The Infinitesimal Method

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    Chapter 6

    The Infinitesimal Method

    Up to now, with exception of the Schrodinger representation of the Heisenberg group,all irreducible representations treated here were finite-dimensional. Before we go furtherinto the construction of infinite-dimensional representations, we discuss a method whichallows a linearization of our objects and hence simplifies the task to determine the struc-ture of the representations and classify them. As we shall see, this is helpful for compactand noncompact groups as well. The idea is to associate to the linear topological groupG a linear object, its Lie algebra g= Lie G, and study representations of these alge-bras, which are open to purely algebraic and combinatorial studies. One can associateto each representation ofG an infinitesimal representation d ofg. Conversely, onecan ask which representations may be integrated to a unitary representation ofG,

    i.e., which are of the form = d, G. As we will see in several examples, thismethod allows us to classify the (unitary) irreducible representations and, hence, gives aparametrization ofG. It will further prove to be helpful for the construction of explicitmodels for representations (, H).

    6.1 Lie Algebras and their Representations

    We introduce some algebraic notions, which will become important in the sequel. Theinterested reader can learn more about these in any text book on Algebra (e.g. LangsAlgebra[La]).

    Definition 6.1: Let K be a field. A K-algebra is a K-vector spaceA provided with aK-bilinear map (multiplication)

    AAA, (x, y) xy.Ais called

    associativeiff one has

    (xy)z= x(yz) for all x,y,z A, commutativeiff one has

    xy= yx for all x, y A,

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    64 6. The Infinitesimal Method

    a Lie algebra iff the multiplication xy, usually written as xy =: [x, y] (the Liebracket), is anti-symmetric, i.e.

    [x, y] = [y, x] for all x, y A,

    and fulfills the Jacobi identity, i.e. one has

    [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 for all x,y,z A.

    a Jordan algebraiff one has

    xy= yx andx2(xy) =x(x2y) for all x, y A,

    a Poisson algebraiffA is a commutative ring with multiplication

    AAA, (x, y) xy

    and one has a bilinear map

    AAA, (x, y) {x, y}

    (the Poisson bracket) fulfilling the identities

    {x, y} = {y, x},{x,yz} = {x, y}z+ y{x, z},

    {x, {y, z}} = {{x, y}, z} + {y, {x, z}} for all x, y, z A.The dimension ofAas a K-vector space is called the dimension of the algebra.

    A map :A A is an algebra homomorphism iff it is a K-linear map and respectsthe respective composition, i.e. if one has

    (xy) =(x)(y) for all x, y A.

    Exercise 6.1: Show thatC(R2n) is a Poisson algebra with

    {f, g} :=ni=1

    ( fqi

    gpi

    fpi

    gqi

    ) for all f, g C(R2n),

    where the coordinates are denoted by (q1, . . . , q n, p1, . . . , pn) R2n.

    Exercise 6.2: Verify that each Poisson algebra is also a Lie algebra.

    In the following, we are mainly interested in Lie algebras over K= R or K= C, whichwe usually denote by the letter g. We recommend as background any text book on thistopic, for instance Humphreys Introduction to Lie Algebras and Representation Theory

    [Hu].

    Example 6.1: Every Mn(K) is an associative noncommutative algebra with composi-tion XYof two elements X, Y Mn(K) given by matrix multiplication.

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    6.1 Lie Algebras and their Representations 65

    Example 6.2: Every associative algebraAis a Lie algebra with

    [X, Y] =X Y Y X for all X, Y A,

    the commutatorofX andY. In particular, one denotes

    gl(n, K) :=Mn(K).

    Example 6.3: IfV is any K-vector space, the space End Vof linear mapsF fromV toV is a Lie algebra with

    [F, G] :=F G GF for all F, G End V.

    Example 6.4: Ifgis a Lie algebra and g0is a subspace with [X, Y] g0for allX, Y g0,g0 is again a Lie algebra. One easily verifies this and that

    sl(n, K) := {X gl(n, K); Tr X= 0},so(n) := {X gl(n, K); X= tX},su(n) := {X gl(n, K); X= t X, Tr X= 0}

    are examples of Lie algebras.

    Example 6.5: IfAis a K-algebra,

    DerA := {D: AA K linear;D(ab) =a Db + Da b for all a, b A}

    is a Lie algebra, the algebra of derivations.

    Exercise 6.3: Verify that the matrices

    F := ( 1

    ), G:= (1

    ), H := ( 1

    1 )

    are a basis ofsl2(K) with the relations

    [H, F] = 2F, [H, G] = 2G, [F, G] =H.

    Exercise 6.4: Verify that the matrices

    X1 := (1/2)( i

    i ), X2 := (1/2)( 1

    1 ), X3 := (1/2)(

    ii )

    are a basis ofsu(2) with the relations

    [Xi, Xj ] = Xk, (i,j,k) = (1, 2, 3), (2, 3, 1), (3, 1, 2).

    Exercise 6.5: Verify that the matrices

    X1 :=

    0 1

    1

    , X2 :=

    10

    1

    , X3 :=

    11

    0

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    66 6. The Infinitesimal Method

    are a basis ofso(3) with the relations

    [Xi, Xj ] = Xk, (i,j,k) = (1, 2, 3), (2, 3, 1), (3, 1, 2).

    Exercise 6.6: Show that g = R3 provided with the usual vector product as compositionis a Lie algebra isomorphic to so(3).

    Exercise 6.7: Show that g= R3 provided with the composition

    ((p, q, r), (p, q, r)) (0, 0, 2(pq pq))

    is a Lie algebra, the Heisenberg algebraheis(R), with basis

    P = (1, 0, 0), Q= (0, 1, 0), R= (0, 0, 1)

    and relations[R, P] = [R, Q] = 0, [P, Q] = 2R.

    Remark 6.1: It will soon be clear that gl(n, R),sl(n, R),so(n),su(n), and heis(R) areassociated to the groups GL(n, R), SL(n, R), SO(n), SU(n), resp. Heis(R).

    In analogy with the concept of the linear representation of a group G in a vector spaceV, one introduces the following notion.

    Definition 6.2: A representation of the Lie algebragin Vis a Lie algebra homorphism: g

    End V, i.e. is K-linear and one has

    ([X, Y]) = [(X), (Y)] for all X, Y g.

    Here we have no topology and, hence, no continuity condition.

    Each Lie algebra has a trivial representation = 0 with 0(X) = 0 for all X gandan adjoint representation = ad given by

    ad X(Y) := [X, Y] for all X, Y g.

    Exercise 6.8: Verify this.

    In analogy with the respective notions for group representations, we have the followingconcepts. (0, V0) is asubrepresentationof (, V) iffV0 Vis a -invariant subspace, i.e. (X)v0V0 for all v0 V0, and 0 = |V . is irreducibleiff has no nontrivial subrepresentation.If 1 and 2 are representations ofgwe have the direct sum1 2 as the representation on V1 V2 given by

    (1

    2)(X)(v1, v2) := (1(X)v1, 2(X)v2) for all v1

    V1, v2

    V2,

    and the tensor product1 2 as the representation on V1 V2 given by

    (1 2)(X)(v1 v2) := 1(X)v1 v2+ v1 2(X)v2 for all v1 V1, v2 V2.

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    6.2 The Lie Algebra of a Linear Group 67

    Exercise 6.9: Verify that these are in fact representations.

    Exercise 6.10: Find the prescription to define a contragredient representation (, V)to a given representation (, V) ofg.

    There is a lot of material about the structure and representations of Lie algebras, muchof it going back to Elie and Henri Cartan. As already indicated, a good reference for thisis the book by Humphreys [Hu]. We will later need some of this and develop parts of it.Here we only mention that one comes up with facts like this: Ifgis semisimple (i.e. hasno nontrivial ideals), then each representation ofg is completely reducible.Before we discuss more examples, we establish the relation between linear groups andtheirLie algebras.

    6.2 The Lie Algebra of a Linear GroupIn Section 3.1 we introduced the concept of a linear group G as a closed subgroup of amatrix group GL(n, R) or GL(n, C) for a suitable n N. To such a group we associatea Lie algebra g= Lie Gusing as essential tool the exponential function for matrices. Werely again on some experience from calculus in several complex variables and we definethe exponential function exp by

    exp X :=k=0

    (1/k!)Xk for all X Mn(C).

    Remark 6.2: One has the following facts:

    1. exp X is absolutely convergent for each X Mn(C).2. exp X :Mn(C) Mn(C) is continuously differentiable.3. There is an open neighbourhood U of 0 Mn(R), which is diffeomorphic to an openneighbourhood V ofE GL(n, R).4. log X:= ((1)k/k)(X E)k converges for X E < 1 and we have

    exp log X = X for X E

    1/2

    for all X, Y Mn(C),and that one has the fundamental inequalities

    X+ Y X + Y, thetriangle inequality,|< X, Y >| X Y, the Cauchy Schwarz inequality.

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    68 6. The Infinitesimal Method

    Everyone interested in concrete examples could do the following Exercise 6.12 preparingthe ground for the next definition: Verify

    exp(tX) =

    1 t

    1

    , forX=

    1

    ,

    exp(tX) =

    cos t sin t sin t cos t

    , forX=

    11

    ,

    exp(tX) =

    et

    et

    , forX=

    1

    1

    ,

    exp(tX) =

    cos t sin t sin t cos t

    1

    , forX=

    11

    0

    .

    Exercise 6.13: Verify that one has the following relations

    {exp(tX), X

    so(3); t

    R

    } = SO(3),

    {exp(tX), X su(2); t R} = SU(2),{exp(tX), X sl(2, R); t R} = SL(2, R).

    This exercise shows that there are groups, which are examples for groups of exponentialtype, i.e. images under the expontial map of their Lie algebras. For G = SL(2, R) onehas

    {exp(tX), X sl(2, R), t R} = SL(2, R),but the matrices exp(tX), X sl(2, R), t R, generate the identity component ofG= SL(2, R).

    We take this as motivation to associate a Lie algebra to each linear group by a kind ofinversion of the exponential map:

    Definition 6.3: Let G be a linear group contained in GL(n, K), K = R or K = C.Then we denote

    g= Lie G:= {X Mn(K); exp(tX) G for all t R}.Theorem 6.1: gis a real Lie algebra.

    A hurried or less curious reader may simply believe this and waive the proof. (See forinstance [He] p.114: it is, in principle, not too difficult but lengthy. It shows why (in ourapproach) we had to restrict ourselves to closed matrix groups.) We emphasize that it isessentially the heart of the whole theory that not only a linear group but also every Liegroup has an associated Lie algebra. This is based on another important notion, whichwe now introduce because it gives us a tool to determine explicitly the Lie algebras forthose groups we are particularly interested in (even if we we have skipped the proof ofthe central theorem above).

    Definition 6.4: A one-parameter subgroup of a topological group G is a continuous

    homomorphism :R G, t (t).This notation is a bit misleading: to be precise, G0 =(R) is indeed a subgroup ofG.is often abbreviated as a 1-PUG (from the German term Untergruppe).

