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Non-standard parametrizations and adjoint invariants of classical groups1
Physics Letters B 465 Ž1999. 169–176
Non-standard parametrizations and adjoint invariants of classical groups 1 Adrian ´ R. Lugo
2
Departmento de Fısica, Facultad de Ciencias Exactas, UniÕersidad Nacional de La Plata, C.C. 67, (1900) La Plata, Argentina ´ Received 5 May 1999; accepted 30 July 1999 Editor: L. Alvarez-Gaume´
Abstract We obtain local parametrizations of classical non-compact Lie groups where adjoint invariants under maximal compact subgroups are manifest. Extension to non-compact subgroups is straightforward. As a by-product parametrizations of the same type are obtained for compact groups. They are of physical interest in any theory gauge invariant under the adjoint action, typical examples being the two dimensional gauged Wess-Zumino-Witten-Novikov models where these coordinatizations become of extreme usefulness to get the background fields representing the vacuum expectation values of the massless modes of the associated Žsuper. string theory. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction It is none to say what group theory meant Žand means. for theoretical physics during this century, in particular the theory of continuous groups or Lie groups Ž see e.g. w1,2x and references therein.. The properties of their Lie algebras, easier to hand than the groups themselves, define them locally and in fact most part of the textbooks are dedicated to them w3x. However depending on the problem at hand, explicit parametrizations of the group manifold become necessary. Many of them are widely known, example of them the SUŽ2. or more generally the Euler angles of orthogonal groups. Of course that we can always write locally a group element as a prod-
1 2
This work was partially supported by CONICET, Argentina. E-mail: [email protected]
uct of one-generator exponentials, or simply as the exponential of an arbitrary Lie algebra element. But in most cases theses obvious parametrizations are of few usefulness because they obscure the global properties of the group and generally lead to untractable computations. An interesting parametrization is suggested by the Mackey theorem, the well-known coset decomposition: let G be a group and H a subgroup of it. Then for any g g G, g s k h,
k g GrH ,
hgH
Ž 1.1 .
in a unique way. This leads to the theory of homogeneous spaces Žor Žleft or right. coset spaces. GrH, good references on the subject being w4,5x. Among others physical applications Ž1.1. is fundamental in the treatment of effective field theories with spontaneously broken symmetries w6x.
0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 9 3 6 - 3
But let us assume instead that we have a field theory including maps from ‘‘space-time’’ on a group manifold G among its degrees of freedom, and gauge invariant under the adjoint action of a subgroup H of G g ™ h g s h g hy1 ,
hgH
Ž 1.2 .
This means that effectively the theory depends on the invariants of the group under the adjoint action of the subgroup. That is, if we were able to write uniquely any element of G as g ' hy1 g h ,
hgH
where u p is a p-dimensional real vector, Pp g SO Ž p . and generically we will mean H Ž P ,Q . s
2. The orthogonal groups We consider in this section the pseudo-orthogonal groups, G ' SO Ž p,q .. Its maximal compact subgroup is H ' SO Ž p . = SO Ž q .. In order to get its decomposition Ž1.3. we need as a first step to get the
P 0
/
Ž 2.2 .
K pq 1 Ž u p . s
1 y Ž 1 q u pq 1 . yu p
1 s u p t u p q Ž u pq1 .
y1
up up t
t
up u pq1
2
0
,
Ž 2.3 .
Under an adjoint transformation h
Ppq 1 s H Ž h,1 . Ppq1 H Ž h t ,1 . ,
h g SO Ž p .
Ž 2.4 . the parameters of Ppq 1 get transformed as: h
up sh up ,
h
Pp s h Pp h t
Ž 2.5 .
The procedure will be constructive. Les us pick an arbitrary matrix Vp g SO Ž p . decomposed this time as a left coset w.r.t. SO Ž p y 1. Vp s H Ž Vpy1 ,1 . K p Ž zpy1 .
Ž 2.6 .
and rewrite Ž2.1. for any such a Vp with the help of Ž2.5. as Ppq 1 s H Ž Vp t ,1 . Ppq1 H Ž Vp ,1 . , Ppq 1 s K pq1 Ž Vp u p . H Ž Vp PpVp t ,1 .
Ž 2.7 .
The general idea to apply here and in the subsequent cases is to fix the whole matrix Vp Ž' zpy1 ,Vpy1 . in terms of variables of Ppq 1 Ž' u p , Pp ., leaving aside only precisely the invariants together with the matrix Vp as parameters of Ppq1. Evidently this procedure is equivalent to make a change of variables from the non-invariant parameters in Ppq 1 to Vp . Eq. Ž2.7. suggests to put u p in some standard form by a specific choice of Ža part of. Vp . In fact it is easy to show that the choice 3 up zpy1 sy yÕp < up <
ž /
2.1. Reduction of SO(p q 1) under SO(p)
0 Q
In what follows the dimensionalities of matrices should be understood from the context when not stated explicitly. The right coset element in SO Ž p q 1.rSO Ž p . ; S p is given by
Ž 1.3 .
then clearly making a gauge transformation Ž1.2. identifying the h’s the theory will depend only on g that encloses the invariants mentioned above. We would like to remark that at difference of Ž1.1. there no exists any general theorem assuring the decomposition Ž1.3.; in fact it is not difficult to find examples where it is not possible to write it. It is the aim of this paper to get the class of local parametrizations of the type Ž1.3. in a whole set of cases of physical interest. Specific examples where they must be used Žand were used in the lower dimensionality cases where parametrizations were available. are the two dimensional gauged WessZumino-Witten-Novikov models w7,8x. It is worth to say however that the results, being explicit parametrizations of classical groups, are valuable on their own right independently of the applications.
ž
Ž 2.8 .
We start by writing w5x Ppq 1 Ž u p , Pp . s K p Ž u p . H Ž Pp ,1 .
Ž 2.1 .
3 As usual we will use the notation Ž eˇi . j s d i j ,Ž Ei j . k l s d i k d jl , A i j ' Ei j y E ji ,Si j ' Ei j q E ji .
defines the rotation K p Ž zpy1 . u p s < u p < eˇ p
Ž 2.9 .
Note that we have changed the Ž p y 1. parameters from u p indicating its direction in terms of the Ž p y 1. parameters in zpy 1. With such a choice of zpy 1 Ž Vpy1 not fixed yet. we can write
where Pp Ž Q q . g Ž SO Ž p .Ž SO Ž q .. and the coset element is given by K p , q Ž S . s exp
s
Ppq 1 s K pq1 Ž < u p < eˇ p . t
=H H Ž Vpy 1 ,1 . Pp H Ž Vpy1 ,1 . ,1
ž
/
0 Nt
N 0
Ž 1 q SS t .
