arXiv:q-alg/9709006v2 2 Sep 1997
IT EP-T H -28/97
-Conform alAlgebras
M aria I.G olenishcheva-K utuzova ,
V ictorG .K acy
A bstract
-conform al algebra is an axiom atic description of the operator
product expansion ofchiral elds w ith sim ple poles at nitely m any
points. W e classify these algebras and their representations in term s
ofLie algebras and their representations w ith an action ofthe group
. To every -conform al algebra and a character of we associate
a Lie algebra generated by elds w ith the O PE w ith sim ple poles.
Exam ples include tw isted a ne K ac-M oody algebras,the sin algebra
(w hich is a \ -conform al" analodue ofthe generallinear algebra) and
itsanalogues,the algebra ofpseudodi erentialoperatorson the circle,
etc.
0
Introduction
In the past years there has been a num ber of papers, w here the e ect of
splitting ofa pole oforder N in the operator product expansion (O PE) occurs. Exam plesinclude representationsoftw isted a neK ac-M oody algebras
[K -K -L-W ],oftheLiealgebrasofQ uantum torus[G -K -L],ofQ uantum a ne
algebras[F-J],and ofthecentralextension ofthedoubleofYangian [K h-L-P].
Institut G irard D esarques,U R A 746,U niversite Lyon-1,< kutuzova@ geom etrie.univlyon1.fr> and InternationalInstitute ofN onlinear Science (M oscow )
y
D epartm ent of M athem atics, M IT , C am bridge, M A
02139, U SA ,
< kac@ m ath.m it.edu>
1
T hese constructionsfound m any applicationsin the theory ofintegrable system s and exactly solvable m odels in quantum eld theory.
T he sim plest m otivation to consider the O PE w ith sim ple poles,but in
the shifted pointsisthe follow ing. C onsider the O PE oftwo local eldsa(z)
and b(z):
NX 1
cj(w )
a(z)b(w )
(0.1)
w )j+ 1
j= 0 (z
and replace (z w )n by its q-analogue (z w )nq = (z w )(z qw )
(z
qn 1w ). T he result ofthis q-deform ation is an O PE w ith only sim ple poles.
A naturalgeneralization isto consider -deform ation,w here isa subgroup
ofthe m ultiplication group ofnon-zero com plex num bers C .N am ely,foran
n-elem ent subset S of we let
(z
w )nS =
Y
(z
w)
2S
and consider a collection of elds satisfying the condition of -locality,i.e.
(z
w )nS [a(z);b(w )]= 0
(0.2)
T he study ofalgebraic properties of -locality leads us to a form alde nition ofa -conform alalgebra given below . Property (0.2) is equivalent to
the follow ing form ofthe O PE (Proposition 1.1):
X
a(z)b(w )
2S
c (w )
:
z
w
(0.3)
W e view the eld c (w ) as an \ -product" ofthe elds a(z) and b(z):
c (w )= (a( ) b)(w ):
(0.4)
Introduce operators T ( 2 C ) on the space of elds by
T a(z)=
a( z):
(0.5)
LetR be a space over C of -local eldsw hich isclosed underproducts(0.4)
and invariantunderoperators(0.5)forall 2 T hen one can show (Proposition 2.1) that the follow ing properties hold,w here a;b;c 2 R , ; 2 :
(C 0) a( ) b= 0 for allbut nitely m any
2 ,
2
(C 1) (T a)( ) b= a(
(C 2) a( ) b=
) b,
T (b(
(C 3) [a( ) ;b( ) ]c = (a(
a),
1)
1
) b)( ) c.
N ow we observe that one can forgetthat is a subgroup ofC and that
R is a space of elds. W hat rem ains is a (not necessarily abelian) group ,
itsrepresentation space R and C -bilinearproductsa( ) bon R foreach 2 ,
such that the axiom s (C 0){(C 3) hold. T his is the basic object ofour study,
called a -conform alalgebra.
R ecall that the O PE (0.1) of local elds sim ilarly leads to the notion
of a conform alalgebra [K 1]. H owever, conform alalgebra and -conform al
algebras have dram atically di erent properties.