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    6.2 The Lie Algebra of a Linear Group 69

    Remark 6.3: Each X gproduces a 1-PUG , namely the subgroup given by(t) := exp(tX).

    This is obvious since exp(tX) is continuous (and even differentiable) in tand fulfills the

    functional equation i) in part 5 of the Remark 6.2 above.

    If the 1-PUG = (t) is differentiable in t, we assign to it an infinitesimal generatorX=X by

    X := d

    dt(t) |t=0 .

    Ifis given as (t) = exp(tX), we have Xas infinitesimal generator.

    This suggests the following procedure to determine the Lie algebra g of a given matrixgroupG (which is also the background of our Exercise 6.13 above): Look for sufficiently

    many independent one-parameter subgroups 1, . . . , d and determine their generatorsX1, . . . , X d. Theng will come out as the R-vector space generated by these matricesXi.Obviously, one needs some explanation: Two 1-PUGs and are independent iff their infinitesimal generators are linearlyindependent over Ras matrices in Mn(C). The number d of the sufficiently manyindependent (i) is just the dimensionof ourgroup G as a topological manifold: Up to now we could work without this notion, butlater on we will even have to consider differentiablemanifolds. Therefore, let us recall (orintroduce) that the notion of a real manifold M of dimensiond includes the conditionthat each point mMhas a neighbourhood which is homeomorphic to an open set inR

    d

    . This amounts to the practical recipe that the dimension of a group G is the numberof real parameters needed to describe the elements ofG.For instance for G = SU(2) and SO(3) we have as parameters the three Euler angles,, , which we introduced in 4.3 and 4.2. Hence, here we haved = 3, as also in thenext example.

    Example 6.6: For G = SL(2, R) we take as a standard parametrization (a specialinstance of the important Iwasawa decomposition)

    (6.1) SL(2, R) g= ( a bc d

    ) =n(x)t(y)r()

    with

    n(x) := ( 1 x

    1 ), t(y) := (

    y1/2

    y1/2 ), r() := (

    cos sin sin cos ),

    wherex R, y >0, [0, 2). Therefore, we take

    1(t) := ( 1 t

    1 ), 2(t) := (

    et

    et ), 3(t) :=r(t),

    and get by differentiating and then putting t = 0

    X1 = ( 1 ), X2 = ( 1 1 ), X3 = ( 11 ).

    Obviously, we have with the notation introduced in Exercise 6.3 in 6.1

    Lie SL(2, R) = < X1, X2, X3 > = < F, G, H > = sl2(R).

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    70 6. The Infinitesimal Method

    Example 6.7: For

    G= Heis(R) := {

    1 x z1 y

    1

    ; x,y,z R}

    we take

    1(t) :=

    1 t1

    1

    , 2(t) :=

    1 1 t

    1

    , 3(t) :=

    1 t1

    1

    ,

    and get

    X1 =

    1

    , X2 =

    1

    , X3 =

    1

    .

    By an easy computation of the commutators we get theHeisenberg commutation relations

    [X1, X2] =X3, [X1, X3] = [X2, X3] = 0.

    Exercise 6.14: Repeat this for G= Heis(R) as a subgroup of GL(4, R).

    Exercise 6.15: Determine Lie G forG= {g= n(x)t(y); x R, y > 0}.

    6.3 Derived RepresentationsNow that we have associated a Lie algebra g = Lie G to a given linear group (using adifferentiation procedure), we want to associate an algebra representation of g to agiven representation (, H) ofG. Here we will have to use differentiation again. To makethis possible, we have to modify the notion of a group representation:

    Definition 6.5: Let (, H) be a continuous representation of a linear group G. Thenits associated smoothrepresentation (, H) is the restriction of to the spaceH ofsmoothvectors, i.e. those v

    Hfor which the map

    G g (g)v H

    is differentiable.

    This definition makes sense if we accept that a linear group is a differentiable manifoldand extend the notion of differentiability to vector valued functions. In our examplesthings are fairly easy as we have functions depending on the parameters of the groupwhere differentiability is not difficult to check.

    Definition 6.6: Let (, H) be a continuous representation of a linear group G. Then itsassociated derived representation is the representation d ofg given on the spaceHof smooth vectors by the prescription

    d(X)v:= d

    dt(exp(tX))v|t=0 for all v H.

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    6.3 Derived Representations 71

    If it is clear, which representation is meant, one abbreviates Xv:= d(X)v.It is not difficult to verify that d(X) stays inH and rather obvious that d(X) islinear. But there is some more trouble to prove (see [Kn] p.53, [BR] p.320, or [FH] p.16)that one has

    [d(X), d(Y)] =d([X, Y]) for all X, Y

    g.

    Example 6.8: We take G= SU(2) and = 1 given by

    (g)f(x, y) :=f(ax by, bx + ay)

    for

    f V(1) = < f1, f0, f1 >with

    f1(x, y) = (1/

    2)y2, f0(x, y) =xy, f1(x, y) = (1/

    2)x2.

    As we did in Exercise 6.4 in 6.1, we take as basis ofsu(2)

    X1 := (1/2)( i

    i ), X2 := (1/2)( 1

    1 ), X3 := (1/2)(

    ii ).

    Then we have the one-parameter subgroups

    exp(tX1) = ( cos(t/2) i sin(t/2)i sin(t/2) cos (t/2) ),

    exp(tX2) = (

    cos(t/2)

    sin(t/2)

    sin(t/2) cos (t/2) ),

    exp(tX3) = ( eit/2

    eit/2 ).

    It is clear that V(1) consists of smooth vectors and hence we can compute

    X1f1 = ddt(exp tX1)f1|t=0= ddt(1/

    2)(i sin(t/2)x + cos(t/2)y)2 |t=0

    = (i/

    2)xy= (i/

    2)f0,

    andX1f0 =

    ddt(exp tX1)f0|t=0

    = ddt (cos(t/2)x + i sin(t/2)y)(i sin(t/2)x + cos(t/2)y) |t=0= (i/2)(x2 + y2) = (i/

    2)(f1+ f1),

    X1f1 = ddt(exp tX1)f1|t=0

    = ddt (1/

    2)(cos(t/2)x + i sin(t/2)y)2 |t=0= (i/

    2)xy= (i/

    2)f0.

    Similarly we get

    X2f1 = (1/

    2)f0, X2f0 = (1/

    2)(f1 f1), X2f1 = (1/

    2)f0,

    X3f1 = if1, X3f0 = 0, X3f1 = if1.

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    72 6. The Infinitesimal Method

    Exercise 6.16: i) Verify this and show that for

    X0 :=iX3, X:= (1/

    2)(iX1 X2)

    one has

    X0fp = pfp, Xfp=fp1, p= 1, 0, 1 (f2 = f2 = 0).ii) Do the same for = j , j >1.

    Example 6.9: We take the Heisenberg group G = Heis(R) GL(4, R) and theSchrodinger representation = m, m R \ {0}, given by

    (g)f(x) =em( + (2x + ))f(x + )

    withg= (,,)

    G, x

    R, f

    H=L2(R), em(u) :=e2imu.

    Here H is the (dense) subspace of differentiable functions inL2(R). We realize the Liealgebra g= heis(R) by the space spanned by the four-by-four matrices

    P=

    1 1

    , Q=

    11

    , R=

    1

    .

    Then we have the Heisenberg commutation relations

    [P, Q] = 2R, [R, P] = [R, Q] = 0

    and (using again the notation introduced in 0.1)

    exp tP = (t, 0, 0), exp tQ= (0, t, 0), exp tR= (0, 0, t).

    For differentiable fwe compute

    P f = ddt(t, 0, 0)f|t=0 = ddtf(x + t) |t=0 = xf,Qf = ddt(0, t, 0)f|t=0 = ddtem(2xt)f(x) |t=0 = 4imxf(x),

    Rf = d

    dt(0, 0, t)f|t=0 = d

    dtem

    (t)f(x) |t=0 = 2imf(x),i.e. we have

    P =x, Q= 4imx, R= 2im.

    Exercise 6.17: i) Verify that these operators really fulfill the same commutation rela-tions as P, Q, R.ii) Take f0(x) :=e

    mx2 and

    Y0 := iR, Y:= (1/2)(P iQ)

    and determinefp:= Y

    p+f0, Yfp and Y0fp for p= 0, 1, 2, . . . .

    These Examples show how to associate a Lie algebra representation to a given grouprepresentation. Now the following question arises naturally: Is there always a nontrivialsubspace H of smooth vectors in a given representation space H? We would even want

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    6.4 Unitarily Integrable Representations ofsl(2, R) 73

    that it is a dense subspace. The answer to this question is positive. This is easy in ourexamples but rather delicate in general and one has to rely on work by Garding, Nelsonand Harish Chandra. It is not very surprising that equivalent group representations areinfinitesimally equivalent, i.e. their derived representations are equivalent algebra repre-sentations. The converse is not true in general. But as stated in [Kn] p.209, at least for

    semisimple groups one has that infinitesimally equivalent irreducible representations areunitarily equivalent.Taking these facts for granted, a discussion of the unitary irreducible representations ofa given linear group G can proceed as follows: We try to classify all irreducible repre-sentations of the Lie algebra g= Lie G ofG and determine those, which can be realizedby derivation from a unitary representation of the group. In general, this demands forsome more information about the structure of the Lie algebra at hand. Before going abit into this structure theory, we discuss in extenso three fundamental examples whoseunderstanding prepares the ground for the more general cases.

    6.4 Unitarily Integrable Representations of sl(2, R)

    A basic reference for this example is the book SL(2, R) [La1] by Serge Lang but in someway or other our discussion can be found in nearly any text treating the representationtheory of semisimple algebras or groups.As we already know from Exercise 6.3 in 6.1 and Example 6.5 in 6.2, g= sl(2, R) is theLie algebra ofG= SL(2, R) and we have

    g= sl(2, R) ={

    X

    M2(R); Tr X= 0}

    = < F,G,H >

    with

    F = ( 1

    ), G= (1

    ), H= ( 1

    1 )

    and the relations[H, F] = 2F, [H, G] = 2G,[F, G] =H.

    The exercises at the end of the examples in the previous section should give a feelingthat it is convenient to complexify the Lie algebra, i.e. here we go over to

    gc:=gRC = < F,G,H >C = < X+, X

    , Z >C

    where

    (6.2) X:= (1/2)(H i(F+ G)) = (1/2)( 1 ii 1 ), Z := i(F G) = ( i

    i )

    with[Z, X] = 2X, [X+, X] =Z.

    This normalization will soon prove to be adequate and the operators X will get ameaning as ladder operators: We consider a representation of gc =< X, Z > on a

    C-vector spaceV =< vi >iIwhere we abbreviateXv:= (X)v for all v V, X gc.

    We are looking for representations (, V) ofgc, which may be integrated, i.e. which arecomplexifications of derived representationsd for a representation of SL(2, R).

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    74 6. The Infinitesimal Method

    This has the following consequence: |K is a representation of K := SO(2). FromExample 3.1 in 3.2 we know that all irreducible unitary representations of SO(2) aregiven by

    k(r()) =eik, k Z,

    hence one has the derived representation dk with

    dk(Y) =ik forY = ( 1

    1 ) =iZ.