/
1r2
S
St
S s Ž NN t .
Ž 2.10 .
ž
y1 r2
Ž 1 q S tS .
sinh Ž NN t .
1r2
Under an adjoint transformation
Ppq 1 s K pq1 Ž < u p < eˇ p . H Pp ,1
h
/
Ž 2.11 .
Inspection of this formula indicates an iterative process, the next step being to write the analogous expression for Pp and so on; after p steps we get § p
Ppq 1 s
L p , q s H Ž h p ,h q . L p , q H Ž h p ,h q .
Ž 2.12 .
t
< u l < s sin u l ,
0 Ful Fp
Ž 2.13 .
S s hp S hq t ,
h
Pp s h p Pp h tp ,
h
Q q s h q Q q h tq
Ž 2.18 . By following the strategy pursued in the past subsection we introduce two matrices Vp ,Vq belonging to SO Ž p .,SO Ž q . respectively and rewrite Ž2.15.
and write the SO Ž p q 1. element in the final form
t
L p , q s H Ž Vp ,Vq . L p , q H Ž Vp ,Vq . ,
t
Ppq 1 s H Ž Vp ,1 . Ppq1 H Ž Vp ,1 . , § p
Ppq 1 s
Ł
exp Ž u l A l ,lq1 .
p=q
Ž 2.17 .
ls1
It is convenient to introduce the angular variables
,
with h p Ž h q . g SO Ž p .Ž SO Ž q .., the parameters of L p, q transforms as h
Ł H Ž K lq1 Ž < u l < eˇl . ,1 pyl .
gR
N
0
Ž 2.16 .
where we have redefined Pp ™ K p Ž zpy1 . t Pp K p Ž zpy1 .. But according to Ž2.7. we can rewrite it as
ž
1r2
Ž 2.14 .
ls1
L p , q s exp
0
Vp NVq
Ž Vp NVq t .
t
0
t
0
which displays explicitly the p invariants u l ,l s 1, . . . , p4 under the adjoint action of SO Ž p .. Note that the number of parameters trivially matches: p p Ž . Ž . 2 p y 1 q p s 2 p q 1 , as should; this is the first of our results, to be used in the following.
As in Ž2.6. we consider left coset parametrizations
2.2. Reduction of SO(p,q) under SO(p) = SO(q)
Vq s H Ž Vqy1 ,1 . K q Ž zqy1 .
Our starting point is again the right coset parametrization w5x 4
and try to totally fix zpy 1 and zqy1 to put N in a standard form. It turns out that it is possible to choose these vectors in such a way that
L p , q Ž S, Pp ,Q q . s K p , q Ž S . H Ž Pp ,Q q .
Ž 2.15 .
=H Ž Vp PpVp t ,Vq Q q Vq
t
.
Ž 2.19 .
Vp s H Ž Vpy1 ,1 . K p Ž zpy1 . ,
Ž 2.20 .
Vp N Vq t ' H Ž Vpy1 ,1 . K p Ž zpy1 . N t
=K q Ž zqy1 . H Ž Vqy1 t ,1 . 4
An explicit derivation from the definition of pseudo unitary groups is given in Appendix A of w10x.
s
ž
Nr 0
0 n
/
Ž 2.21 .
where Nr g R Ž py1.=Ž qy1. and n g R . In fact if we write Ns
ž
NrX
n py1
n qy 1 t
nX
/
Ž 2.22 .
then is straightforward to verify that the gauge fixing condition n py 1 s n qy1 s 0, i.e. t
K p Ž zpy1 . NK q Ž zqy1 . s
ž
Nr 0
0 n
/
Ž 2.23 .
holds if we choose zpy 1 ' 1 q < t py1 < 2 zqy 1
ž ' ž1 q < t
<2 qy1
Again we verify the matching in the number of parameters Vp ,Vq , Ni j ,n, u k , u k , p q Ž p y 1. q Ž q y 1. q Ž p y 1. Ž q y 1. q 1 2 2 q Ž p y 1 . q Ž q y 1 . s 12 Ž p q q . Ž p q q y 1 . Ž 2.27 . It is worth to say that the first term in Ž2.26. can be computed as in Ž2.16. with N as in the r.h.s. of Ž2.21.; however this is a formal expression for which, to our knowledge, only the ‘‘minkowskian’’ cases p s 1 or q s 1 admit an explicit form.
y1 r2
/ /
t py1 , 3. The unitary groups
y1 r2
t qy1
with t py1 , t qy1 satisfying
Ž 2.24 .
5
t py 1 s Ž n qy1 t t qy1 y n .
y1
t qy 1 s Ž n py1 t t py1 y n .
y1
Ž n py1 y NrX t qy1 . ,
žn
Xt qy1 y Nr t py1
The treatment of these groups parallels that made in the case of the orthogonal ones, with some additional complications due to the complex character of them. As before we start considering 3.1. Reduction of SU(p q 1) under U(p)
/ Ž 2.25 .
t
A final redefinition Nr ™ Vpy1 Nr Vqy1 leads to Ž2.21 .. Finally Žafter reparametrizing Pp ™ K p Ž zpy 1 . t Pp K p Ž zpy1 . , Q q ™ K q Ž zqy1 . t Q q =K q Ž zqy1 .. we use Ž2.14. to fix Vpy1 and Vqy1; the result can be recast in the form t
L p , q s H Ž Vp ,Vq . L p , q H Ž Vp ,Vq . , py1 qy1
L p , q s exp
žÝ Ý
Ni j Si , pqj q nS p , pqq
is1 js1
/
§ py1
=
Ł
An element of SUŽ p q 1. can be written as Ppq 1 Ž u p ,Up . s K pq1 Ž u p . H Ž Up ,u )p . s K pq 1 Ž u p . H Ž Pp ,1 . exp Ž i f p Tp .
Ž 3.1 . p
where u p ' detUp s expŽ ipf p . , u p g C , Up g UŽ p ., Pp g SUŽ p ., and we have introduced a convenient basis Tk s Ý kls1Ž El l y Ekq1, kq1 ., k s 1, . . . , p4 in the Cartan subalgebra of suŽ p q 1.. The right coset element in Ž3.1. belonging to SUŽ p q 1.rUŽ p . ; CP p is given by w5x K pq 1 Ž u p .
exp Ž u k A k , kq1 .
ks1
s
§ qy1
=
Ł
ks1
exp u k A pqk , pqkq1 ,
ž
/
Ž 2.26 .