O ur rstresulton -conform alalgebrasisT heorem 3.1 w hich gives their
classi cation in term sofadm issible pairs(g;’),w here g isa Lie algebra and
’ is a hom om orphism of to the group A utg ofautom orphism s of g such
that for any a;b2 g one has
[T a;b]= 0 for allbut nitely m any
2
:
(0.6)
O fcourse,if is a nite subgroup ofA utg,then (0.6) holds. T he associated -conform alalgebra is called a tw isted current conform alalgebra. A
m ore interesting exam ple corresponds to the adm issible pair(g‘1 ;Z ),w here
g‘1 is the Lie algebra ofallZ Z -m atrices (aij)over C w ith a nite num ber
ofnon-zero entries and the action ofZ is by translations along the diagonal:
n (a
i;j)= (ai+ n;j+ n ),n 2 Z . T his Z -conform alalgebra is denoted by gc1(Z )
for reasons explained below .
To a -conform alalgebra and a hom om orphism
of to C (or,m ore
generally,to the group ofm erom orphic transform ations,see x 5) one canonically associates an in nite-dim ensionalLie algebra. T his construction produces m any know n (and a lot ofnew ) exam ples ofin nite-dim ensionalLie
algebras w hich appearin deform ation ofconform al eld theories. Forexam ple,the Lie algebra ofq-di erentialoperators on the circle (the sin-algebra)
isobtained from the Z -conform alalgebra gc1(Z )by using :Z ! C de ned
by (n)= qn .Taking the sam e Z -conform alalgebra,butthe hom om orphism
:Z ! G L 2(C ) de ned by (n)z = z + n,z 2 C ,instead,produces the Lie
algebra ofpseudodi erentialoperators on the circle (see x 5).
R epresentations of -conform al algebras are classi ed by equivariant
(g; )-m odules satisfying a niteness condition (T heorem 4.1). O n the other
3
hand, to any representation ofthe group we canonically associate the
general -conform al algebra gc( ; ), w hich plays the role of the general
linear group in the sense that representations of a -conform alalgebra R
correspond to hom om orphism s R ! gc( ; ) (T heorem 4.2). T he m ost im portant case is w hen is a free C [ ]-m odule ofrank N . T he corresponding
general -conform alalgebra,denoted by gcN ( ),isa generalization ofgc1(Z )
m entioned above.Itisan analogueofgcN in thetheory ofconform alalgebras
[K 2].
In the present paperwe consider the sim plest case ofsim ple poles and of
one indeterm inate z. It is straightforward to generalize this to the case of
m ultiple polesand the case ofseveralindeterm inates. T his w illbe discussed
elsew here.
1
S-localform aldistributions
R ecallthat a form aldistribution w ith coe cients in a vector space U is an
expression ofthe form
X
a(z)=
an z
n 1
; w here an 2 Z :
n2 Z
T hey form a vector space denoted by U [[z;z 1]]. W e shalluse the standard
notation:
R esa(z)= a0 :
z= 0
Sim ilarly,a form aldistribution in z and w is an expression ofthe form
X
a(z;w )=
am ;n z
m
1
w
n 1
; w here am ;n 2 U :
m ;n2 Z
Itm ay beviewed asa form aldistribution in z w ith coe cientsin U [[w ;w 1]].
R ecallthatthe form al -function isthe follow ing form aldistribution in z
and w w ith coe cients in C :
X
(z
w )=
zn 1w
n
:
n2 Z
It has the follow ing two basic properties:
(z
w ) (z
R esa(z) (z
z= 0
w) = 0
h
i
w ) = a(w ) for a(z)2 U [z;z 1] :
4
(1.1)
(1.2)
W e shalluse the form aldistribution
X
( z
( z)n 1( w ) n ;
w )=
; 2C ;
n2 Z
and the follow ing obvious property
( z
1
w )=
1
(z
1
w )=
1
(
z
w ):
(1.3)
W e shalluse also the follow ing equality ofdistribution in z1,z2 and z3:
(z1
z2) (z2
z3)=
(z1
z2) (z1
z3)
(1.4)
Let S be an N -elem ent set of(distinct) non-zero com plex num bers. W e
shalluse the follow ing notation:
(z
w )NS =
Y
(z
w)
(1.5)
2S
D e nition 1.1 T wo form aldistributions a(z) and b(z) with coe cients is
Lie algebra g are called S-localifin g [[z;z 1;w ;w 1]]one has:
(z
w )NS [a(z);b(w )]= 0:
(1.6)
N ote thatS-localform aldistributionsare S 0-localforany nite subsetS 0
of C containing S.
