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Appendix B Linear Representations of a Lie Group
Appendix B
Linear Representations of a Lie Group
Consider a set <§ = {T(a)} of linear operators' T(a), labeled by m real essential parameters, the components of the m-tuple a = (a l , ... , am)' <§ is a linear representation of an m-parameter Lie group ie: 1. T(a) is an analytic function of a about a identity operator; thus, T(a)
=
0 with T(O)
=
1, the
= 1 + a . G plus terms quadratic and of higher order in a (B.1)
where the generators appear as (B.2) The m linear operators (B.2) are linearly independent. 2. Each T(a) has a unique inverse T(a)-l == T(ii) T(ii)T(a)
= T(a)T(ii) = 1.
E <§ such
that (B.3)
1 See, for example, N. 1. Akhiezer and 1. M. Glazman, "Theory of Linear Operators in Hilbert Space," pp. 30-39. Ungar, New York, 1961. In physical applications, T(et) is either a square-matrix that acts on a finite-dimensional vector space or a differential operator that acts on an infinite-dimensional function space. • 2 For the sake of practical transparency, our conditions on the group elements T(et) are somewhat more specific than necessary.
101
102
Appendix B
3. For any pair of elements T(a) and T(f3) product T(a)T(f3) = T(y)
In
'fl, their ordered
(B.4)
is contained in 'fl with the m-tuple of parameters y = Yea, fi) = a
+ 13
plus
(8.5)
(terms bilinear, quadratic, and of higher order in a and 13) as a consequence of (B.I). Because of (B.4), we have
so by setting rx = f3 = (B.2), we obtain
°and making use of
2 a T (Y)) G·G= ( - . J DYj OYj. y=O L
+
(8.5) and the definition
Im ( -a2Yk -) Gko k= 1 oaj of3j a=p=O
(8.7)
From (B. 7) it follows that m
c,c, - c,c, = I
k=1
CijkGko
(B.8)
where the structure constants appear as Cjjk ==
(a:i2~pj - O:;~kp)FP=O == -Cjjko
(8.9)
It is readily seen that the structure constants satisfy the quadratic Lie
identities,
m
I
"=1
(Cjjl,Chkl
+ CjkhChil + CkihChj/) == 0,
(8.10)
by adding the i,j, k cyclic permutations of the commutator of (B.8) with G, and evoking the linear independence of the C'5. Lie's main theorem is that an array of constants Cijk == - Cjik' [i, i. k = 1, . ° • , m], satisfying (B.lO) permits the Eqs. (B.8) to be solved for m linear operators G i related by (B.2) to a T(a) satisfying Eqs. (B. 1), (B.3), and (B.4); furthermore, by evoking a trivial relabeling of the elements of q; with a parameter transformation of the form a - t a + (terms quadratic and of higher order in a), elements of q; can be expressed canonically as
Linear Representations of a Lie Group T(a)
= (exp
a . C)
==
103
L (N !)-l(a . ct. 00
N=O
(B.ll)
The set of all linear combinations of the generators, Lcd = {a' G} with elements parametrized by a, is closed with respect to ordinary operator addition and operator commutation, and hence constitutes a Lie algebra with the Lie product prescribed as the commutator of a pair of elements. A solution to Eqs. (B.8) for the generators, such that no G, is the zero operator, yields a representation of the Lie algebra." If the structure constants are such that C ij k does not vanish for all values of j and k with i any fixed value, we have the so-called adjoint representation with the generators m-dimensional matrices composed of the elements (GJjk = - Cijk, as readily seen by writing (B.8) with matrix indices and recalling (B.10). The associated adjoint representation gives the generators as the differential operators G, = - L},k=l CjjkX j %xk on the space of infinitely differentiable functions of x = (x, ... , x m ) . Lie's main theorem asserts that a representation of the Lie algebra provides a linear representation of the Lie group '§ for an appropriate set of a in R m , with elements in '§ given canonically by (B. 11). If the appropriate set of a in R m is a compact point set, the Lie group is said to be compact; otherwise, the Lie group is noncompact. To obtain all elements for certain Lie groups, it is necessary to augment the set {(exp a' G)} in an obvious way (see Example 4 below). Linear representations of Lie groups are illustrated by the following examples: 1. Cijk = 0, the m-dimensional Abelian Lie group. According to (B.8), all m generators commute with one another. Of particular interest is the differential operator realization C i = 0/ ox i v for which the set of differential operator group elements (B.11), T(a) = expect . %x), 3
(B.12)
All Lie algebra representations are isomorphic to the Lie algebra of real
m-tuples with the addition
and the Lie product
a
+ (3 == (rXl + (31 ..... rXm + (3m) [a.