    This motivates the idea to look for each k Z at the subspace ofV consisting of theeigenvectors with eigenvalue k with respect to the action ofZ, namely

    Vk := {v V; Zv= kv}.This space Vk is called a K-isotypic subspaceofV. Each v Vk is said to have weightk. Now we can explain the meaning of the term ladder operator: The isotypic spacesVk

    may be seen as rungs of a ladder numerated by the weight k and the operators Xchange the weights in a controlled way:

    Remark 6.4: We haveXVk Vk2.

    Proof: Forv Vk we putv :=Xvand check thatv is again aZ-eigenvector of weightk 2 :

    Zv=Z Xv= (XZ 2X)v= X(k 2)v= (k 2)v.Here we used the Lie algebra relation [Z, X] =2X, which for the operators actingon Vis transformed into the relation

    ZX XZ= 2X.This remark has immediate consequences:

    i) IfV = Vk is the space of an irreducible gc-representation, there are only nontrivialVk wherek is throughout odd or even.ii) If there is a Vk0 = {0}, then all Vk withk k0 or allVk with k k0 have to be zero.Hence, there can be only four different configurations for our irreduciblegc-representations.The nontrivial weights k, i.e. those for which Vk= {0}, compose case 1: a finite interval

    Inm:= {m k n; m k n mod 2}of even or odd integers, case 2: a half line with an upper bound

    In := {k n; k n mod 2}consisting of even or odd integers, case 3: a half line with lower bound

    Im:= {m k; m k mod 2}

    consisting of even or odd integers, case 4: a full chain of even or odd integers

    I+:= {k 2Z} orI:= {k 2Z + 1}.

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    6.4 Unitarily Integrable Representations ofsl(2, R) 75

    case 1 case 2

    case 3

    case 4

    X+ X

    It is quite natural to call an element 0= v Vk with k = n in configuration 1 or 2 avector ofhighest weightand similarly fork = m in configuration 1 or 3 a vector oflowestweight.

    The following statement offers the decisive point for the reasoning that we really getall representations by our procedure. It is a special case of a more general theorem byGodement onsphericalfunctions. It also is a first example for the famous and importantmultiplicity-onestatements.

    Lemma: If is irreducible, we have dim Vk 1.

    For a proof we refer to Lang [La1] p.24 or the article by van Dijk [vD].

    We look at the configurations 1 or 2: Let 0= v Vn be a vector of highest weight,i.e. with

    Zvn=nvn, X+vn= 0,

    and put

    vn2k :=Xkvn.

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    76 6. The Infinitesimal Method

    As we want an irreducible representation and we have accepted to take the muliplicity-one statement from the Lemma for granted, we can take this vn2k as a generator forVn2k with vn2k= 0 for all k N0 in configuration 1 and all k with n 2k m inconfiguration 2. We check what X+ does to vn2k:Remark 6.5: Fork as fixed above, we have

    X+vn2k =akvn2k+2 with ak :=kn k(k 1).Proof: Since X+vn = 0, we know that a0 = 0. As in the proof of Remark 6.4, for theoperators realizing the commutation relation [X+, X] =Zwe calculate

    X+vn2 = X+Xvn= (XX++ Z)vn= nvn.

    Hence we get a1 =n. Then we verify by induction

    X+vn2(k+1) = X+Xvn2k = (XX++ Z)vn2k= X((kn k(k 1))vn2k+2+ (n 2k)vn2k= ((k+ 1)n

    k(k+ 1))vn

    2k =ak+1vn

    2k.

    We see that the irreducibility of forcesak to be nonzero for all k Nin configuration2 and for 2kn m in configuration 1. Sinceak = 0 iffn (k 1) = 0, we concludethatn has to be negative in case 2 and m= n with n N in case 1.Just as well, we can start to treat the configurations 1 and 3 with a vector 0 =vm Vmof lowest weight. Here we put vm+2k :=X

    k+vm. In parallel to Remark 6.5 we get:

    Remark 6.6: Fork N0 in configuration 3 and m + 2k nin configuration 1, we haveXvm+2k =bkvm+2k2 with bk = (km + k(k 1)).

    Exercise 6.18: Prove this.

    As above, we see that we have m = nwith negativen in configuration 1 and that n hasto be positive in configuration 3. For instance, the following are possible configurations.

    case 1

    case 2 case 3

    m= 2

    n= 2

    n= 2

    4

    6

    m= 2

    4

    6

    X+ X

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    6.4 Unitarily Integrable Representations ofsl(2, R) 77

    In configuration 4 we have no distinguished vector of highest or lowest weight. In thiscase we choose for each even (or odd) integer n a nonzero vn Vn, i.e. with Zvn=nvn,such that they are related as follows:

    Remark 6.7: For all even (resp. odd) n

    Z we have

    Xvn=an vn2 withan = (1/2)(s + 1 n),

    wheres C, such that an= 0 for all n 2Z (resp. 2Z + 1), i.e. s 2Z + 1 for n 2Zands 2Z forn 2Z + 1.

    Proof: From irreducibility and multiplicity-one we conclude that each Xvn has to bea nonzero multiple ofvn2. We verify that the choice ofan is consistent with the com-mutation relations:

    [Z, X]vn = (ZX XZ)vn=Z an vn2 an nvn2= 2an vn2= 2Xvn,

    and[X+, X]vn = Zvn=nvn.

    Verify this as Exercise 6.19.

    The conditions for s are obvious since we have an = 0 iffs + 1 = n.

    We chose a symmetric procedure for our configuration 4. But as well we could havechosen any nonzero vn Vn, and then vn+2j :=Xj+vn, vn2j :=Xjvn forj N wouldconstitute an equivalent representation.Exercise 6.20: Check this.

    We denote the representations obtained above as follows

    s,+ or s, for the even resp. odd case in configuration 4 with s= 2Z+ 1 for s,+ands = 2Z for s,, s C, +n for configuration 3 with lowest weight n, n N0, n for configuration 2 with highest weight

    n, n

    N0,

    n for configuration 1 with highest weight n and lowest weightn, n N0.

    Remark 6.8: If in s,+ and s, we do not exclude the values ofsas above, we get allrepresentations as sub - resp. quotient representations of these s, .

    Proposition 6.1: We have the following equivalences

    s,+s,+ and s,s,.

    All other representations listed above are inequivalent.

    Proof: i) LetV = < vn> andV= < vn>be representation spaces for s,+resp. s,+

    andF :V V an intertwining operator. Because ofZvn= nvn, we have for eachn

    ZF vn=F Zvn=nF vn.

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    78 6. The Infinitesimal Method

    Hence one has F vn Vn, i.e. F vn=bnvn with bn C. Moreover, fromXF vn=F Xvn

    we deduce

    (s+ 1 n)bnvn2 = bn2(s + 1 n)vn2,i.e.

    bn/bn2 = (s + 1 n)/(s+ 1 n)and

    bn+2/bn= (s+ 1 + n)/(s + 1 + n)

    resp.bn/bn2 = (s+ n 1)/(s + n 1).

    Now we see that we have

    (s + 1 n)(s + n 1) = (s+ n 1)(s+ 1 n), i. e. s = s.The same conclusions can be made for s,.ii) The inequivalence of the ns among themselves and with respect to all the otherrepresentations is clear because the dimensions are different. The other inequivalenceshave to be checked individually. As an example let V =< vn > and V

    =< vn > bespaces of representations +k and s,+ and F : V V an intertwining operator. Asabove, fork nwe have the relation F vn=bnvnwithbn C. HenceFis not surjectiveand both representations are not equivalent. The same reasoning goes through in all theother cases.

    Exercise 6.21: Realize this.

    Classification of the Unitarily Integrable Representations ofsl(2, R)

    At first, we derive a necessary condition for a general representation of a Lie algebrag= Lie Gto be integrable to a unitary representation ofG.

    Proposition 6.2: If (, H) is a unitary representation ofG and X g = Lie G, theoperator X :=d(X) onH is skew-Hermitian, i.e. one has X= X.

    Proof: Forv, w H, unitarity of implies< v, (g)w > = < (g1)v,w > for all g G

    and hence, fort R, t = 0,

    < v, i(exp(tX))w w

    t > = < i

    (exp(tX))v vt , w > .

    Recalling the definition

    d(X)v:= ddt (exp(tX))v|t=0= limt0 (exp tX)v vt ,

    we get from the last equation in the limit t 0< v,i d(X)w > = < i d(X)v,w > .

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    6.4 Unitarily Integrable Representations ofsl(2, R) 79

    Therefore we have< v, i d(X)w > = < v,i d(X)w >,

    i.e. X=d(X) is skew-Hermitian.

    Now, we apply this to g= sl(2, R). The Proposition leads to the condition:

    Remark 6.9: We have d(X)= d(X).

    Proof: Using the notation from the beginning of this section we have

    X= (1/2)(H i(F+ G)) = (1/2)( 1 ii 1 ) =: (X1 iX2)

    withX1 =H, X2 =F+ G sl(2, R). Hence, the skew-Hermiticity ofXi leads to

    (X1 iX2)= (X1 iX2).Now, if we ask for unitarity, we have to check whether we find a scalar product defined on the representation space Vsuch that we get

    < Xv,w > = < v, Xw > for all v, w V.ForV = < vj >jJ this condition implies

    (6.3) < X+vj , vj+2> = < vj , Xvj+2 > for all j J.i) In configuration 1 and 2 we have

    Xvj+2 = vj

    and from Remark 6.5 for j =n 2kX+vj =akvj+2 with ak =k(n k+ 1),

    i.e.ak vj+22= vj2 .

    An equation like this is only possible for ak j2Z. Condition (6.4) requires forj = 0

    (s + 1) v22= (s 1) v02 .

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    80 6. The Infinitesimal Method

    As the norm has to be positive, a relation like this is only possible if one has

    (1 s)/(s + 1)> 0, i. e. (1 s)(1 + s)> 0.

    We puts= + i,,

    R, and get the condition

    s2 =2 2 + 2i j2Z+1 we get for j= 1 from (j)

    s v12= s v12 .

    This is possible only for

    s/s > 0, i. e. s2 =2 2 2i

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    6.4 Unitarily Integrable Representations ofsl(2, R) 81

    k +k

    is,+

    is,

    s

    iR

    123 1 2 3

    We can observe a very nice fact, which will become important later and is the startingpoint for a general procedure in the construction of representations:

    Exercise 6.23:Show that the operator

    :=X+X+ XX++ (1/2)Z2

    acts as a scalar for the three types of representations in Theorem 6.2 and determine thesescalars.

    Our infinitesimal considerations showed that there is no nontrivial finite-dimensional uni-tary representation of SL(2, R). There are other proofs for this fact. Because it is soelegant and instructive, we repeat the one from [KT] p.16.

    Theorem 6.3: Every finite-dimensional representation ofG= SL(2, R) is trivial.

    Proof: IfGhas an n-dimensional representation space, we have a homomorphism

    : SL(2, R) U(n).