1 y Ž 1 q u 2 pq1 .
y1
up up †
yu p †
1 s u p † u p q Ž u 2 pq1 .
up u 2 pq1
2
0
,
Ž 3.2 .
The adjoint action under Hp g UŽ p . is Ž h p ' det Hp . H
Ppq 1 s H Ž Hp ,h )p . Ppq1 H Ž Hp ,h )p .
°u V
5
This set of equations can be reduced to a system of two Žquadratic. equations with two unknowns; the important thing for us is that solutions exist and define the change of variables Ž n py 1 ,n qy1 . ™ Ž zpy1 ,zqy1 . .
~
l
H
s
h p Hp u p
Pp
s
Hp Pp Hp†
p
s
fp
¢f H
p
†
Ž 3.3 .
As before we pick an arbitrary Vp g UŽ p . left coset parametrized ) Vp s H Ž Vpy1 ,Õpy1 . K p Ž zpy1 . exp Ž i b p . ,
Vpy 1 g U Ž p y 1 .
Ž 3.4 .
SUŽ p,q . can be left coset decomposed under SŽUŽ p . = UŽ q .. as L p , q Ž S,Up ,Uq . s K p , q Ž S . H Ž Up ,Uq . , u p s u q ) ,
K p,q Ž S . s
and write Ž3.1. with the help of Ž3.3. as
S s Ž NN † .
Ppq 1 s K pq1 Ž ÕpVp u p . H Ž Vp PpVp † ,1 .
Ž 3.5 .
By choosing zpy 1 and b p Ž Vpy1 free. as
Ž
°
ž /
~
l
1
p
.
0 N†
S
Ž 1 q S †S .
/
sinh Ž NN † .
1r2
we have Õp Vp u p s < u p < eˇ p
Ž 3.7 .
and identifying Vpy 1 as the SUŽ p y 1. matrix corresponding to the Pp decomposition in Ž3.5. Žprevious redefinition Pp ™ K p Ž zpy1 . †Pp K p Ž zpy1 .. we get Ppq 1 s K pq1 Ž < u p < eˇ p . H Pp ,1 exp Ž i f p Tp .
/
Ž 3.8 .
S, N g C p=q Ž 3.10 .
N,
H
s
hp S hq †
Up
s
h p Up h†p
s
h q Uq h q †
q
Ž 3.6 .
<Ž u p . <
0
N , 0
S
¢U
pq1
ž
H
H
p
1r2
The adjoint action under H Ž h p ,h q . g SŽUŽ p . = UŽ q .. is
/
exp Ž i b p . s
y1 r2
†
L p , q s H Ž h p ,h q . L p , q H Ž h p ,h q .
zpy1 Ž u )p . u p sy , p yÕ 2 py1 <Ž u p . < < u p < u )p
ž
1r2
H
p
ž
S
s exp
†
Ppq 1 s H Ž Vp ,Õp) . Ppq1 H Ž Vp ,Õp ) . ,
=exp Ž i f p Tp .
Ž 1 q SS † .
†
Ž 3.11 .
Two matrices Vp ,Vq belonging to UŽ p .,UŽ q . respectively with Õp Õq s 1 are introduced and following similar steps as in Subsection 2.2 we find that it is possible to fix N in the way Nr 0 Ž 3.12 . 0 n where now Nr g C Ž py1.=Ž qy1. and n g R . Then by using the results of the past subsection we get the final result for the parametrization Ns
ž
/
†
By repeating the analysis for Pp and after p steps we arrive to the final result
L p , q s H Ž Vp ,Vq . L p , q H Ž Vp ,Vq . , L p , q s exp
†
Ppq 1 s H Ž Vp ,Õp) . Ppq1 H Ž Vp ,Õp) . ,
Ł
0 N†
N 0
/Ł
exp Ž u l A l ,lq1 .
ls1
§ qy1
§ p
Ppq 1 s
ž
§ py1
exp Ž u l A l ,lq1 . C pq1 Ž F .
Ž 3.9 .
ls1
It differs from Ž2.14. from the unitary character of Vp and the comparison of the arbitrary Cartan elep ment C pq 1ŽF . s expŽ iÝ ls1 f l Tl . at right in Ppq1.
=
Ł
ls1
exp u l A pql , pqlq1 C Ž F .
ž
/
Ž 3.13 .
where H Ž Vp ,Vq . g SŽUŽ p . = UŽ q .. and we denote by C ŽF . an arbitrary element in the Cartan subalgebra of the Lie algebra of SŽUŽ p . = UŽ q ... 3.3. Decomposition under the maximal torus
3.2. Reduction of SU(p,q) under S(U(p) = U(q)) In order not to be repetitive we will skip some steps in what follows. An arbitrary element in
Some times is useful to have the adjoint decomposition of unitary groups under the Cartan subalgebra. We will work out for definiteness the case of SUŽ n q 1.; the non-compact versions differ as usual
by signs in the coset elements and Wick rotations of some compact generators. To this end we begin by searching for the coset decomposition of SUŽ n q 1. under C Ž SUŽ n q 1..; from Ž3.1.
It is easy to show that the versors rˇl get transformed as l a
žÝ /
rˇl s exp Ž yi a˜ lq1 . exp i l s 1, . . . ,n
Pnq 1 Ž u n ,Un . s K nq1 Ž u n . H Ž Un ,u n) . , K nq 1 Ž u n . s expun
s
ž
ž
rˇn
0 yrˇn
1 y Ž 1 q u 2 nq1 .
†
y1
0
/
< u n < q Ž u 2 nq1 . s 1 , rˇn † rˇn s 1 ,
°a ¢ya
un un †
un
a˜ s~ya k
/
,
u n s sin un rˇn ,
un g w 0,p x
Ž 3.14 .
Un s Pn
ž
0 , un
/
Pn g SU Ž n . ,
u n ' detUn s exp Ž i w n .
Ž 3.15 .
we can rewrite Ž3.14. as
ky 1 q a k n
2 a1 y a2
s
yphase Ž rˇ1 .
1
ya 1 q 2 a 2 y a 3 ...
s
yphase Ž rˇ2 .
2
ya ny 1 q 2 a n
s
yphase Ž rˇn .
Pnq 1 Ž u n ,Un . s K nq1 Ž u n . H Ž Pn ,1 . exp Ž i w Hn . Ž 3.16 .