(31. =
m
L
sJ> 1
C'jk rX , (3 j
The integrability property (1.30) is satisfied for this Lie product because of (B.IO). Notice that for m = 3 and e'jk = e'jk. this Lie product is the 3-tuple cross-product of Cartesian vector analysis.
Appendix B
104
translates the m-tuple argument of a C'" functionf(x) by a, T(a)f(x) = f(x + a). In order for all elements in the group to be included, each of the m parameters ai must assume all finite real values, and hence the m-dimensional Abelian Lie group is noncompact. The special case m = 1 is closely associated with solutions of first-order (ordinary or partial) linear differential equations. 2. m = 2 with Cl2 1 = and CI22 = I, the proper affine Lie group on real numbers. It is easy to verify that the quadratic Lie identities (B.IO) are satisfied. The generator equations (B.8) produce the single relation GI G2 - G2 GI = G 2 , which admits the two-dimensional matrix representation
°
(R13)
By putting this two-dimensional representation into (B.11), we obtain T(a) = exp
(~
(R14)
The group parameters a l and a2 assume all finite real values, so this Lie group is noncompact. Note that the representation (B.14) acts on
(7) to produce proper affine transfor-
the space of vectors of the form
mations of the real number x--+ [ea1x + az(e at - l)/a l ]. 3. m = 3 with C i j k = 8 i j k, the Levi-Civita symbol, the Lie groups SU(2) and SO(3). That such an array of structure constants satisfies the quadratic Lie identities (RIO) is verified immediately by recalling the relation I,l=l 8ijh8hkl = ~ik ~jl - ~il ~jk' With C i j k = 8 i j k, generator equations (R8) become GI G2
-
G2 GI = G3
G 2G 3
-
G 3G 2
= GI
G 3 GI
-
GI G 3
= G2 •
(B.l5)
We have the two-dimensional matrix representation G2
1(01 -1)o '
=2
G3
1
=2
(-i
0
(B.16)
Linear Representations of a Lie Group
105
as well as the three-dimensional adjoint representation GI
=
(~ ~ - ~), o
1
(~ ~
o. =
0
-1
G,~ (! -~ ~)
0
(B.17)
By putting the two-dimensional representation (B.l6) into (B.1l), we obtain I ( T(a) = exp -2
- ia
-
.
la l
+3 az
0) _i(sin tlal) ( I
lal
a3
al + iaz
(B.I8)
lal == (a/ + a/ + a/)l/Z, a parametric representation of all 2 x 2 unitary matrices of determinant one, the Lie group SU(2).4 All elements in the group are included if a is restricted to a sphere about the origin of radius 2n, lal ::::; 2n, with all points on the spherical surface lal = 2n being identified with the group element
0). (-1 o -1 '
thus, SU(2) is compact and simply connected. By putting the threedimensional adjoint representation (B.l7) into (B.ll), we obtain
-a 3 a z) (RI9) 0 -aI' ( -a a 0 z l a parametric representation of all 3 x 3 real orthogonal matrices of determinant one, the Lie group SO(3).4 All elements in the group are T(a)=exp
0 a3
4 The nomenclature symbol S stands for .. special" and means .. determinant equal to one." The determinant of an exponentiated traceless matrix [such as (B.l8), (B.l9), or (B.26)] equals one because we have
det(exp M) = limN_oo deteCt + N- 1M )N) = limN_oo (det(t + N- ' M » N = limN_oo (1 + N- 1 (tr M) + O(N - 2»N =exp(trM)
Appendix B
106
included if o: is restricted to a sphere about the origin of radius n:, I'l.l == (:x/ + :x/ + 'l./)1/2.( tt, with each pair of opposite points on the spherical surface l:xl = tt being identified with a single group element (which squares to the identity because T(:x)T( -'l.) = 1); hence, SO(3) is compact and doubly connected. Since linear representations of both SU(2) and SO(3) follow from the structure constants Cijk = 8 i jk' the Lie algebras for SU(2) and SO(3) are isomorphic, which implies that the Lie groups SU(2) and SO(3) are isomorphic in the neighborhood of the identity. On the other hand, the appropriate parameter set domains and topological character (manifest by the identification of points with group elements on the critical spherical surfaces) are different for the complete sets of group elements, and so the isomorphism between the Lie groups SU(2) and SO(3) is local but not global. We also note that the differential operator solution to (B.15),