    Since

    ( a

    a1 )( 1 b

    1 )(

    aa1 )

    1 = ( 1 a2b

    1 )

    alln(b) = ( 1 b

    1) are conjugate. So all (n(b)) withb >0 are conjugate. It is a not too

    difficult topological statement that in a compact group all conjugacy classes are closed.As U(n) is compact, we can deduce

    limb0+

    (n(b)) =(E2) =En,

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    6.5 The Examples su(2) and heis(R) 83

    with the Heisenberg commutation relations

    [P, Q] = 2R, [R, P] = [R, Q] = 0.

    We complexify gand take as a C-basis

    Y0 := iR, Y:= (1/2)(P iQ)to get the relations [Y+, Y] =Y0, [Y0, Y] = 0.

    We want to construct a representation of g on a space V = < vj >jJ, (which wecontinue linearly to a representation of gc,) such that can be unitarily integrated,i.e. we can find a unitary representation of Heis(R) with d= : In every group onehas as a distinguished subgroup, the centerC(G), defined by

    C(G) :=

    {g

    G; gg0 =g0g for all g0

    G

    }.

    Obviously for G= Heis(R) we have

    C :=C(Heis(R)) = {(0, 0, ); R} R.Hence, a representation of ourG restricted to the centerCis built up (see Chapter 5)from the characters ofR. We write here

    () :=m() := exp(2im) =em(), m R \ {0}

    with derived representation d() = 2im. Having this in mind (and the discussion of

    Example 6.9 in 6.3), we propose the following construction of a representation (, V),where we abbreviate again (Y)v=: Y v.

    We suppose that we have a vacuum vector, i.e. an element v0 V withY0v0 =v0, Yv0 = 0, = 2m.

    The first relation is inspired by differentiation of the prescription of the Schrodingerrepresentation

    ((0, 0, ))f=()f

    where we have (0, 0, ) = exp R andY0 =iR.

    We put vj := Yj+v0. As Y0 commutes with Y, we have Y0vj = vj . Then we try to

    realize a relationYvj =ajvj1 :

    From the commutation relation [Y+, Y] =Y0 we deduce

    Yv1 = YY+v0 = (Y+Y Y0)v0 = ,Yvj = YY+vj1 = (Y+Y Y0)vj1 = (aj1 )vj1 = ajvj1,

    and, hence, by induction aj = j.Thus we have V spanned byv0, v1, v2, . . . with

    Y0vj =vj , Y+vj =vj+1, Yvj = jvj1.

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    84 6. The Infinitesimal Method

    We check whether this representation can be unitarized: We look at the necessary con-dition in Proposition 6.2 in 6.4. The skew-Hermiticity ofP, Q, R leads to

    Y=d(Y)= (1/2)(P iQ) = (1/2)(P iQ) = Y.

    Hence, as in the sl-discussion, we get the necessary condition for a scalar product on V

    < Yv,w >= < v, Yw >,in particular, forv= vj1, w= vj , we have

    vj2=j vj12 .

    We see that unitarity is possible iff = 2m > 0. A model, i.e. a realization of thisrepresentation by integration, is given by the Schrodinger representationm, which wediscussed already in several occasions (see Example 6.9 and Exercise 6.17 in 6.3).

    Remark 6.11: Our infinitesimal considerations show that m is irreducible.

    Remark 6.12: This discussion above is already a good part of the way to a proof of thefamous Stone-von Neumann Theoremstating that up to equivalence there is no otherirreducible unitary representation of Heis(R) with|C=m for m = 0 (for a completeproof see e.g. [LV] p.19ff).

    Remark 6.13: What we did here is essentially the same as that appears in the physicsliterature under the heading ofharmonic oscillator. Our v0 =f0 with f0(x) =e

    2mx2

    describes thevacuum, andY+andYarecreationresp.annihilation operatorsproducinghigher resp. lower excitations.

    Exercise 6.25: Use Exercise 6.17 in 6.3 to verify the equation

    (Y+Y+ YY+)f= (1/2)(f (4mx)2f) = 2mf

    and show that f0 is a solution. Recover the Hermite polynomials.

    6.6 Roots, Weights and some Structure TheoryIn the examples discussed above in 6.4 and 6.5 we realized the representations of thecomplexificationgc of the given Lie algebra g= Lie G by ladder operatorsX resp. Ychanging generators vj of our representation space V by going to vj2 resp. vj1 in acontrolled way. ForG = SL(2, R) and SU(2) the indices of the generators were related tothe weights, i.e. the eigenvalues of the operator Z. ForG = Heis(R) we observe a differ-ent behaviour: We have an operatorY0 producing the same eigenvalue when applied toall generators of our representation space. This is expression of the fact that sl(2, R) andsu(2) resp. the groups SL(2, R) and SU(2) on the one hand and heis(R) resp. Heis(R) on

    the other are examples of two different types for which we have to expect different waysof generalizations. Let us already guess that we expect to generalize the machinery of theweights in the first case (the semisimpleone) by assuming to have a greater number ofcommuting operators whose eigenvalues constitute the weights and to have two differentkinds of operators raising or lowering the weights (ordered, say, lexicographically).

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    6.6 Some Structure Theory 85

    We have to introduce some more notions to get the tools for an appropriate structuretheory. The reader interested mainly in special examples and eager to see the repre-sentation theory of the other groups mentioned in the introduction may want to skipthis section. Otherwise she or he is strongly recommended to a parallel study of a moredetailed source like [Hu], [HN] or [Ja]. Here we follow [Ki] p.92f, [KT], and [Kn1], which

    dont give proofs neither (for these see [Kn] p.113ff, and/or [Kn2] Ch.I - VI).

    6.6.1 Specifications of Groups and Lie Algebras

    In this section we mainly treat Lie algebras. But we start with some definitions forgroups, which often correspond to related definitions for algebras.

    The reports [Kn1] and [KT] take as their central objects real or complex linear connectedreductivegroups. The definition of a reductive group is not the same with all authors

    but the following definition ([Kn] p.3, [KT] p.25) is very comprehensive and practical forour purposes.

    Definition 6.7: A linear connected reductivegroup is a closed connected group of realor complex matrices that is stable under conjugate transpose, i.e. under the Cartaninvolution

    :Mn(C) Mn(C), X (t X)1.A linear connected semisimple group is a linear connected reductive group with finitecenter.

    More standard is the following definition: A group is called simple iff it is non-trivial,and has no normal subgroups other than{e} andGitself.

    Example 6.10: The following groups are reductive

    a)G= GL(n, C),

    b)G= SL(n, C),

    c)G = SO(n, C),

    d)G= Sp(n, C) := {g SL(2n, C); tgJ g= J := ( EnEn )}.

    The center of GL(n, C) is isomorphic to C, so GL(n, C) is not semisimple. The othergroups are semisimple (with exception n = 2 for c)). More examples come up by thegroups of real matrices in the above complex groups. GL(n, R) is disconnected and there-fore not reductive in the sense of the above definition, however its identity component is.We will come back to a classification scheme behind these examples in the discussion ofthe corresponding notions for Lie algebras.

    For the moment we follow another approach (as in [Ki] p.17). Non-commutative groupscan be classified according to the degree of their non-commutativity: Given two subsetsA and B of the group G, let [A, B] denote the set of all elements of the form aba1b1

    asa runs through Aandb runs through B. We define two sequences of subgroups:

    LetG0 :=G and for n N let Gn be the subgroup generated by the set [Gn1, Gn1].Gn is called the n-th derived group ofG.

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    86 6. The Infinitesimal Method

    LetG0 :=Gand forn NassumeGn to be the subgroup generated by the set [G, Gn1].We obviously obtain the following inclusions

    G= G0 G1 Gn . . . ,

    G= G

    0

    G1

    Gn

    . . . .For a commutative group these sequences are trivial, we have Gn = Gn ={e} for alln N.

    Definition 6.8: G is called solvable of class kiffGn= {e} beginning with n= k.G is called nilpotent of class kiffGn = {e} beginning with n = k.

    Exercise 6.26: Show that Heis(R) is nilpotent and solvable of class 1.

    Now we pass to the corresponding concepts for Lie algebras (following [Ki] p.89).

    Definition 6.9: A linear subspace g0 in a Lie algebra is called a subalgebraiff

    [X, Y] g0 for all X, Y g0.A linear subspace a in a Lie algebra is called an idealiff

    [X, Y] a for all X g andY a.If a is an ideal in g, then the factor space g/a is provided in a natural way with thestructure of a Lie algebra, which is called the factor algebraofgbya.

    In every Lie algebragwe can define two sequences of subspaces, where here the expression[a, b] denotes the linear hull of all [X, Y], X a, Y b:

    g1 := [g, g], g2 := [g, g1], . . . , gn+1 := [g, gn],

    g1 := [g, g], g2 := [g1, g1], . . . , gn+1 := [gn, gn].

    It is clear that one has the following inclusions

    gn gn+1, gn gn+1, gn gn, n N.Exercise 6.27: Prove that all gn and gn are ideals in gand that gn/gn+1 and g

    n/gn+1

    are commutative.

    For dim g j} is solvable,nn(K) := {X= (xij) Mn(K); , xij = 0 for i j}is nilpotent.It is clear that everyX nnis nilpotent, i.e. has the propertyXn = 0. This is generalizedby the following statement justifying the name nilpotent algebra:

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    6.6 Some Structure Theory 87

    Theorem 6.4 (Engel): g is nilpotent iff the operator ad X is nilpotent for all X g,i.e. for every Xthere is an n Nwith (ad X)n = 0.We recall that ad X is defined by ad X(Y) := [X, Y] for all Y g.

    Now we pass to the other side of Lie algebra structure theory. As in [Kn1] p.1 ff we

    restrict ourselves to the treatment of finite dimensional real or complex algebras g.

    Definition 6.11: gis said to besimpleiffg is non-abelian and g has no proper non-zeroideals.

    In this case one has [g, g] =g, which shows that we are as far from solvability as possible.

    Definition 6.12: gis said to be semisimpleiffghas no non-zero abelian ideal.

    There are other (equivalent) definitions, for instance g is said to be semisimple iff one

    has rad g= 0. In this definition the radical rad g is the sum of all solvable ideals ofg.

    Semisimple and simple algebras are related a follows.

    Theorem 6.5: g is semisimple iffg is the sum of simple ideals. In this case there areno other simple ideals, the direct sum decomposition is unique up to order of summandsand every ideal is the sum of the simple ideals. Also in this case [g, g] =g.

    Finally, from [KT] p.29 we take over

    Definition 6.13: A reductiveLie algebra is a Lie algebra that is the direct sum of twoideals, one equal to a semisimple algebra and the other to an abelian Lie algebra.

    We have a practical criterium:

    Theorem 6.6: Ifg is a real Lie algebra of real or complex (even quaternion) matricesclosed under conjugate transposition, then g is reductive. If moreover the center ofg istrivial, i.e. Zg:= {X g; [X, Y] = 0 for all Y g} = 0, then g is semisimple.