Pnq 1 s Cnq1 Ž a . Pnq1 Cnq1 Ž a . ,
where this time is convenient to introduce the basis Hk s Ek k y Ekq1, kq1 ,k s 1, . . . ,n4 in the Cartan subalgebra of suŽ n q 1.. By repeating with Pn and iterating we get
Pnq 1 s
Pnq 1 s
Ł
ls1
ž
K lq1 Ž u l . 0
...
Ž 3.21 .
n
In other words, from Ž3.18. we get the final result n
§ n
Ž 3.20 .
Then we can put the phases of the Ž a rˇl . l components, l s 1, . . . ,n, to zero by choosing the a X s such that 6
By introducing 1 ny 1 0
if k s 1 if k s 2, . . . ,n if k s n q 1
1
u 2 nq1
2
Ž 3.19 .
where
yu n †
2
a˜ i Ei i rˇn ,
is1
0 1 ny l
/
exp Ž i w P H .
†
§ n
Ł
ls1
ž
K lq1 Ž u l . 0
0 1 ny l
/
Cnq1 Ž w .
Ž 3.22 .
with the constraints implied by Ž3.21.: Ž rˇl . l g R , l s 1, . . . ,n.
4. The decomposition of Sl( n ) under SU( n )
Ž 3.17 . Now let us pick an element Cnq 1Ž a . ' expŽ a i Hi . g C Ž SUŽ n q 1.. and write as usual § n
Pnq 1 s Cnq1 Ž a .
†
Ł
ls1
=
ž
K lq1 Ž u l . 0
=Cnq 1 Ž a .
†
/
ž
Cnq1 Ž a . 0 1 ny l
This is one of the two irreducible riemannian cases 7 in the sense that the coset element is not of the form exp
ž
0 "N †
N 0
/
/
Cnq1 Ž w . Cnq1 Ž a .
Ž 3.18 .
We see that the Ž w l . variables are invariant; we must choose the a i ’s to kill parameters in the productory.
6 Eqs. Ž 3.21 . can be formally solved by a i s y Ky1 i j phaseŽ rˇj . j where K is the Killing-Cartan matrix of the A n algebra; it is probable that this fact does not be an accident but occurs in other cases Gr C Ž G .. 7 The other one is SU ) Ž2 n.r USpŽ2 n. and will not be considered here.
for some matrix N Žoff-diagonal cosets.. We start from the well-known coset decomposition under SO Ž n. of any unimodular real n = n matrix g n s Sn Pn ,
Sn t s Sn ,
Pn g SO Ž n .
Ž 4.1 .
Also Sn is positive definite and det Sn s 1. But we know from elementary linear algebra that any such a matrix is diagonalizable by an orthogonal one Q n completely determined Sn s Q n t Diag Ž l1 2 , . . . , l n 2 . Q n , n
Ł li s 1 ,
l1 G l2 G . . . G l n G 0
Ž 4.2 .
is1
from where after a redefinition Pn ™ Q n t Pn Q n we get g n s Qn t g n Qn ,
Q n g SO Ž n . ,
g n s Diag Ž l1 2 , . . . , l n 2 . Pn
Ž 4.3 .
Analogous steps using well known results yield the complexification of Ž4.3., namely the decomposition C . under SUŽ n,C C . that we quote without of Sl Ž n,C proof g n s Vn
†
g n Vn ,
C. , Vn g SU Ž n,C
§ ny1
g n s Cn Ž a .
Ł
ls1
seems to suggest that the corresponding decomposition under USpŽ2 p . = USpŽ2 q . could follow from replacing in Ž3.13. E by Z generators 8 and respective Cartan subalgebras, but we have not checked this. We remark the locality of the parametrizations obtained; the changes of variables needed to carry out the job are singular in some points of the group manifold as can be seen by direct inspection of Ž2.8., Ž3.6. for example. Possible applications in physical problems of these parametrizations are in the context of GWZW models as models of strings moving on background fields. Eq. Ž2.26. can be used to treat systematically all the models in w9x where the lowest dimensional cases were considered. Also the measure on the group can be computed straightforwardly through the Maurer-Cartan forms without introducing FadeedPopov ghost due to constraints because they were solved once for all. For p q q s 2 Ž A1 algebras. parametrization Ž3.13. is widely known; for p s 2,q s 1 was introduced in w10x; it allows to extend the study of coset models in the search of physically relevant string backgrounds represented by exact conformal field theories.
ž
K lq1 Ž u l . 0
0 1 ny 1yl
/
Cn Ž b .
Ž 4.4 . C .. and the vectors u l are where Cn g C Ž SUŽ n,C l Ž . constrained by rˇl g R as in Section 3.3.
5. Conclusions We have obtained in this paper adjoint parametrizations defined by Ž1.3. w.r.t. maximally compact subgroups for a large set of non-compact groups, the classically riemannian cosets. The constructive procedure used allows to extend them straightforwardly to non-riemannian decompositions, for example SO Ž p q n,q . under SO Ž n,q .. Few words about symplectic groups: these groups can be treated in the same way as made here; the fact that C . F Sp Ž 2 p q 2 q;C C. USp Ž 2 p,2 q . ; U Ž 2 p,2 q;C
Acknowledgements It is a pleasure to thank to Loriano Bonora for useful correspondence.
References w1x J.F. Cornwell, Group theory in Physics, vols. 1r2, Academic Press Limited, London, 1984. w2x A.O. Barut, R. Raczka, Theory of group representations and applications, World Scientific Publishing Co., Singapore, 1986. w3x J.E. Humphreys, Introduction to Lie algebras and representation theory, Springer, Heidelberg, 1972. w4x S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, 1962. w5x R. Gilmore, Lie Groups, Lie algebras, and some of their applications, Wiley, New York, 1974.
8
See Ref. w5x, chapter 5 for definitions.
w6x S. Weinberg, The quantum theory of fields, Cambridge University Press, Cambridge, 1996. w7x D. Karabali, Gauged WZW models and the coset construction of CFT, Brandeis preprint BRX TH-275, July 1989, and references therein. w8x K. Gawedzki, Non compact WZM CFT, in: J. Frolich et al. ¨
ŽEds.., Proc. NATO Advanced Study Institute, Cargese, ` France, NATO ASI Series B: Physics, vol. 295, Plenum, New York, 1992, p. 247. w9x I. Bars, K. Sfetsos, Phys. Rev. D 46 Ž1992. 4510. w10x A.R. Lugo, Phys. Rev. D 52 Ž1995. 2266.