(B.20)
the associated adjoint representation, yields the set of differential operator group elements T(:x)
=
exp (-
.I_
I,
J, k - 1
8ijk'l.iXP3/0Xk»)
(B.21)
eYe
that rotates the 3-tuple argument of a I-tuple function f(x), T(ex)f(x) = f(x'), where x' is related to x by the inverse of the associated SO(3) matrix of the form (B.19),
(B.22) 4. m = 3 with Cijk = 8 ij k T k ,Tk == (_I)k-t, the Lie group SL(2, R). To show that this array of structure constants satisfies the quadratic Lie identities (B.I 0), we first establish the formula 3
I
h=l
CijhChkl
= (-l)k(b ik b j l
-
b il b j k),
(B.23)
Linear Representations of a Lie Group
107
from which (RIO) follows by cyclic permutation of i, j, k and addition. The generator equations (B.8), G1 G z -
c, G1 = G3
Gz G 3
-
G3 Gz
=
G1
G 3G1
-
G1 G 3
=
-G z ,
(B.24)
admit the two-dimensional matrix representation
-I)o '
G3
="21(10
(8.25)
By putting the latter set of generators into (Bi l l), we obtain
T(a)=ex p -1( (;(3 :x1-(2) 2 :Xl + :Xz -a3
=(COSh~(;(/_(;(/+(;(/)I/Z)(~ ~) Z + (Sinh Ha/ - a/ + a 3 )1/2) ( a3 (:x1 2 - a/ + a/)1/2 , :Xl + a2 for = (cos
~ (-:x/ + a/ - a/)1/2)(~
~)
+ (Sin -!( -a/ + a/ - a/)l/Z) (;(3 (-a 1 z+IX/-a3 2)I/Z
(a/ - a/ + :x/) ~ 0,
a 1+a Z for
a1 -
:xz) -IX 3 (IX 1 Z - az 2 + IX3Z) ~ 0,
(8.26) a parametric representation of 2 x 2 real matrices of determinant one, a part of the Lie group SL(2, R).5 All elements of SL(2, R) are included in the augmented set {texp « . G), -(exp a . G)} with the components of IX assuming all finite real values for which (:x/ - IX/ + IX/)~ _7[z with opposite points on the hyperboloid surfaces IX 2 =
±(a/
+
a/ +
7[Z)I/Z
5 The trace of matrices of the form (B.26) is greater or equal to - 2; to get the matrices with trace less than - 2, we must also consider the derived set {-(exp ex' G)}.
Appendix B
108
identified with a traceless matrix that occurs in both subsets {(exp (X • G)} and {- (exp (X' G)}; thus, SL(2, R) is noncompact and has the connectivity of a periodic slab. By making the formal parameter replacement ((Xl' (X2' (X3)--+ (-i(Xl' (X2' -i(X3), one obtains (B.18) from (B.26), and so SU(2) is the (unique) compact complex-extension of SL(2, R). An immediate general classification of m-parameter Lie groups follows from Cartan's metric, defined in terms of the structure constants as the symmetric m x m array gij
==
m
k,
It> 1Cik1Cj 1k•
(B.27)
We have (gij) = 0 for the m-dimensional Abelian Lie group (Example 1 above), (gij) = diagf l , 0...1 for the proper affine group on real numbers (Example 2 above), (gij) = diag'" -2, -2, -2...1 for the Lie groups SU(2) and SO(3) (Example 3 above), and (gij) = diag r2, -2, 2...1for the Lie group SL(2, R) (Example 4 above). Cartan's metric (B.27) is negative-definite for compact Lie groups and indefinite for noncompact Lie groups. By making use of (B.1O), we find that the quantity aijk
==
m
I
h=l
Cijhghk
=
m
I
r,s,t==l
(CrisCSjtCtkr -
CrjsCsitCtkr)
(B.28) is, in general, a totally antisymmetric array of real constants, changing its sign under transposition of any two indices. An m-parameter Lie group is semisimple if the m x m array (B.27) constitutes a nonsingular matrix, det(g i) =I- O.