    Reductive Lie algebras have a very convenient property:

    Proposition 6.3: A Lie algebra gis reductive iff each ideal ain ghas a complementaryideal, i.e. an ideal bwith g= a b.

    As to be expected, to some extent the notions just defined for groups and algebras fittogether and one has statements like the following:

    Proposition 6.4 ([KT] p.29): IfG is a linear connected semisimple group, then g= Lie Gis semisimple. More generally, ifG is linear connected reductive, then g is reductive withg= Zg

    [g, g] as a direct sum of ideals. Here Zg denotes the center ofg, and the com-

    mutator ideal is semisimple.

    Example 6.12: gl(n, R) = {scalars} sl(n, R).

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    88 6. The Infinitesimal Method

    The main tool for the structure theory of semisimple Lie algebras is the following bilinearform that was first defined by Killing and then extensively used by E. Cartan.

    Definition 6.14: The Killing formB is the symmetric bilinear form on gdefined by

    B(X, Y) := Tr (adXadY) for all X, Y g.Remark 6.14: B is invariant in the sense that

    B([X, Y], Z]) =B(X, [Y, Z]) for all X,Y,Z g.Exercise 6.28: Determine the matrix ofB with respect to the basisa)F, G, H(from Exercise 6.3 in 6.1) for g= sl(2, R),b) P,Q, R (from Exercise 6.7 in 6.1) for g= heis(R).

    The starting point for structure theory of semisimple Lie algebras is Cartans Criterium

    for Semisimplicity.

    Theorem 6.7: g is semisimple if and only if the Killing form is nondegenerate, that isB(X, Y) = 0 for all Y gimplies X= 0.

    The proof uses the remarkable fact that ker B is an ideal in g.There is another important and perhaps a bit more accessible bilinear form on g: Thetrace formB0 is given by

    B0(X, Y) := Tr (XY) for all X, Y g.The trace form is invariant in the same sense as the Killing form. Both forms are related(see e.g. [Fog] IV.4):

    Remark 6.15: We have

    B(X, Y) = 2nTr (XY) for g = sl(n, K) and n 2,= (n 2)Tr (XY) for g = so(n, K) and n 3,= 2(n + 1)Tr (XY) for g = sp(2n, K) and n 1.

    A variant of the trace form already appeared in our definition of topology in matrix

    spaces. We introduced forX, Y Mn(C)< X, Y >:= Re Tr t XY, and X2 := Re Tr t XX.

    As infinitesimal object corresponding to the Cartan involution for our matrix groups G

    : GL(n, C) GL(n, C), g (tg)1,we have the map for the Lie algebra g= Lie G

    : Mn(C) Mn(C), X t X.Hence we have

    < X, Y > := Re B0(X,Y)as a scalar product on gas a real vector space. This will become important later on.

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    6.6 Some Structure Theory 89

    6.6.2 Structure Theory for Complex Semisimple Lie Algebras

    A complete classification exists for semisimple Lie algebras, as one knows their buildingblocks, the simple algebras: Over Cthere exist four infinite series of classicalsimple Liealgebras and five exceptionalsimple Lie algebras. Over R we have 12 infinite series and

    23 exceptional simple algebras. From all these, here we reproduce only the followingstandard list of classical complex simple algebras

    1) An:= {X Mn+1(C); Tr X= 0}, n= 1, 2, 3, . . .2) Bn:= {X M2n+1(C); X= tX}, n= 2, 3, . . .3) Cn:= {X M2n(C); XJ2n+ J2nX= 0}, n= 3, 4, . . . , J 2n:= ( EnEn ),4) Dn:= {X M2n(C); X= tX}, n= 4, 5, . . . .

    We have the isomorphisms

    B1 A1 C1, C2 B2, D2 A1 A1, D3 A3and D1 is commutative. In the sequel we shall in most cases return to our previousnotation and write

    An=sl(n + 1, C), Bn=so(2n + 1, C), Cn=sp(n, C), Dn= so(2n, C).

    All these algebras are also simple algebras over R. The remaining real classical algebrasremain simple upon complexification. For a complete list see for instance [Ki] p.92/3.By the way, we introduce here the following also otherwise important concept:

    Definition 6.15: Given a complex Lie algebra g, a real algebra g0 is called a real formofgiff (g0)c:= g0 C= g.

    Example 6.13: g= sl(2, C) has just two real forms g0 = sl(2, R) and su(2).More about this is to be found in [Kn1] p.15.

    The main ingredients for a classification scheme are the rootand Dynkin diagrams. Wegive a sketch starting by the case of complex semisimple algebras where the theory ismost accessible, and then go to compact real and finally briefly to noncompact real al-

    gebras. The central tool to define roots and their diagrams is a certain distinguishedabelian subalgebra.

    Definition 6.16: Let gbe a complex semisimple Lie algebra. A Cartan subalgebrah isa maximal abelian subspace ofgin which every ad H, H h, is diagonizable.

    There are other equivalent ways to characterize a Cartan subalgebra h, for instance (see[Kn1] p.2): h is a nilpotent subalgebra whose normalizerNg(h) satisfies

    Ng(h) := {X g, [X, H] h for all H h} =h.

    Each semisimple complex algebra has a Cartan subalgebra. Any two are conjugate viaIntg ([Kn1] p.24), where Int g is a certain analytic subgroup of GL(g) with Lie algebraad g. We refer to [Kn2] p.69/70 for the notion of an analytic subgroup, which comes upquite naturally when one analyzes the relation between Lie algebras and Lie groups.

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    90 6. The Infinitesimal Method

    The theory of Cartan subalgebras for the complex semisimple case extends to a complexreductive Lie algebra gby just saying that the center ofgis to be adjoined to a Cartansubalgebra of the semisimple part ofg.

    Example 6.14: For g= sl(n, C) we have as Cartan subalgebra

    h:= {diagonal matrices ing}.Using our Cartan subalgebra h, we generalize the decomposition we introduced at thebeginning of Section 6.4 for g= sl(2, C)

    g= < Z0 > + < X+>+ < X>,

    [Z0, X] = 2X, [X+, X] =Z0,to the root space decomposition

    g= h

    g

    as follows: For all H h the maps ad Hare diagonizable, and, as the elements of hall commute, they are simultaneously diagonizable (by the finite-dimensional spectraltheorem from Linear Algebra). LetV1, . . . , V be the eigenspaces in g for the differentsystems of eigentupels. Ifh := < H1, . . . , H r > and ad Hi acts as ij id on Vj , definea linear functional j on h by j(Hi) = ij . If H := ciHi, then ad H acts on Vj bymultiplication with

    i

    ciij =i

    cij(Hi) =:j(H).

    In other words, ad h acts in simultaneously diagonal fashion on g and the simultaneouseigenvalues are members of the dual vector space h := HomC(h, C). There are finitelymany such simultaneous eigenvalues and we write

    g:= {X g; [H, X] =(H)X for all H h}for the eigenspace corresponding to h. The nonzero such are called rootsand thecorresponding g a root space. The (finite) set of all roots is denoted by .

    The following are some elementary properties of root space decompositions (for proofs

    see for instance [Kn2] II.4):

    Proposition 6.5: We have

    a)[g, g] g+.b) If, \ {0} and += 0, then B(g, g) = 0, i.e. root spaces are orthogonalwith respect to the Killing form.

    c)B is nonsingular on g g if .d) if .e)B|hh is nondegenerate. We define H to be the element ofh paired with .f) spans h.

    Some deeper properties of root space decompositions are assembled in the following state-ment (see again [Kn2] II.4).

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    6.6 Some Structure Theory 91

    Theorem 6.8: Root space decompositions have the following properties:

    a) dim g= 1 for .b)n for and any integer n 2.

    c) [g, g] =g+ for + = 0.d) The real subspace h0 ofh on which all roots are real is a real form ofh andB|h0h0is an inner product.

    We transfer B| h0h0 to the real span h0 of the roots obtaining a scalar product and a norm .. It is not too difficult to see that for there is an orthogonaltransformations given on h0 by

    s() :=

    2< , >

    2 for

    h0.

    s is called a root reflectionin and the hyperplane a mirror.

    The analysis of the root space shows that it has very nice geometrical and combinatoricalproperties. The abstraction of these is the following:

    Definition 6.17: An abstract root systemis a finite set of nonzero elements in a realinner product spaceV such that

    a) spans V,

    b) all reflections s for carry to itself,c) 2< , > / 2 Z for all , .The abstract root system is called reducediff implies 2 .And it is called reducibleiff = with , otherwise it is irreducible.

    The root system of a complex semisimple Lie algebra gwith respect to a Cartan subal-gebra h forms a reduced abstract root system in h0. And a semisimple Lie algebrag issimple if the corresponding root system is irreducible.

    Definition 6.18: The dimension of the underlying space Vof an abstract root system is called its rank, and if is the root system of a semisimple Lie algebra g, we alsorefer to r= dim h as the rankofg.

    To give an illustration, we sketch the reduced root systems of rank 2.

    case 1 case 2 case 3 case 4

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    92 6. The Infinitesimal Method

    Ordering of the Roots, Cartan Matrices, and Dynkin Diagrams

    We fix an ordered basis 1, . . . , r ofh0 and define = cii to be positiveif the first

    nonzero ci is positive. The ordering comes from saying > if is positive. Let+ be the set of positive members in .

    A root is called simpleif >0 and does not decompose as = 1+ 2 with1 and2 positive roots. And a root is called reducedif (1/2) is not a root.Relative to a given simple system 1, . . . , r, the Cartan matrixC is the r r-matrixwith entries

    cij = 2< i, j > / i2 .It has the following propertiesa)cij Z for all i, j,b) cii= 2 for all i,

    c)cij 0 for all i =j ,d) cij = 0 iffcji = 0,e) there exists a diagonal matrix D with positive diagonal entries such that DCD1 issymmetric positive definite.

    Anabstract Cartan matrixis a square matrix satisfying properties a) through e) as above.To such a matrix we associate a diagram usually called a Dynkin diagram (historicallymore correct is the term CDW-diagram, C indicating Coxeter and W pointing to Witt):To the elements 1, . . . , r of a basis of a root system correspond bijectively r points ofa plane, which are also called1, . . . , r. Fori

    =j one joins the point i toj bycijcji

    lines, which do not touch any k, k= i,j. For cij= cji the lines are drawn as arrowspointing in the direction to j ifcij < cji .

    The main facts are that a Cartan matrix and equivalently the CDW-diagram determinethe Lie algebra uniquely (up to isomorphism). In particular this correspondence doesnot depend on the choice of a basis of the root system.