Non-standard parametrizations and adjoint invariants of classical groups 1 Adrian ´ R. Lugo
2
Departmento de Fısica, Facultad de Ciencias Exactas, UniÕersidad Nacional de La Plata, C.C. 67, (1900) La Plata, Argentina ´ Received 5 May 1999; accepted 30 July 1999 Editor: L. Alvarez-Gaume´
Abstract We obtain local parametrizations of classical non-compact Lie groups where adjoint invariants under maximal compact subgroups are manifest. Extension to non-compact subgroups is straightforward. As a by-product parametrizations of the same type are obtained for compact groups. They are of physical interest in any theory gauge invariant under the adjoint action, typical examples being the two dimensional gauged Wess-Zumino-Witten-Novikov models where these coordinatizations become of extreme usefulness to get the background fields representing the vacuum expectation values of the massless modes of the associated Žsuper. string theory. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction It is none to say what group theory meant Žand means. for theoretical physics during this century, in particular the theory of continuous groups or Lie groups Ž see e.g. w1,2x and references therein.. The properties of their Lie algebras, easier to hand than the groups themselves, define them locally and in fact most part of the textbooks are dedicated to them w3x. However depending on the problem at hand, explicit parametrizations of the group manifold become necessary. Many of them are widely known, example of them the SUŽ2. or more generally the Euler angles of orthogonal groups. Of course that we can always write locally a group element as a prod-
1 2
This work was partially supported by CONICET, Argentina. E-mail: [email protected]
uct of one-generator exponentials, or simply as the exponential of an arbitrary Lie algebra element. But in most cases theses obvious parametrizations are of few usefulness because they obscure the global properties of the group and generally lead to untractable computations. An interesting parametrization is suggested by the Mackey theorem, the well-known coset decomposition: let G be a group and H a subgroup of it. Then for any g g G, g s k h,
k g GrH ,
hgH
Ž 1.1 .
in a unique way. This leads to the theory of homogeneous spaces Žor Žleft or right. coset spaces. GrH, good references on the subject being w4,5x. Among others physical applications Ž1.1. is fundamental in the treatment of effective field theories with spontaneously broken symmetries w6x.
0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 9 3 6 - 3
But let us assume instead that we have a field theory including maps from ‘‘space-time’’ on a group manifold G among its degrees of freedom, and gauge invariant under the adjoint action of a subgroup H of G g ™ h g s h g hy1 ,
hgH
Ž 1.2 .
This means that effectively the theory depends on the invariants of the group under the adjoint action of the subgroup. That is, if we were able to write uniquely any element of G as g ' hy1 g h ,
hgH
where u p is a p-dimensional real vector, Pp g SO Ž p . and generically we will mean H Ž P ,Q . s
2. The orthogonal groups We consider in this section the pseudo-orthogonal groups, G ' SO Ž p,q .. Its maximal compact subgroup is H ' SO Ž p . = SO Ž q .. In order to get its decomposition Ž1.3. we need as a first step to get the
P 0
/
Ž 2.2 .
K pq 1 Ž u p . s
1 y Ž 1 q u pq 1 . yu p
1 s u p t u p q Ž u pq1 .
y1
up up t
t
up u pq1
2
0
,
Ž 2.3 .
Under an adjoint transformation h
Ppq 1 s H Ž h,1 . Ppq1 H Ž h t ,1 . ,
h g SO Ž p .
Ž 2.4 . the parameters of Ppq 1 get transformed as: h
up sh up ,
h
Pp s h Pp h t
Ž 2.5 .
The procedure will be constructive. Les us pick an arbitrary matrix Vp g SO Ž p . decomposed this time as a left coset w.r.t. SO Ž p y 1. Vp s H Ž Vpy1 ,1 . K p Ž zpy1 .
Ž 2.6 .
and rewrite Ž2.1. for any such a Vp with the help of Ž2.5. as Ppq 1 s H Ž Vp t ,1 . Ppq1 H Ž Vp ,1 . , Ppq 1 s K pq1 Ž Vp u p . H Ž Vp PpVp t ,1 .
Ž 2.7 .
The general idea to apply here and in the subsequent cases is to fix the whole matrix Vp Ž' zpy1 ,Vpy1 . in terms of variables of Ppq 1 Ž' u p , Pp ., leaving aside only precisely the invariants together with the matrix Vp as parameters of Ppq1. Evidently this procedure is equivalent to make a change of variables from the non-invariant parameters in Ppq 1 to Vp . Eq. Ž2.7. suggests to put u p in some standard form by a specific choice of Ža part of. Vp . In fact it is easy to show that the choice 3 up zpy1 sy yÕp < up <
ž /
2.1. Reduction of SO(p q 1) under SO(p)
0 Q
In what follows the dimensionalities of matrices should be understood from the context when not stated explicitly. The right coset element in SO Ž p q 1.rSO Ž p . ; S p is given by
Ž 1.3 .
then clearly making a gauge transformation Ž1.2. identifying the h’s the theory will depend only on g that encloses the invariants mentioned above. We would like to remark that at difference of Ž1.1. there no exists any general theorem assuring the decomposition Ž1.3.; in fact it is not difficult to find examples where it is not possible to write it. It is the aim of this paper to get the class of local parametrizations of the type Ž1.3. in a whole set of cases of physical interest. Specific examples where they must be used Žand were used in the lower dimensionality cases where parametrizations were available. are the two dimensional gauged WessZumino-Witten-Novikov models w7,8x. It is worth to say however that the results, being explicit parametrizations of classical groups, are valuable on their own right independently of the applications.
ž
Ž 2.8 .
We start by writing w5x Ppq 1 Ž u p , Pp . s K p Ž u p . H Ž Pp ,1 .
Ž 2.1 .
3 As usual we will use the notation Ž eˇi . j s d i j ,Ž Ei j . k l s d i k d jl , A i j ' Ei j y E ji ,Si j ' Ei j q E ji .
defines the rotation K p Ž zpy1 . u p s < u p < eˇ p
Ž 2.9 .
Note that we have changed the Ž p y 1. parameters from u p indicating its direction in terms of the Ž p y 1. parameters in zpy 1. With such a choice of zpy 1 Ž Vpy1 not fixed yet. we can write
where Pp Ž Q q . g Ž SO Ž p .Ž SO Ž q .. and the coset element is given by K p , q Ž S . s exp
s
Ppq 1 s K pq1 Ž < u p < eˇ p . t
=H H Ž Vpy 1 ,1 . Pp H Ž Vpy1 ,1 . ,1
ž
/
0 Nt
N 0
Ž 1 q SS t .