P roposition 1.1 Ifa(z)and b(z)are S-localform aldistributions,then there
exists a unique decom position ofthe form
X
[a(z);b(w )]=
c (w ) (z
w ):
(1.7)
2S
T he form aldistributions c (w ) are given by the form ula
c (w )= R esP S;
z= 0
where
z
[a(z);b(w )];
w
Y
P S; (u)=
u
(1.8)
(1.9)
2S
=
6
5
Proofofthis proposition is im m ediate by the follow ing lem m a.
Lem m a 1.1 (a) Each form aldistribution a(z;w ) can be uniquely written in
the form :
X
a(z;w ) =
c (w ) (z
w )+ b(z;w )
(1.10)
2S
where
z
a(z;w )
w
c (w ) = R esP S;
z= 0
and
R esP S;
z= 0
z
b(z;w ) = 0 for all 2 S :
w
(b) Ifb(z;w ) satis es (1.12) and (z
N
w)
S b(z;w )= 0, then b(z;w )
(1.11)
(1.12)
0.
Proof: C onsider the follow ing operator on the space ofform aldistribution
in z and w :
X
c (w ) (z
w );
S a(z;w )=
2S
w here c (w ) are given by (1.11). It is clear using (1.2) that S2 = S ,w hich
im plies (a). Furtherm ore,suppose that (z w )NS b(z;w )= 0. It follow s from
[K 1,corollary 2.2]that for each 2 S:
P S; b(z;w )= d (w ) (z
w)
for som e form aldistribution d (w ). Ifalso (1.12) holds,then R esz= 0 d (w )
(z
w )= 0,hence d (w )= 0 by (1.2),and
P S;
z
b(z;w )= 0 for all
w
2 S:
(1.13)
T hisim pliesthatb(z;w )= 0 since thepolynom ials,P S; arerelatively prim e.
P
(Indeed 2 S P S; u = 1 forsom e polynom ialsu .M ultiplying both sidesof
(1.13) by u wz and sum m ing over ,we get b(z;w )= 0.) T his proves (b).
C orollary 1.1 T he form aldistributions a(z) and b(z) are S-locali there
exists a decom position ofthe form (1.7).
Proof: Follow s from Proposition 1.1 and (1.1).
6
R em ark 1.1 T he coe cients c (w ) are independent ofthe choice ofS for
w hich (1.6) holds. T his follow s from the uniqueness of the decom position
(1.10).
R em ark 1.2 Form ula (1.7),w ritten out in m odes,looks as follow s:
X
[am ;bn ]=
m
c ;m + n :
2S
2
-products of -localform aldistributions
Let be a subgroup ofC .T wo form aldistributionsa(z)and b(z)are called
-localifthey are S-localforsom e nite subset S of .For 2 de ne the
-product(a( ) b)(w )asthe coe cient c (z) in the expansion (1.7).In other
words,
z
a( ) b (w )= R esP S;
[a(z);b(w )]:
(2.1)
z= 0
w
D ue to R em ark 1.1,the -product is independent ofthe choice ofS.
For 2 introduce the follow ing operator:
T (a(z))=
a( z):
It is clear that T preserves -locality.
W e collect below the m ain properties of -products.
P roposition 2.1 For -localform aldistributionsonehasthefollowingproperties ( ; 2 ):
(a) (Translation invariance)(T a)( ) b= a(
(b) (Skewsym m etry) a( ) b=
T (b(
(c) (Jacobiidentity)a( ) (b( ) c)= a(
the suitable localities hold.