    Example 6.15: For g= sl(r+ 1, C), r 1 and its standard Cartan subalgebra

    h:=

    {D(d1, . . . , dr); d1, . . . , dr

    C, di = 0

    }we have the Cartan matrix

    C:=

    2 11 2

    2 11 2

    and the corresponding CDW-diagram

    Ar r 1.. . .1 2 r

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    6.6 Some Structure Theory 93

    The diagrams for the other classical algebras are the following

    Br r 2,. . .1 2 r

    1 r

    Cr r 3,. . .1 2 r1 r

    Dr r 4.. . .1 2 r2

    r1

    r

    The remaining exceptionalgraphs of indecomposable reduced root systems are the fol-lowing

    E6

    E7

    E8

    F4 G2

    6.6.3 Structure Theory for Compact Real Lie Algebras

    Up to now, we treated complex Lie algebras. We go over to a real Lie algebra g0 and(following [Kn1] p.16) we call a subalgebra h0 ofg0 aCartan subalgebraif its complexifi-cation is a Cartan subalgebra of the complex algebra g= g0 C.At first we look at thecompact case: Ifg0 is the Lie algebra of a compact Lie group G and if t0 is a maximal

    abelian subspace ofg0, then t0 is a Cartan subalgebra. We have already discussed theexampleg0 = su(2) in 4.2 and we will discuss g0 = su(3) in the next section and see thatwe have dim t0 = 2 in this case. To illustrate the general setting, we collect some moreremarks about the background notions (from [Kn1] p.15):

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    94 6. The Infinitesimal Method

    Theorem 6.9: Ifg0 is semisimple, then the following conditions are equivalent:a) g0 is the Lie algebra of some compact Lie group.b) Int g0 is compact.c) The Killing form ofg0 is negative definite.

    If g is semisimple complex, a real form g0 of g is said to be compact if the equivalentconditions of the theorem hold. Our main example su(n) is a compact form ofsl(n, C).The fundamental result is here:

    Theorem 6.10: Each complex semisimple Lie algebra has a compact real form.

    Another important topic is maximal tori.

    Definition 6.19: LetG be a compact connected linear group. A maximal torusin G isa subgroupT maximal with respect to the property of being compact connected abelian.

    From [Kn2] Proposition 4.30, Theorem 4.34 and 4.36 we take over:

    Theorem 6.11: IfGis a compact connected linear group, thena) the maximal tori inGare exactly the analytic subgoups corresponding to the maximalabelian subalgebras ofg0 = Lie G;b) any two maximal abelian subalgebras ofg0 are conjugate via Ad Gand hence any twomaximal tori are conjugate via G.

    Theorem 6.12: IfG is compact connected and Ta maximal torus, then each element

    ofGis conjugate to a member ofT.

    Example 6.16:

    1. For G= SU(n) one has as a maximal torus, its Lie algebra and its complexified Liealgebra

    T = {D(ei1 , . . . , ein); j = 0},t0 = {D(i1, . . . , i n); j = 0},t = {D(z1, . . . , zn); zj = 0}.

    2. For G = SO(2n) and SO(2n+ 1) one has as a maximal torus T the diagonal blockmatrices with 2 2-blocksr(j), j = 1, . . . , n ,in the diagonal, resp. additionally 1 in thesecond case.

    We go back to the general case and use the notation we have introduced. In this setting,we can form a root-space decomposition

    g= t

    g.

    Each root is the complexified differential of a multiplicative character of the maximaltorusTthat corresponds to t0, with

    Ad(t)X=(t)X for all X g.

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    6.6 Some Structure Theory 95

    Another central concept is the one of the Weyl group. We keep the notation used above:Gis compact connected,g0 = Lie G,Ta maximal torus,t0 = Lie T,t= t0 C, (g, t) isthe set of roots, and B is the negative of a G invariant scalar product on g0. We definetR = it0. As roots are real on tR, they are in t

    R

    . The form B, when extended to becomplex bilinear, is positive definite on tR, yielding a scalar product ont

    R

    . Now, theWeyl groupW =W((g, t)) is in this context defined as the group generated by the rootreflectionss, (g, t),given (as already fixed above) on tR bys() = 2||2 .This is a finite group, which also can be characterized like this:One defines W(G, T) as the quotient of the normalizer by the centralizer

    W(G, T) :=NG(T)/ZG(T).

    By Corollary 4.52 in [Kn2] p.260 one has ZG(T) = T and hence also the formulaW(G, T) =NG(T)/T.

    Theorem 6.13: The group W(G, T), when considered as acting on tR, coincides withW((g, t)).

    6.6.4 Structure Theory for Noncompact Real Lie Algebras

    Now, in the final step, we briefly treat the general case of a real Lie algebra g0 ofa noncompact semisimple group G. We already introduced the Killing form B, theCartan involution with X=t X for a matrix X, and the fact that ifg0 consists ofreal, complex or quaternion matrices and is closed under conjugate transpose, then it isreductive. More generally we call here a Cartan involution any involution ofg0 suchthat the symmetric bilinear form

    B(X, Y) := B(X,Y)

    is positive definite. Then we have the Cartan decomposition

    g0 =k0 p0, k0 := {X g0; X=X}, p0 := {X g0; X= X}

    with the bracket relations

    [k0, k0] k0, [k0, p0] p0, [p0, p0] k0.

    One has the following useful facts ([Kn 2] VI.2):a) g0 has a Cartan involution.b) Any two Cartan involutions ofg0 are conjugate via Int g0.c) If g is a complex semisimple Lie algebra, then any two compact real forms of g areconjugate via Int g.d) Ifg is a complex semisimple Lie algebra and is considered as a real Lie algebra, thenthe only Cartan involutions ofg are the conjugations with respect to the compact real

    forms ofg.

    The Cartan decomposition of Lie algebras has a global counterpart (see for instance[Kn1] Theorem 4.3). Its most rudimentary form is the following:

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    96 6. The Infinitesimal Method

    Proposition 6.6: For every A GL(n, C) resp. GL(n, R), there are S1, S2 O(n)resp. U(n) and a diagonal matrixD with real positive elements in the diagonal such thatA= S1DS2. This decomposition is not unique.

    Proofas Exercise 6.29 (use the fact that to each positive definite matrix B there is a

    unique positive definite matrixP with B =P2 and/or see [He] p.73).

    Restricted Roots

    We want to find a way to describe Cartan subalgebras in our general situation: LetG be semisimple linear group with Lie G = g0, g = g0 C, a Cartan involution ofg0, and g0 = k0 p0 the corresponding Cartan decomposition. Let B be the Killingform (or more generally any nondegenerate symmetric invariant bilinear form on g0 withB(X,Y) =B(X, Y) such that B(X, Y) = B(X,Y) is positive definite).

    Definition 6.20: Let a0 be a maximal abelian subspace ofp0. Restricted rootsare thenonzero a0 such that

    (g0):= {X g0; (ad H)X=(H)X for all H a0} = {0}.

    Let be the set of restricted roots and m0 :=Zk0 . We fix a basis ofa0 and an associatedlexicographic ordering in a0 and define

    + as the set ofpositive resticted roots. Then

    n0 =

    +

    (g0)

    is a nilpotent Lie subalgebra, and we have the following Iwasawa decomposition:

    Theorem 6.14: The semisimple Lie algebra g0 is a vector space direct sum

    g0 = k0 a0 n0.

    Here a0 is abelian, n0 is nilpotent, a0 n0 is a solvable subalgebra ofg0, and a0 n0 has[a0 n0, a0 n0] =n0.

    For a proof see for instance [Kn2] Proposition 6.43. There is also a global version ([Kn2]Theorem 6.46) stating (roughly) that one has a diffeomorphism of the correspondinggroups K A N G given by (k,a,n) kan. We already know all this in ourstandard example G= SL(2, R): If we take X= tX, we have

    g0 = sl(2, R) =k0 a0 n0,

    with

    k0 =, a0 =, n0 =

    and as in Example 6.6 in 6.2

    G= KAN with K= SO(2), A= {t(y); y>0}, N = {n(x); x R}.

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    6.6 Some Structure Theory 97

    As to be expected, roots and restricted roots are related to each other. Ift0 is a maximalabelian subspace ofm0 := Zk0(a0), then h0 := a0 t0 is a Cartan subalgebra of g0 inthe sense we defined at the beginning ([Kn2] Proposition 6.47). Roots are real valued ona0 and imaginary valued on t0. The nonzero restrictions to a0 of the roots turn out tobe restricted roots. Roots and restricted roots can be ordered compatibly by taking a0before it0. Cartan subalgebras in this setting are not always unique up to conjugacy.

    Exercise 6.30: Show that h0 :=< ( 1

    1 ) > and h0 :=< (

    11 ) > are Cartan

    subagebras ofg0 = sl(2, R) and determine the corresponding Iwasawa decomposition.

    Every Cartan subalgebra ofg0 is conjugate (via Intg0) to this h0 or the h0 above.

    6.6.5 Representations of Highest Weight

    After this excursion into structure theory of complex and real Lie algebras, we finallycome back to show a bit more of the general representation theory, which is behind theexamples discussed in our sections 6.4 and 6.5 and which we shall apply in our nextsection to the example g0 = su(3). We start by following the presentation in [Kn1] p.8.Let at first g be a complex Lie algebra, h a Cartan subalgebra, = (g, h) the set ofroots, h0 the real form ofh where roots are real valued, B the Killing form (or a moregeneral form as explained above), and H h0 corresponding to h0.Let : g End Vbe a representation. For h we put

    V:= {v V; ((H) (H)1)n

    v= 0 for all H h and some n= n(H, V) N}.If V ={0}, V is called a generalized weight space and a weight. If V is finitedimensional, Vis the direct sum of its generalized weight spaces. This is a generalizationof the fact from linear algebra about eigenspace decompositions of a linear transformationon a finite-dimensional vector space. If is a weight, then the subspace

    V0 := {v V; (H)v= (H)v for all H h}

    is nonzero and called the weight spacecorresponding to. One introduces a lexicographicordering among the weights and hence has the notion ofhighestand lowest weights. The

    set of weights belonging to a representation is denoted by (). For dim h = 1 wehave h h C and so the weights are simply the complex numbers we met in ourexamples in 6.4 and 6.5, for instance for g= sl(2, C) we found representations N withdim VN = N+ 1 and weightsN, N+ 2, . . . , N 2, N. In continuation of this, wetreat a more general example.

    Example 6.17: Let G = SU(n) and, hence, g = su(n)c = sl(n, C) and the Cartansubalgebra the diagonal subalgebra h:=< H; Tr H= 0>. We choose as generators ofhthe matrices Hj :=Ejj Ej+1,j+1 and (slightly misusing) we write also Hj =ej ej+1whereej denote the canonical basis vectors of C

    n1

    h. We have three natural types

    of representations :At first, letVbe the space of homogeneous polynomials Pof degree N inz1, . . . , zn andtheir conjugates and take the action ofg SL(n, C) given by

    ((g)P)(z, z) :=P(g1z, g1z), z= t(z1, . . . , zn).

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    98 6. The Infinitesimal Method

    Hence for H=D(i1, . . . . i n) with j R, j = 0, we come up with

    (d(H)P)(z, z) =

    nj=1

    (ijzj)zjP(z, z) +n

    j=1

    (ij zj)zjP(z, z).

    IfP is a monomial

    P(z, z) =zk11 zknn zl11 zlnn withn

    j=1

    (kj+ lj) =N,

    then we get

    d(H)P =n

    j=1

    (lj kj)(ij)P.