/
1r2
S
St
S s Ž NN t .
Ž 2.10 .
ž
y1 r2
Ž 1 q S tS .
sinh Ž NN t .
1r2
Under an adjoint transformation
Ppq 1 s K pq1 Ž < u p < eˇ p . H Pp ,1
h
/
Ž 2.11 .
Inspection of this formula indicates an iterative process, the next step being to write the analogous expression for Pp and so on; after p steps we get § p
Ppq 1 s
L p , q s H Ž h p ,h q . L p , q H Ž h p ,h q .
Ž 2.12 .
t
< u l < s sin u l ,
0 Ful Fp
Ž 2.13 .
S s hp S hq t ,
h
Pp s h p Pp h tp ,
h
Q q s h q Q q h tq
Ž 2.18 . By following the strategy pursued in the past subsection we introduce two matrices Vp ,Vq belonging to SO Ž p .,SO Ž q . respectively and rewrite Ž2.15.
and write the SO Ž p q 1. element in the final form
t
L p , q s H Ž Vp ,Vq . L p , q H Ž Vp ,Vq . ,
t
Ppq 1 s H Ž Vp ,1 . Ppq1 H Ž Vp ,1 . , § p
Ppq 1 s
Ł
exp Ž u l A l ,lq1 .
p=q
Ž 2.17 .
ls1
It is convenient to introduce the angular variables
,
with h p Ž h q . g SO Ž p .Ž SO Ž q .., the parameters of L p, q transforms as h
Ł H Ž K lq1 Ž < u l < eˇl . ,1 pyl .
gR
N
0
Ž 2.16 .
where we have redefined Pp ™ K p Ž zpy1 . t Pp K p Ž zpy1 .. But according to Ž2.7. we can rewrite it as
ž
1r2
Ž 2.14 .
ls1
L p , q s exp
0
Vp NVq
Ž Vp NVq t .
t
0
t
0
which displays explicitly the p invariants u l ,l s 1, . . . , p4 under the adjoint action of SO Ž p .. Note that the number of parameters trivially matches: p p Ž . Ž . 2 p y 1 q p s 2 p q 1 , as should; this is the first of our results, to be used in the following.
As in Ž2.6. we consider left coset parametrizations
2.2. Reduction of SO(p,q) under SO(p) = SO(q)
Vq s H Ž Vqy1 ,1 . K q Ž zqy1 .
Our starting point is again the right coset parametrization w5x 4
and try to totally fix zpy 1 and zqy1 to put N in a standard form. It turns out that it is possible to choose these vectors in such a way that
L p , q Ž S, Pp ,Q q . s K p , q Ž S . H Ž Pp ,Q q .
Ž 2.15 .
=H Ž Vp PpVp t ,Vq Q q Vq
t
.
Ž 2.19 .
Vp s H Ž Vpy1 ,1 . K p Ž zpy1 . ,
Ž 2.20 .
Vp N Vq t ' H Ž Vpy1 ,1 . K p Ž zpy1 . N t
=K q Ž zqy1 . H Ž Vqy1 t ,1 . 4
An explicit derivation from the definition of pseudo unitary groups is given in Appendix A of w10x.
s
ž
Nr 0
0 n
/
Ž 2.21 .
where Nr g R Ž py1.=Ž qy1. and n g R . In fact if we write Ns
ž
NrX
n py1
n qy 1 t
nX
/
Ž 2.22 .
then is straightforward to verify that the gauge fixing condition n py 1 s n qy1 s 0, i.e. t
K p Ž zpy1 . NK q Ž zqy1 . s
ž
Nr 0
0 n
/
Ž 2.23 .
holds if we choose zpy 1 ' 1 q < t py1 < 2 zqy 1
ž ' ž1 q < t
<2 qy1
Again we verify the matching in the number of parameters Vp ,Vq , Ni j ,n, u k , u k , p q Ž p y 1. q Ž q y 1. q Ž p y 1. Ž q y 1. q 1 2 2 q Ž p y 1 . q Ž q y 1 . s 12 Ž p q q . Ž p q q y 1 . Ž 2.27 . It is worth to say that the first term in Ž2.26. can be computed as in Ž2.16. with N as in the r.h.s. of Ž2.21.; however this is a formal expression for which, to our knowledge, only the ‘‘minkowskian’’ cases p s 1 or q s 1 admit an explicit form.
y1 r2
/ /
t py1 , 3. The unitary groups
y1 r2
t qy1
with t py1 , t qy1 satisfying
Ž 2.24 .
5
t py 1 s Ž n qy1 t t qy1 y n .
y1
t qy 1 s Ž n py1 t t py1 y n .
y1
Ž n py1 y NrX t qy1 . ,
žn
Xt qy1 y Nr t py1
The treatment of these groups parallels that made in the case of the orthogonal ones, with some additional complications due to the complex character of them. As before we start considering 3.1. Reduction of SU(p q 1) under U(p)
/ Ž 2.25 .
t
A final redefinition Nr ™ Vpy1 Nr Vqy1 leads to Ž2.21 .. Finally Žafter reparametrizing Pp ™ K p Ž zpy 1 . t Pp K p Ž zpy1 . , Q q ™ K q Ž zqy1 . t Q q =K q Ž zqy1 .. we use Ž2.14. to fix Vpy1 and Vqy1; the result can be recast in the form t
L p , q s H Ž Vp ,Vq . L p , q H Ž Vp ,Vq . , py1 qy1
L p , q s exp
žÝ Ý
Ni j Si , pqj q nS p , pqq
is1 js1
/
§ py1
=
Ł
An element of SUŽ p q 1. can be written as Ppq 1 Ž u p ,Up . s K pq1 Ž u p . H Ž Up ,u )p . s K pq 1 Ž u p . H Ž Pp ,1 . exp Ž i f p Tp .
Ž 3.1 . p
where u p ' detUp s expŽ ipf p . , u p g C , Up g UŽ p ., Pp g SUŽ p ., and we have introduced a convenient basis Tk s Ý kls1Ž El l y Ekq1, kq1 ., k s 1, . . . , p4 in the Cartan subalgebra of suŽ p q 1.. The right coset element in Ž3.1. belonging to SUŽ p q 1.rUŽ p . ; CP p is given by w5x K pq 1 Ž u p .
exp Ž u k A k , kq1 .
ks1
s
§ qy1
=
Ł
ks1
exp u k A pqk , pqkq1 ,
ž
/
Ž 2.26 .