(B.29)
Thus, the m-dimensional Abelian Lie group and the proper affine Lie group on real numbers are not semisimple, while the Lie groups SU(2), SO(3), and SL(2, R) are semisimple. The symmetric matrix inverse of Caftan's metric (B.27), customarily denoted gij, exists for a semisimple Lie group and satisfies m '" f...,g ikgkj=b j.i
(B.30)
k~l
In terms of s" and the generators for a linear representation of a semisimple Lie group, the quadratic Casimir invariant is defined as C ==
m
L
gijGiG j •
i,j= 1
(B.31)
Linear Representations of a Lie Group
109
Because it commutes with all the generators, CGk-GkC=
m
I
i, j, Il= 1
gii(CjkIlGiGh+CikhGhG)
m
I
(gii Cikll i,j, h> 1 m
I
(gii o"
t, j, h, 1= 1
+ gillCik)Gj Gh + gihgjl)aikl Gj Gh (8.32)
as a consequence of the antisymmetry of (B.28), the quadratic Casimir invariant plays a central role in the representation theory for semisimple Lie groups." 6 For general discussions of Lie group representation theory, see L. S. Pontrjagin, "Topological Groups." Princeton Univ. Press, Princeton, New Jersey, 1946; R. E. Behrends, et al., Rev. Modern Phys. 34, I (1962) and works cited therein.
Linear Representations of a Lie Group
Consider a set <§ = {T(a)} of linear operators' T(a), labeled by m real essential parameters, the components of the m-tuple a = (a l , ... , am)' <§ is a linear representation of an m-parameter Lie group ie: 1. T(a) is an analytic function of a about a identity operator; thus, T(a)
=
0 with T(O)
=
1, the
= 1 + a . G plus terms quadratic and of higher order in a (B.1)
where the generators appear as (B.2) The m linear operators (B.2) are linearly independent. 2. Each T(a) has a unique inverse T(a)-l == T(ii) T(ii)T(a)
= T(a)T(ii) = 1.
E <§ such
that (B.3)
1 See, for example, N. 1. Akhiezer and 1. M. Glazman, "Theory of Linear Operators in Hilbert Space," pp. 30-39. Ungar, New York, 1961. In physical applications, T(et) is either a square-matrix that acts on a finite-dimensional vector space or a differential operator that acts on an infinite-dimensional function space. • 2 For the sake of practical transparency, our conditions on the group elements T(et) are somewhat more specific than necessary.