    Exercise 6.31: Describe the weights for this representation with respect to the lexico-graphic ordering for the basis elements ofh0 =< iH1, . . . , i H n1 >R. Do this also for thetwo other natural representations, namely on the subspaces V1 of holomorphic andV2of antiholomorphic polynomials in V(i.e. polyomials only in z1, . . . , zn resp. z1, . . . , zn).We will come back to this for the case n= 3 in the next section.

    In our examples g0 = sl(2, R) and g0 = su(2) in 6.4 resp. 6.5 we saw that the finite-dimensionality of the representation space V was equivalent to an integrality conditionfor the weight. Now we want to look at the general case where the weights are l-tupelsif one fixes a basis of a Cartan subalgebra h resp. its dual h and the real form h0.

    Definition 6.21: h is said to be algebraically integralif

    2< , > /||2 Z for all .

    Then as a generalization of what we saw in 6.4 and 6.5, we have the elementary propertiesof the weights for a finite-dimensional representation on a vector space V:

    Proposition 6.7: a) (h) acts diagonally on V, so that every generalized weight vectoris a weight vector and V is the direct sum of all weight spaces.b) Every weight is real valued on h0 and algebraically integral.

    c) Roots and weights are related by (g)V V+.We fix a lexicographical ordering and take + to be the set of positive roots with ={1, . . . , l} as the corresponding simple system of roots. We say that V isdominantif< , j > 0 for all j . The central fact is here the beautiful Theoremof the Highest Weight([Kn2] Th. 5.5):

    Theorem 6.15: The irreducible finite-dimensional representations ofg stand (up toequivalence) in one-one correspondence with the algebraically integral dominant linearfunctionalson h, the correspondence being that is the highest weight of .

    The highest weight of has these additional properties:a) depends only on the simple system and not on the ordering used to define .b) The weight space V for is one-dimensional.c) Each root vector E for arbitrary + annihilates the members ofV, and themembers ofV are the only vectors with this property.

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    6.6 Some Structure Theory 99

    d) Every weight of is of the form l

    i=1 nii with ni N0 and i .e) Each weight space V for has dimVw = dim V for allw in the Weyl group W(),and each weighthas| || | with equality only if is in the orbit W().

    We already introduced in 3.2 the concept of complete reducibility. Here we can state the

    following fact ([Kn2] Th. 5.29).

    Theorem 6.16: Let be a complex linear representation ofg on a finite-dimensionalcomplex vector space V. Then V is completely reducible in the sense that there existinvariant subspaces U1, . . . , U r of V such that V = U1 Ur and such that therestriction of the representation to each Ui is irreducible.

    The proofs of these two theorems use three tools, which are useful also in other contexts: Universal enveloping algebras, Casimir elements,

    Verma modules.Again following [Kn1] p.10f, we briefly present these as they will help us to a betterunderstanding in our later chapters.

    The Universal Enveloping Algebra

    This is a general and far reaching concept applicable for any complex Lie algebra g:

    We take the tensor algebra

    T(g) :=C g (g g) . . . .

    and the two-sided ideal a in T(g) generated by all

    X Y Y X [X, Y], X , Y T1(g).

    HereT1(g) denotes the space of first order tensors.

    Then the universal enveloping algebrais the associative algebra with identity given by

    U(g) :=T(g)/a.

    This formal definition has the practical consequence that U(g) consists of sums of mono-mials usually written (slightly misusing) as a(j)X

    j11 . . . X

    jnn , a(j) C, X1, . . . , X n g

    with the usual addition and a multiplication coming up from the Lie algebra relationsbyXiXj =XjXi+ [Xi, Xj ]. More carefully and formally correct, one has the following:Let: g U(g) be the composition of natural maps

    : g T1(g) T(g) U(g),

    so that

    ([X, Y]) =(X)(Y) (Y)(X).

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    100 6. The Infinitesimal Method

    is in fact injective as a consequence of the fundamental Poincare-Birkhoff-Witt Theorem:

    Theorem 6.17: Let{Xi}iI be a basis of g, and suppose that a simple ordering hasbeen imposed on the index set I. Then the set of all monomials

    (Xi1)j1 (Xin)jn

    with i1 0, and H

    k11 . . . H

    kll will act as a scalar. Thus one has only to sort out

    the effect ofEp11 . . . E pmm and most of the conclusions of the Theorem of the HighestWeight follow readily. If one looks at our examples in 6.4 and 6.5, one can get an ideahow this works even in the case of representations, which are not finite-dimensional, andhow the integrality of the weight leads to finite-dimensionality.

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    6.6 Some Structure Theory 101

    The Casimir Element

    For a complex semisimple Lie algebra g with Killing form B, the Casimir element isthe element

    0 :=i,j

    B(Xi, Xj)Xi Xj

    ofU(g), where (Xi) is a basis ofg and (Xi) is the dual basis relative to B. One can showthat 0 is defined independently of the basis (Xi) and is an element of the center Z(g)ofU(g).

    Exercise 6.32: Check this for the case g = sl(2, C) and determine the Casimir element.

    In the general case one has the following statement.

    Theorem 6.20: Let 0 be the Casimir element, (Hi)i=1,.,l an orthogonal basis of h0relative toB, and choose root vectors E so that B(E, E) = 1 for all roots . Thena) 0 =

    li=1 H

    2i +

    EE.b) 0 operates by the scalar||2 + 2 < , > =|+|2 ||2 in an irreducible finite-dimensional representation ofgof highest weight, whereis half the sum of the positiveroots.c) The scalar by which 0 operates in an irreducible finite-dimensional representation ofgis nonzero if the representation is not trivial.

    The main point is that ker 0 is an invariant subspace ofV ifVis not irreducible.

    Remark 6.16: The center ofU(g) is important also in the context of the determinationof infinite-dimensional representations. We observed in Exercise 6.23 in 6.4 that , amultiple of the Casimir element 0,acts as a scalar for the unitary irreducible represen-tations ofsl(2, R). For more general information we recommend [Kn] p.214 where as aspecial case of Corollary 8.14 one finds the statement: If is unitary, then each mem-ber of the center Z(gc) ofU(gc) acts as a scalar operator on the Kfinite vectors of.We will come back to this in 7.2 while explicitly constructing representations of SL(2 , R).

    The Verma Module

    We fix a lexicographic ordering and introduce

    b:= h >0

    g.

    For h, make C into a one-dimensional U(b) module Cby defining the action ofH h byHz = (H)z for z C and the action of>0g by zero. For h, wedefine the Verma moduleV() by

    V() :=U(g) U(h)C.whereis again half the sum of the positive roots and this term is introduced to simplifycalculations with the Weyl group.

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    102 6. The Infinitesimal Method

    Verma modules are essential for the construction of representations. They have the fol-lowing elementary properties:a)V() = 0.b) V() is a universal highest weight module for highest weight modules ofU(g) withhighest weight

    .

    c) Each weight space ofV() is finite-dimensional.d) V() has a unique irreducible quotient L().

    If is dominant and algebraically integral, thenL( +) is the irreducible representationof highest weight looked for in the theorem of the Highest Weight (Theorem 6.15).

    In our treatment of finite and compact groups we already saw the effectiveness of thetheory of characters of representations. As for the moment we look at finite-dimensionalrepresentations we can use characters here too. To allow for more generalization, wetreat them for now as formal exponential sums (again following [Kn1] p.12/3):

    Let againg be a semisimple Lie algebra, h a Cartan subalgebra, a set of roots providedwith a lexicographic ordering, 1, . . . , l the simple roots, and W() the Weyl group.We regard the set Zh

    of functions f from h to Z as an abelian group under pointwiseaddition. We write

    f=h

    f()e.

    The support off is defined to be the set of h for which f()= 0. Within Zh ,let Z[h] be the subgroup of all fof finite support. The subgroup Z[h] has a naturalcommutative ring structure, which is determined byee =e+.Moreover, we introduce

    a larger ring Z< h> with

    Z[h] Z< h> Zh

    consisting of all f Zh whose support is contained in the union of finitely many setsi Q+, i h and

    Q+ := {l

    i=1

    nii; ni N0}.

    Multiplication in Z< h> is given by

    (h

    ce)(h

    ce) :=

    h

    (

    +=

    cc)e.

    IfV is a representation ofg (not necessarily finite-dimensional), one says that V has acharacterifV is the direct sum of its weight spaces under h, i.e., V = hV , and ifdim V< for h. In this case the characteris

    char(V) := h

    (dim V)e

    as an element ofZh

    . This definition is meaningful ifVis finite-dimensional or ifV is aVerma module.

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    6.6 Some Structure Theory 103

    We have two more important notions:The Weyl denominatoris the element d Z[h] given by

    d:= e+(1 e).

    Here is again half the sum of the positive roots.The Kostant partition functionPis the function fromQ+ to Nthat tells the number ofways, apart from order, that a member ofQ+ can be written as the sum of positve roots.We put P(0) = 1 and define K:= Q+P()e Z< h>. Then one can prove thatone has Ked= 1 in the ring Z< h > , hence d1 Z< h>. Then we have as thelast main theorem in this context the famous Weyl Character Formula:

    Theorem 6.21: Let (, V) be an irreducible finite-dimensional representation of thecomplex semisimple Lie algebra gwith highest weight. Then

    char(V) =d1

    wW()(det w)ew(+).

    Now we leave the treatment of the complex semisimple case and go over to the follow-ing situation: G is compact connected, g0 := Lie G, g the complexification of g0, T amaximal torus, t0 := Lie T, (g, t) the set of roots, and B the negative of a G invariantinner product on g0, and tR := it0. As we know, roots are real on tR, hence are in t

    R

    .The formB, when extended to be complex bilinear, is positive definite ontR, yielding aninner producton t

    R. W((g, t)) is the Weyl group generated by the root reflections

    s for (g, t). Besides the notion of algebraic integrality already exploited in thecomplex semisimple case above, we have here still another notion of integrality:We say that t is analytically integral if the following equivalent conditions hold:1) Whenever H t0 satisfies exp H= 1, then (H) is in 2iZ.2) There is a multiplicative character ofT with (exp H) =e

    (H) for all H t0.In [Kn1] p.18 one finds a list of properties of these notions. We cite part of it:a) Weights of finite-dimensional representations ofG are analytically integral. In partic-ular every root is analytically integral.b) Analytically integral implies algebraically integral.c) IfG is simply connected and semisimple, then algebraically integral implies analyti-

    cally integral.

    For instance, the half sumof positve roots is algebraically integral but not analyticallyifG= SO(3).

    In our situation the Theorem 6.15 (Theorem of the Highest Weight) comes in the followingform:

    Theorem 6.22: Let G be a compact connected Lie group with complexified Lie alge-bra g, let T be a maximal torus with complexified Lie algebra t, and let +(g, t) be a

    positive system for the roots. Apart from equivalence the irreducible finite-dimensionalrepresentations ofG stand in one-one correspondence with the dominant analyticallyintegral linear functionals on t, the correspondence being that is the highest weightof.