1 y Ž 1 q u 2 pq1 .
y1
up up †
yu p †
1 s u p † u p q Ž u 2 pq1 .
up u 2 pq1
2
0
,
Ž 3.2 .
The adjoint action under Hp g UŽ p . is Ž h p ' det Hp . H
Ppq 1 s H Ž Hp ,h )p . Ppq1 H Ž Hp ,h )p .
°u V
5
This set of equations can be reduced to a system of two Žquadratic. equations with two unknowns; the important thing for us is that solutions exist and define the change of variables Ž n py 1 ,n qy1 . ™ Ž zpy1 ,zqy1 . .
~
l
H
s
h p Hp u p
Pp
s
Hp Pp Hp†
p
s
fp
¢f H
p
†
Ž 3.3 .
As before we pick an arbitrary Vp g UŽ p . left coset parametrized ) Vp s H Ž Vpy1 ,Õpy1 . K p Ž zpy1 . exp Ž i b p . ,
Vpy 1 g U Ž p y 1 .
Ž 3.4 .
SUŽ p,q . can be left coset decomposed under SŽUŽ p . = UŽ q .. as L p , q Ž S,Up ,Uq . s K p , q Ž S . H Ž Up ,Uq . , u p s u q ) ,
K p,q Ž S . s
and write Ž3.1. with the help of Ž3.3. as
S s Ž NN † .
Ppq 1 s K pq1 Ž ÕpVp u p . H Ž Vp PpVp † ,1 .
Ž 3.5 .
By choosing zpy 1 and b p Ž Vpy1 free. as
Ž
°
ž /
~
l
1
p
.
0 N†
S
Ž 1 q S †S .
/
sinh Ž NN † .
1r2
we have Õp Vp u p s < u p < eˇ p
Ž 3.7 .
and identifying Vpy 1 as the SUŽ p y 1. matrix corresponding to the Pp decomposition in Ž3.5. Žprevious redefinition Pp ™ K p Ž zpy1 . †Pp K p Ž zpy1 .. we get Ppq 1 s K pq1 Ž < u p < eˇ p . H Pp ,1 exp Ž i f p Tp .
/
Ž 3.8 .
S, N g C p=q Ž 3.10 .
N,
H
s
hp S hq †
Up
s
h p Up h†p
s
h q Uq h q †
q
Ž 3.6 .
<Ž u p . <
0
N , 0
S
¢U
pq1
ž
H
H
p
1r2
The adjoint action under H Ž h p ,h q . g SŽUŽ p . = UŽ q .. is
/
exp Ž i b p . s
y1 r2
†
L p , q s H Ž h p ,h q . L p , q H Ž h p ,h q .
zpy1 Ž u )p . u p sy , p yÕ 2 py1 <Ž u p . < < u p < u )p
ž
1r2
H
p
ž
S
s exp
†
Ppq 1 s H Ž Vp ,Õp) . Ppq1 H Ž Vp ,Õp ) . ,
=exp Ž i f p Tp .
Ž 1 q SS † .
†
Ž 3.11 .
Two matrices Vp ,Vq belonging to UŽ p .,UŽ q . respectively with Õp Õq s 1 are introduced and following similar steps as in Subsection 2.2 we find that it is possible to fix N in the way Nr 0 Ž 3.12 . 0 n where now Nr g C Ž py1.=Ž qy1. and n g R . Then by using the results of the past subsection we get the final result for the parametrization Ns
ž
/
†
By repeating the analysis for Pp and after p steps we arrive to the final result
L p , q s H Ž Vp ,Vq . L p , q H Ž Vp ,Vq . , L p , q s exp
†
Ppq 1 s H Ž Vp ,Õp) . Ppq1 H Ž Vp ,Õp) . ,
Ł
0 N†
N 0
/Ł
exp Ž u l A l ,lq1 .
ls1
§ qy1
§ p
Ppq 1 s
ž
§ py1
exp Ž u l A l ,lq1 . C pq1 Ž F .
Ž 3.9 .
ls1
It differs from Ž2.14. from the unitary character of Vp and the comparison of the arbitrary Cartan elep ment C pq 1ŽF . s expŽ iÝ ls1 f l Tl . at right in Ppq1.
=
Ł
ls1
exp u l A pql , pqlq1 C Ž F .
ž
/
Ž 3.13 .
where H Ž Vp ,Vq . g SŽUŽ p . = UŽ q .. and we denote by C ŽF . an arbitrary element in the Cartan subalgebra of the Lie algebra of SŽUŽ p . = UŽ q ... 3.3. Decomposition under the maximal torus
3.2. Reduction of SU(p,q) under S(U(p) = U(q)) In order not to be repetitive we will skip some steps in what follows. An arbitrary element in
Some times is useful to have the adjoint decomposition of unitary groups under the Cartan subalgebra. We will work out for definiteness the case of SUŽ n q 1.; the non-compact versions differ as usual
by signs in the coset elements and Wick rotations of some compact generators. To this end we begin by searching for the coset decomposition of SUŽ n q 1. under C Ž SUŽ n q 1..; from Ž3.1.
It is easy to show that the versors rˇl get transformed as l a
žÝ /
rˇl s exp Ž yi a˜ lq1 . exp i l s 1, . . . ,n
Pnq 1 Ž u n ,Un . s K nq1 Ž u n . H Ž Un ,u n) . , K nq 1 Ž u n . s expun
s
ž
ž
rˇn
0 yrˇn
1 y Ž 1 q u 2 nq1 .
†
y1
0
/
< u n < q Ž u 2 nq1 . s 1 , rˇn † rˇn s 1 ,
°a ¢ya
un un †
un
a˜ s~ya k
/
,
u n s sin un rˇn ,
un g w 0,p x
Ž 3.14 .
Un s Pn
ž
0 , un
/
Pn g SU Ž n . ,
u n ' detUn s exp Ž i w n .
Ž 3.15 .
we can rewrite Ž3.14. as
ky 1 q a k n
2 a1 y a2
s
yphase Ž rˇ1 .
1
ya 1 q 2 a 2 y a 3 ...
s
yphase Ž rˇ2 .
2
ya ny 1 q 2 a n
s
yphase Ž rˇn .
Pnq 1 Ž u n ,Un . s K nq1 Ž u n . H Ž Pn ,1 . exp Ž i w Hn . Ž 3.16 .