101
102
Appendix B
3. For any pair of elements T(a) and T(f3) product T(a)T(f3) = T(y)
In
'fl, their ordered
(B.4)
is contained in 'fl with the m-tuple of parameters y = Yea, fi) = a
+ 13
plus
(8.5)
(terms bilinear, quadratic, and of higher order in a and 13) as a consequence of (B.I). Because of (B.4), we have
so by setting rx = f3 = (B.2), we obtain
°and making use of
2 a T (Y)) G·G= ( - . J DYj OYj. y=O L
+
(8.5) and the definition
Im ( -a2Yk -) Gko k= 1 oaj of3j a=p=O
(8.7)
From (B. 7) it follows that m
c,c, - c,c, = I
k=1
CijkGko
(B.8)
where the structure constants appear as Cjjk ==
(a:i2~pj - O:;~kp)FP=O == -Cjjko
(8.9)
It is readily seen that the structure constants satisfy the quadratic Lie
identities,
m
I
"=1
(Cjjl,Chkl
+ CjkhChil + CkihChj/) == 0,
(8.10)
by adding the i,j, k cyclic permutations of the commutator of (B.8) with G, and evoking the linear independence of the C'5. Lie's main theorem is that an array of constants Cijk == - Cjik' [i, i. k = 1, . ° • , m], satisfying (B.lO) permits the Eqs. (B.8) to be solved for m linear operators G i related by (B.2) to a T(a) satisfying Eqs. (B. 1), (B.3), and (B.4); furthermore, by evoking a trivial relabeling of the elements of q; with a parameter transformation of the form a - t a + (terms quadratic and of higher order in a), elements of q; can be expressed canonically as
Linear Representations of a Lie Group T(a)
= (exp
a . C)
==
103
L (N !)-l(a . ct. 00
N=O
(B.ll)
The set of all linear combinations of the generators, Lcd = {a' G} with elements parametrized by a, is closed with respect to ordinary operator addition and operator commutation, and hence constitutes a Lie algebra with the Lie product prescribed as the commutator of a pair of elements. A solution to Eqs. (B.8) for the generators, such that no G, is the zero operator, yields a representation of the Lie algebra." If the structure constants are such that C ij k does not vanish for all values of j and k with i any fixed value, we have the so-called adjoint representation with the generators m-dimensional matrices composed of the elements (GJjk = - Cijk, as readily seen by writing (B.8) with matrix indices and recalling (B.10). The associated adjoint representation gives the generators as the differential operators G, = - L},k=l CjjkX j %xk on the space of infinitely differentiable functions of x = (x, ... , x m ) . Lie's main theorem asserts that a representation of the Lie algebra provides a linear representation of the Lie group '§ for an appropriate set of a in R m , with elements in '§ given canonically by (B. 11). If the appropriate set of a in R m is a compact point set, the Lie group is said to be compact; otherwise, the Lie group is noncompact. To obtain all elements for certain Lie groups, it is necessary to augment the set {(exp a' G)} in an obvious way (see Example 4 below). Linear representations of Lie groups are illustrated by the following examples: 1. Cijk = 0, the m-dimensional Abelian Lie group. According to (B.8), all m generators commute with one another. Of particular interest is the differential operator realization C i = 0/ ox i v for which the set of differential operator group elements (B.11), T(a) = expect . %x), 3
(B.12)
All Lie algebra representations are isomorphic to the Lie algebra of real
m-tuples with the addition
and the Lie product
a
+ (3 == (rXl + (31 ..... rXm + (3m) [a.
(31. =
m
L
sJ> 1
C'jk rX , (3 j
The integrability property (1.30) is satisfied for this Lie product because of (B.IO). Notice that for m = 3 and e'jk = e'jk. this Lie product is the 3-tuple cross-product of Cartesian vector analysis.
Appendix B
104
translates the m-tuple argument of a C'" functionf(x) by a, T(a)f(x) = f(x + a). In order for all elements in the group to be included, each of the m parameters ai must assume all finite real values, and hence the m-dimensional Abelian Lie group is noncompact. The special case m = 1 is closely associated with solutions of first-order (ordinary or partial) linear differential equations. 2. m = 2 with Cl2 1 = and CI22 = I, the proper affine Lie group on real numbers. It is easy to verify that the quadratic Lie identities (B.IO) are satisfied. The generator equations (B.8) produce the single relation GI G2 - G2 GI = G 2 , which admits the two-dimensional matrix representation
°
(R13)
By putting this two-dimensional representation into (B.11), we obtain T(a) = exp
(~
(R14)
The group parameters a l and a2 assume all finite real values, so this Lie group is noncompact. Note that the representation (B.14) acts on
(7) to produce proper affine transfor-
the space of vectors of the form
mations of the real number x--+ [ea1x + az(e at - l)/a l ]. 3. m = 3 with C i j k = 8 i j k, the Levi-Civita symbol, the Lie groups SU(2) and SO(3). That such an array of structure constants satisfies the quadratic Lie identities (RIO) is verified immediately by recalling the relation I,l=l 8ijh8hkl = ~ik ~jl - ~il ~jk' With C i j k = 8 i j k, generator equations (R8) become GI G2
-
G2 GI = G3
G 2G 3
-
G 3G 2
= GI
G 3 GI
-
GI G 3
= G2 •
(B.l5)
We have the two-dimensional matrix representation G2
1(01 -1)o '
=2
G3
1
=2
(-i
0
(B.16)
Linear Representations of a Lie Group
105
as well as the three-dimensional adjoint representation GI
=
(~ ~ - ~), o
1
(~ ~
o. =
0
-1
G,~ (! -~ ~)
0
(B.17)
By putting the two-dimensional representation (B.l6) into (B.1l), we obtain I ( T(a) = exp -2
- ia
-
.