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    104 6. The Infinitesimal Method

    And we restate Theorem 6.21 (Weyls Character Formula) in the form:

    Theorem 6.23: The character of the irreducible finite-dimensional representation ofG with highest weight is given by

    =

    wW(det w)w(+)(t)+(1 (t))

    at every t Twhere no takes the value 1 on t.

    In the next section we shall illustrate this by treating an example, which was a milestonein elementary particle physics.

    6.7 The Example su(3)

    In 1962 Gell-Mann proposed in a seminal paper [Gel] a symmetry scheme for the de-scription of hadrons, i.e. certain elementary particles (defined by interacting by stronginteraction), which had a great influence in elementary physics and beyond, as it ledto the notion of quarksand the eightfold way. We can not go too much into the phys-ical content but discuss the mathematical background to give another example of thegeneral theory. Respecting the historical context, we adopt the notation introduced byGell-Mann and used in Cornwells presentation in [Co] vol II, p.502ff: As su(3) consistsof tracelass skew hermitian three-by-three matrices, one can use as a basis

    A1 :=

    ii , A2 :=

    11 , A3 :=

    i i ,

    A4 :=

    i

    i

    , A5 :=

    1

    1

    , A6 :=

    i

    i

    ,

    A7 :=

    1

    1

    , A8 := 13

    i i

    2i

    .

    This basis is also a basis for the complexification su(3)c = sl(3, C). One may be temptedto use the elementary matrices Eij with entries (Eij)kl =ikjl and take as a basis

    Eij , i =j, (i, j= 1, 2, 3) H1 :=E11 E22, H2 := E22 E33.One has the commutation relations [Eij , Ekl] = jkEil ilEkj . For a diagonal matrixH :=D(h1, h2, h3) we get

    [H, Eij] = (ei(H) ej(H))Eij , ei(H) =hi,and hence

    [H1, E12] = 2E12, [H2, E12] = E12,[H1, E13] = E13, [H2, E13] = E13,[H1, E23] = E23, [H2, E23] = 2E23,

    and[E12, E21] =H1, [E23, E32] =H2, [E13, E31] =E11 E33 =: H3.

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    6.7 The Example su(3) 105

    We see that h:=< H1, H2 > is a Cartan subalgbra,

    g1 := < E12 >, g2 := < E13 >, g3 := < E23 >

    are root spaces, i.e. nontrivial eigenspaces for the roots 1, 2, 3 h given by1 := (1(H1), 1(H2)) = (2, 1),2 := (2(H1), 2(H2)) = (1, 1),3 := (3(H1), 3(H2)) = (1, 2).

    (In the physics literature these tuples j themselves sometimes are called roots.)We choose these roots j as positive rootsand then get in this coordinization a slightlyunsymmetric picture:

    1

    2

    3

    1 2

    -1-2

    2

    -1

    Hence we better use Gell-Manns matrices and change to Xj :=iAj . Then we getH1 =X3, H

    2 := X8.Moreover, forX

    := (1/2)(X1

    iX2), Y

    := (1/2)(X6

    iX7),and

    Z := (1/2)(X4iX5),we haveX+=E12, X=E21etc. and the commutation relations[H1, X] = 2X, [H2, X] = 0,[H1, Y] = Y, [H2, Y] =

    3Y,

    [H1, Z] = Z, [H2, Z] =

    3Z.

    Now the Cartan subalgebra is h=< H1, H2 > and the root spaces are

    g1 =< X, >, g2 =< Y>, g3 =< Z>

    with positive roots given by

    1 := (1(H1), 1(H2)) = (2, 0),

    2 := (2(H1), 2(H2)) = (1,

    3),

    3 := (3(H1), 3(H2)) = (1,

    3).

    1 and 2 are simple roots, we have 1+ 2 =3 and we get a picture with hexagonalsymmetry:

    1

    32

    1 2-1-2

    2

    -1

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    106 6. The Infinitesimal Method

    In the literature (see for instance [Co] p.518) we find still another normalization, whichis motivated like this: The Killing form B for su(3) is given by diagonal matrices

    B(Ap, Aq) = 12pq, resp. B(Xp, Xq) = 12pq.

    Exercise 6.33: Verify this.

    One usesB to introduce a nondegenerate symmetric bilinear form on h as follows:We define a map

    h H h byB(H, H) =(H) for all H h,and then

    < , >:=B(H, H) for all , .This leads to inner products on hR = < H1 , . . . , H l >R (1, . . . , l simple positive

    roots) and its dual space. For su(3) we have l= 2 and with 1, 2 from above

    < 1, 1 >=< 2, 2 >= 1/3, < 1, 2 >=< 2, 1 >= 1/6.It is convenient to introduce an orthonormal basis H1, . . . , Hl ofh, i.e. withB(Hp, Hq) =pq. Then we have the simple rule

    < , > =j

    jj , j =(Hj),j =(Hj).

    In our case this leads to

    H1 = (1/(23))H1, H2 = (1/2)H2 = (1/(23))X8and

    1 = (

    3/6)1 = (1/

    3, 0), 2 = (

    3/6)2 = (1/(2

    3), 1/2).

    We see that, for the Cartan matrixC= (Cpq) = (2< p, q > / < p, p >) from 6.6.2,we have in this case

    C= ( 2 11 2 ).

    The Weyl group W(see the end of 6.6.3) is generated by the reflections

    sp(q) = q Cpqpat the planes in h

    R Rl orthogonal to the p withl = 2 and p, q= 1, 2 in our case. For

    g0 =su(3) we have 6 elements in W.

    Exercise 6.34: Determine these explicitely.

    Now we proceed to the discussion of the representations (, V) of su(3). The generaltheory tells us that we can assume that (H) is a diagonal matrix for each H h. The

    diagonal elements (H)jj =: j(H) fix the weights j , i.e. linear functionals onh

    . Wewrite= ((H1), . . . , (Hl))

    for an ON-basis ofh. By the Theorem of the Highest Weight (Theorem 6.15 and Theorem6.22), an irreducible representation is determined by its highest weight and this highest

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    6.7 The Example su(3) 107

    weight is simple, i.e. has multiplicity one. Moreover the general theory says ([Co] p.568)that these highest weights are linear combinations with non negative integer coefficientsof fundamental weights1, . . . , l defined by

    j(H) =

    lk=1

    (C1)kjk(H)

    where1, . . . , l are the positive simple roots, fixed at the beginning.And with= (1/2)j the half sum of these positive simple roots Weyls dimensionalityformula says ([Co] p.570) that the irreducible finite-dimensional representation (, V)with highest weight has dimension

    d= lj=1

    < ,j > .

    In our case we have

    C1 = ( 2/3 1/31/3 2/3

    ).

    and the fundamental weights

    1 = (2/3)1+ (1/3)2 = (1/6)(

    3, 1), 2 = (1/3)1+ (2/3)2 = (1/6)(0, 2).

    We write (n1, n2) for the representation with highest weight =n11+ n22 and get

    by Weyls formula for its dimension

    (6.6) d= (n1+ 1)(n2+ 1)((1/2)(n1+ n2) + 1).

    In the physics literature one often denotes the representation by its dimension. Then onehas

    a) (0, 0) = {1}, the trivial representation, which has only one weight, namely the high-est weight = 0,b) (1, 0) =

    {3

    }with highest weight = 1 and dimension 3,

    c) (0, 1) = {3}with highest weight = 2 and dimension 3,d) (2, 0) = {6} with highest weight = 21 and dimension 6,e) (0, 2) = {6}with highest weight = 22 and dimension 6,f) (1, 1) = {8} with highest weight = 1+ 2 and dimension 8,g) (3, 0) = {10}with highest weight = 31 and dimension 10,

    and so on. The weights appearing in the representations can be determined by the factthat they are of the form

    =

    m11

    m22, m1, m2

    N0,

    and that the Weyl group transforms weights into weights (preserving the multiplicity).We sketch some of the weight diagrams and later discuss an elementary method to getthese diagrams, which generalizes our discussion in 6.4 and 6.5.

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    108 6. The Infinitesimal Method

    (1, 0) = {3}

    (H1)

    (H2)

    = (1/6)(

    3, 1) = (2/3)1+ (1/3)2

    s1 = (1/6)(

    3, 1) = (1/31+ (1/3)2

    s2s1 = (1/6)(0, 2) = (1/3)1 (2/3)2

    (0, 1) = {3}

    (H1)

    (H2)

    = (1/6)(0, 2) = (1/3)1+ (2/3)2

    s2 = (1/6)(

    3, 1) = (1/31 (1/3)2

    s1s2 = (1/6)(

    3,

    1) =

    (2/3)1

    (1/3)2

    (2, 0) = {6}

    (H1)

    (H2)

    = (1/6)(2

    3, 2) = (4/3)1+ (2/3)2

    1 = (1/6)(0, 2) = (1/3)1+ (2/3)2

    s1 = (2/3)1+ (2/3)2

    s1s2( 1) = (2/3)1 (1/3)2

    s2( 1) = (1/6)(

    3, 1) = (1/3)1 (1/3)2

    s2s1 = (1/6)(0, 4) = (2/3)1 (4/3)2

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    6.7 The Example su(3) 109

    (1, 1) = {8}

    (H1)

    (H2)

    = (1/6)(

    3, 3) =1+ 2s1 =2

    s1s2 = 1 0 = (0, 0) s2 =1

    s1s2s1 = 1 2 s2s1 = (1/6)(

    3, 3) = 2

    (3, 0) = {10}

    (H1)

    (H2)

    = (1/6)(3

    3, 3) = 21+ 21+ 22

    1+ 2

    10 = (0, 0)1

    1 2 2

    1

    22

    In{8} we find a first example of a weight, which has not multiplicity one, namely theweight (0, 0).

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    110 6. The Infinitesimal Method

    Similar to our discussion for G= SU(2) and g= su(2) in 4.2, one has here the problemof the explicit decomposition of representations into irreducible components. This goesunder the heading ofClebsch Gordon seriesand can be found for instance in [Co] p.611ff.As examples we cite

    {3} {3} {8} {1} and{3} {3} {3} {10} 2{8} {1}.

    The proofs of these formulae need some skill or at least patience. But it is very tempt-ing to imagine the bigger weight diagrams in our examples as composition of the tri-angles{3} and{3}. This leads to an important interpretation in the theory of ele-mentary particles: We already remarked that the discussion of the irreducible repre-sentations of SU(2) resp. SO(3) is useful in the description of particles subjected toSO(3)-symmetry (in particular an electron in an hydrogen atom), namely a representa-

    tion j , j = 0, 1/2, 1, 3/2, . . . describes a multiplet of states of the particle with angularmomentum resp. spin 2j+ 1 and the members of the multiplet are distinguished by amagnetic quantum number m Z with m= (2j+ 1), (2j 1), . . . , (2j+ 1). As onetried to describe and classify the heavier elementary particles, one observed patterns,which could be related to the representation theory of certain compact groups, in partic-ular G= SU(3): In Gell-Mannseightfold waywe take SU(3) as an internal symmetrygroup to get the ingredients of an atom, the proton p and the neutron n as members ofa multiplet of eight states, which correspond to our representation{8}. To be