..,x-) ad y = (y 1,..,y;y- 1,... y-) are two colum vectors with dimesio 2 ad R(a) are 2 x 2 matrices, which trasform x ad y ito x' ad y': x' = R(a)x, y' = R(a)y. (5-79a)

11 218 Group Represetatio Theory for Physicists The symplectic group is the set of all 2 x 2 liear trasformatios R( a) which leave the skew-symmetric biliear form a= (5-79b) ivariat. f the 2 x 2 matrices R(a) are complex (real), it is called the complex (real) symplectic group of order 2(2 + 1) ((2 + 1)). f the complex matrices R(a) are uitary, the group is called the uitary symplectic group Sp 2. We have GL(2, C) :J Sp(2, C) :J Sp(2, R), Sp(2, C) :J Sp2i SU2 :J Sp2. The groups Sp(2, C) ad Sp(2, R) are ocompact, while Sp 2 is compact fiitesimal Operators for Liear Trasformatio Groups Cosider subjectig x 1, x 2,, x to a ifiitesimal trasformatio x' = R(a)x, R(a) = 1 + A(a), (5-80a) A(a) = L aa13ea,e, a/3 (5-81) where aa,e are ifiitesimal quatities ad ea/3 are the x matrices defied i (2-4). The ea/3 obey the commutator (5-51) ad the followig relatio Equatio (5-80a) ca also be rewritte as Uder the trasformatio (5-80b), a arbitrary fuctio 'lf;(x) goes over to (5-82) (5-80b) 'lf;'(x) = 'lf;(x') = 'lf;(x ' + a ' 13 x 13 ) = 'lf;(x) + a 'a x 13 ~'lf;(x). ' 8x ' (5-83) Defiig the ifiitesimal operators Eq. (5-83) reads 'lf;'(x) = (1 +a ' 13 E13a)'l/J(x) = (1 +apxp)'lf;(x), (5-84) (5-85) ad ( 5-80b) ca be expressed as x' ' = (1+ l:a ' 13 E13a)x '.,B (5-80c) From this we obtai a simple method for fidig the ifiitesimal operators of the liear trasformatio group: 1. First fid the ifiitesimal matrix A i the ifiitesimal trasformatios (5-80a), that is, (5-86a) Notice that ot all the parameters aa/3 are idepedet, except for the group GL(, R) or GL(,C).