la l
+3 az
0) _i(sin tlal) ( I
lal
a3
al + iaz
(B.I8)
lal == (a/ + a/ + a/)l/Z, a parametric representation of all 2 x 2 unitary matrices of determinant one, the Lie group SU(2).4 All elements in the group are included if a is restricted to a sphere about the origin of radius 2n, lal ::::; 2n, with all points on the spherical surface lal = 2n being identified with the group element
0). (-1 o -1 '
thus, SU(2) is compact and simply connected. By putting the threedimensional adjoint representation (B.l7) into (B.ll), we obtain
-a 3 a z) (RI9) 0 -aI' ( -a a 0 z l a parametric representation of all 3 x 3 real orthogonal matrices of determinant one, the Lie group SO(3).4 All elements in the group are T(a)=exp
0 a3
4 The nomenclature symbol S stands for .. special" and means .. determinant equal to one." The determinant of an exponentiated traceless matrix [such as (B.l8), (B.l9), or (B.26)] equals one because we have
det(exp M) = limN_oo deteCt + N- 1M )N) = limN_oo (det(t + N- ' M » N = limN_oo (1 + N- 1 (tr M) + O(N - 2»N =exp(trM)
Appendix B
106
included if o: is restricted to a sphere about the origin of radius n:, I'l.l == (:x/ + :x/ + 'l./)1/2.( tt, with each pair of opposite points on the spherical surface l:xl = tt being identified with a single group element (which squares to the identity because T(:x)T( -'l.) = 1); hence, SO(3) is compact and doubly connected. Since linear representations of both SU(2) and SO(3) follow from the structure constants Cijk = 8 i jk' the Lie algebras for SU(2) and SO(3) are isomorphic, which implies that the Lie groups SU(2) and SO(3) are isomorphic in the neighborhood of the identity. On the other hand, the appropriate parameter set domains and topological character (manifest by the identification of points with group elements on the critical spherical surfaces) are different for the complete sets of group elements, and so the isomorphism between the Lie groups SU(2) and SO(3) is local but not global. We also note that the differential operator solution to (B.15),
(B.20)
the associated adjoint representation, yields the set of differential operator group elements T(:x)
=
exp (-
.I_
I,
J, k - 1
8ijk'l.iXP3/0Xk»)
(B.21)
eYe
that rotates the 3-tuple argument of a I-tuple function f(x), T(ex)f(x) = f(x'), where x' is related to x by the inverse of the associated SO(3) matrix of the form (B.19),
(B.22) 4. m = 3 with Cijk = 8 ij k T k ,Tk == (_I)k-t, the Lie group SL(2, R). To show that this array of structure constants satisfies the quadratic Lie identities (B.I 0), we first establish the formula 3
I
h=l
CijhChkl
= (-l)k(b ik b j l
-
b il b j k),
(B.23)
Linear Representations of a Lie Group
107
from which (RIO) follows by cyclic permutation of i, j, k and addition. The generator equations (B.8), G1 G z -
c, G1 = G3
Gz G 3
-
G3 Gz
=
G1
G 3G1
-
G1 G 3
=
-G z ,
(B.24)
admit the two-dimensional matrix representation
-I)o '
G3
="21(10
(8.25)
By putting the latter set of generators into (Bi l l), we obtain
T(a)=ex p -1( (;(3 :x1-(2) 2 :Xl + :Xz -a3
=(COSh~(;(/_(;(/+(;(/)I/Z)(~ ~) Z + (Sinh Ha/ - a/ + a 3 )1/2) ( a3 (:x1 2 - a/ + a/)1/2 , :Xl + a2 for = (cos
~ (-:x/ + a/ - a/)1/2)(~
~)
+ (Sin -!( -a/ + a/ - a/)l/Z) (;(3 (-a 1 z+IX/-a3 2)I/Z
(a/ - a/ + :x/) ~ 0,
a 1+a Z for
a1 -
:xz) -IX 3 (IX 1 Z - az 2 + IX3Z) ~ 0,
(8.26) a parametric representation of 2 x 2 real matrices of determinant one, a part of the Lie group SL(2, R).5 All elements of SL(2, R) are included in the augmented set {texp « . G), -(exp a . G)} with the components of IX assuming all finite real values for which (:x/ - IX/ + IX/)~ _7[z with opposite points on the hyperboloid surfaces IX 2 =
±(a/
+
a/ +
7[Z)I/Z
5 The trace of matrices of the form (B.26) is greater or equal to - 2; to get the matrices with trace less than - 2, we must also consider the derived set {-(exp ex' G)}.