Pnq 1 s Cnq1 Ž a . Pnq1 Cnq1 Ž a . ,
where this time is convenient to introduce the basis Hk s Ek k y Ekq1, kq1 ,k s 1, . . . ,n4 in the Cartan subalgebra of suŽ n q 1.. By repeating with Pn and iterating we get
Pnq 1 s
Pnq 1 s
Ł
ls1
ž
K lq1 Ž u l . 0
...
Ž 3.21 .
n
In other words, from Ž3.18. we get the final result n
§ n
Ž 3.20 .
Then we can put the phases of the Ž a rˇl . l components, l s 1, . . . ,n, to zero by choosing the a X s such that 6
By introducing 1 ny 1 0
if k s 1 if k s 2, . . . ,n if k s n q 1
1
u 2 nq1
2
Ž 3.19 .
where
yu n †
2
a˜ i Ei i rˇn ,
is1
0 1 ny l
/
exp Ž i w P H .
†
§ n
Ł
ls1
ž
K lq1 Ž u l . 0
0 1 ny l
/
Cnq1 Ž w .
Ž 3.22 .
with the constraints implied by Ž3.21.: Ž rˇl . l g R , l s 1, . . . ,n.
4. The decomposition of Sl( n ) under SU( n )
Ž 3.17 . Now let us pick an element Cnq 1Ž a . ' expŽ a i Hi . g C Ž SUŽ n q 1.. and write as usual § n
Pnq 1 s Cnq1 Ž a .
†
Ł
ls1
=
ž
K lq1 Ž u l . 0
=Cnq 1 Ž a .
†
/
ž
Cnq1 Ž a . 0 1 ny l
This is one of the two irreducible riemannian cases 7 in the sense that the coset element is not of the form exp
ž
0 "N †
N 0
/
/
Cnq1 Ž w . Cnq1 Ž a .
Ž 3.18 .
We see that the Ž w l . variables are invariant; we must choose the a i ’s to kill parameters in the productory.
6 Eqs. Ž 3.21 . can be formally solved by a i s y Ky1 i j phaseŽ rˇj . j where K is the Killing-Cartan matrix of the A n algebra; it is probable that this fact does not be an accident but occurs in other cases Gr C Ž G .. 7 The other one is SU ) Ž2 n.r USpŽ2 n. and will not be considered here.
for some matrix N Žoff-diagonal cosets.. We start from the well-known coset decomposition under SO Ž n. of any unimodular real n = n matrix g n s Sn Pn ,
Sn t s Sn ,
Pn g SO Ž n .
Ž 4.1 .
Also Sn is positive definite and det Sn s 1. But we know from elementary linear algebra that any such a matrix is diagonalizable by an orthogonal one Q n completely determined Sn s Q n t Diag Ž l1 2 , . . . , l n 2 . Q n , n
Ł li s 1 ,
l1 G l2 G . . . G l n G 0
Ž 4.2 .
is1
from where after a redefinition Pn ™ Q n t Pn Q n we get g n s Qn t g n Qn ,
Q n g SO Ž n . ,
g n s Diag Ž l1 2 , . . . , l n 2 . Pn
Ž 4.3 .
Analogous steps using well known results yield the complexification of Ž4.3., namely the decomposition C . under SUŽ n,C C . that we quote without of Sl Ž n,C proof g n s Vn
†
g n Vn ,
C. , Vn g SU Ž n,C
§ ny1
g n s Cn Ž a .
Ł
ls1
seems to suggest that the corresponding decomposition under USpŽ2 p . = USpŽ2 q . could follow from replacing in Ž3.13. E by Z generators 8 and respective Cartan subalgebras, but we have not checked this. We remark the locality of the parametrizations obtained; the changes of variables needed to carry out the job are singular in some points of the group manifold as can be seen by direct inspection of Ž2.8., Ž3.6. for example. Possible applications in physical problems of these parametrizations are in the context of GWZW models as models of strings moving on background fields. Eq. Ž2.26. can be used to treat systematically all the models in w9x where the lowest dimensional cases were considered. Also the measure on the group can be computed straightforwardly through the Maurer-Cartan forms without introducing FadeedPopov ghost due to constraints because they were solved once for all. For p q q s 2 Ž A1 algebras. parametrization Ž3.13. is widely known; for p s 2,q s 1 was introduced in w10x; it allows to extend the study of coset models in the search of physically relevant string backgrounds represented by exact conformal field theories.
ž
K lq1 Ž u l . 0
0 1 ny 1yl
/
Cn Ž b .
Ž 4.4 . C .. and the vectors u l are where Cn g C Ž SUŽ n,C l Ž . constrained by rˇl g R as in Section 3.3.
5. Conclusions We have obtained in this paper adjoint parametrizations defined by Ž1.3. w.r.t. maximally compact subgroups for a large set of non-compact groups, the classically riemannian cosets. The constructive procedure used allows to extend them straightforwardly to non-riemannian decompositions, for example SO Ž p q n,q . under SO Ž n,q .. Few words about symplectic groups: these groups can be treated in the same way as made here; the fact that C . F Sp Ž 2 p q 2 q;C C. USp Ž 2 p,2 q . ; U Ž 2 p,2 q;C
Acknowledgements It is a pleasure to thank to Loriano Bonora for useful correspondence.
References w1x J.F. Cornwell, Group theory in Physics, vols. 1r2, Academic Press Limited, London, 1984. w2x A.O. Barut, R. Raczka, Theory of group representations and applications, World Scientific Publishing Co., Singapore, 1986. w3x J.E. Humphreys, Introduction to Lie algebras and representation theory, Springer, Heidelberg, 1972. w4x S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, 1962. w5x R. Gilmore, Lie Groups, Lie algebras, and some of their applications, Wiley, New York, 1974.
8
See Ref. w5x, chapter 5 for definitions.
w6x S. Weinberg, The quantum theory of fields, Cambridge University Press, Cambridge, 1996. w7x D. Karabali, Gauged WZW models and the coset construction of CFT, Brandeis preprint BRX TH-275, July 1989, and references therein. w8x K. Gawedzki, Non compact WZM CFT, in: J. Frolich et al. ¨
ŽEds.., Proc. NATO Advanced Study Institute, Cargese, ` France, NATO ASI Series B: Physics, vol. 295, Plenum, New York, 1992, p. 247. w9x I. Bars, K. Sfetsos, Phys. Rev. D 46 Ž1992. 4510. w10x A.R. Lugo, Phys. Rev. D 52 Ž1995. 2266.