Appendix B
108
identified with a traceless matrix that occurs in both subsets {(exp (X • G)} and {- (exp (X' G)}; thus, SL(2, R) is noncompact and has the connectivity of a periodic slab. By making the formal parameter replacement ((Xl' (X2' (X3)--+ (-i(Xl' (X2' -i(X3), one obtains (B.18) from (B.26), and so SU(2) is the (unique) compact complex-extension of SL(2, R). An immediate general classification of m-parameter Lie groups follows from Cartan's metric, defined in terms of the structure constants as the symmetric m x m array gij
==
m
k,
It> 1Cik1Cj 1k•
(B.27)
We have (gij) = 0 for the m-dimensional Abelian Lie group (Example 1 above), (gij) = diagf l , 0...1 for the proper affine group on real numbers (Example 2 above), (gij) = diag'" -2, -2, -2...1 for the Lie groups SU(2) and SO(3) (Example 3 above), and (gij) = diag r2, -2, 2...1for the Lie group SL(2, R) (Example 4 above). Cartan's metric (B.27) is negative-definite for compact Lie groups and indefinite for noncompact Lie groups. By making use of (B.1O), we find that the quantity aijk
==
m
I
h=l
Cijhghk
=
m
I
r,s,t==l
(CrisCSjtCtkr -
CrjsCsitCtkr)
(B.28) is, in general, a totally antisymmetric array of real constants, changing its sign under transposition of any two indices. An m-parameter Lie group is semisimple if the m x m array (B.27) constitutes a nonsingular matrix, det(g i) =I- O.
(B.29)
Thus, the m-dimensional Abelian Lie group and the proper affine Lie group on real numbers are not semisimple, while the Lie groups SU(2), SO(3), and SL(2, R) are semisimple. The symmetric matrix inverse of Caftan's metric (B.27), customarily denoted gij, exists for a semisimple Lie group and satisfies m '" f...,g ikgkj=b j.i
(B.30)
k~l
In terms of s" and the generators for a linear representation of a semisimple Lie group, the quadratic Casimir invariant is defined as C ==
m
L
gijGiG j •
i,j= 1
(B.31)
Linear Representations of a Lie Group
109
Because it commutes with all the generators, CGk-GkC=
m
I
i, j, Il= 1
gii(CjkIlGiGh+CikhGhG)
m
I
(gii Cikll i,j, h> 1 m
I
(gii o"
t, j, h, 1= 1
+ gillCik)Gj Gh + gihgjl)aikl Gj Gh (8.32)
as a consequence of the antisymmetry of (B.28), the quadratic Casimir invariant plays a central role in the representation theory for semisimple Lie groups." 6 For general discussions of Lie group representation theory, see L. S. Pontrjagin, "Topological Groups." Princeton Univ. Press, Princeton, New Jersey, 1946; R. E. Behrends, et al., Rev. Modern Phys. 34, I (1962) and works cited therein.