)
b,and a( ) T b = T
a(
b.
1)
a).
1)
b)( ) c+ b( ) (a( ) c)provided thatall
1)
Proof:
7
(a) W e have:
X
(T a)( ) b (w ) (z
[T a(z);b(w )] =
w );
2
X
[T a(z);b(w )] = [ a( z);b(w )]=
(a( )b)(w ) ( z
w)
2
X
=
1
(a( )b)(w ) (z
w ):
2
C om paring coe cients,we get the rst equation of(a).
(b) W e have using (1.7) and (1.3):
X
[a(z);b(w )] =
[b(w );a(z)]=
(b( ) a)(z) (w
z)
2
X
=
(b(
a)( w ) (z
1)
w)
2
C om paring w ith (1.7),weobtain (b).T hesecond equation of(a)follow s
from the rst and (b)
(c) W e have:
X
[a(z);[b(w );c(t)]] =
a( ) (b( ) c) (t) (z
t) (w
t);
;2
X
(a(
[[a(z);b(w )];c(t)] =
b)( ) c (t) (z
1)
1
w ) (w
t)
;2
X
=
(a(
b)( ) c (t) (z
1)
t) (w
t):
2
W e have used here (1.3) and (1.4). W e also have:
X
[b(w );[a(z);c(t)]]=
b( ) (a( ) c)(t) (w
t) (z
t):
;2
Equating the rst equality w ith the sum of the rem aining two and using
Jacobi identity in g, we obtain equality (c) due to linear independence of
(w
t) (z
t) for ; 2 .
8
3
-conform alalgebras and Lie algebras of localform aldistributions
T he above considerations m otivate the follow ing de nitions. Let be a
group and let C [ ]be the group algebra of w ith basis T ( 2 ),so that
T T = T ,T1 = 1. Since is not necessarily abelian we m ust distinguish
the case ofleft and right C [ ]-m odules.
D e nition 3.1 A left C [ ]-m odule R is called a left -conform alalgebra if
it is equipped with a C -bilinear product a( ) b for each 2 such that the
following axiom s hold (a;b;c 2 R ; ; 2 ):
(C 0) a( ) b= 0 for allbut nitely m any a 2 ,
(C 1) (T a)( ) b = a(
(C 2) a( ) b=
)
T (b(
b,
a),
1)
(C 3) a( ) (b( ) c)= (a(
1
) b)( ) c+
b( ) (a( ) c).
A right C [ ]-m odule R is called a right -conform al algebra if axiom s
(C 1),(C 2) and (C 3) are replaced by:
(C 1)R
(aT )( ) b= a(
(C 2)R
a( ) b=
(C 3)R
a( ) (b( ) c)= (a(
(b(
) b,
a)T ,
1)
b)( ) c+ b( ) (a( ) c).
1)
U nlessotherw ise speci ed,we w illconsiderleft -conform alalgebrasand
w illdrop the adjective left.
R em ark 3.1
(C 10)
(a) Properties(C 1)and (C 2)(resp.(C 1)R and (C 2)R )im ply
a( ) T b= T (a(
1
0
) b) (resp. (C 1 )R
a( ) (bT )= (a(
b)T )
1)
(b) (C 1) and (C 10) (resp. (C 1)R and (C 10)R ) im ply
(T a)( ) (T b)= T (a(
1
) b)
(resp.(aT )( ) (bT )= (a(
b)T ):
1)
In particular,each T isan autom orphism ofthe producta(1)b foreach
2 .
9
(c) W ith respect to the product a(1)b,R is a Lie algebra over C . D ue to
(C 1) allother products are expressed via the 1-product:
a( ) b= (T a)(1)b
D ue to R em ark 3.1,we can associate to a -conform alalgebra R a pair
(g;’),w here g isa Liealgebra over C w hoseunderlying spaceisR and bracket
is[a;b]= a(1)b,and ’ isa hom om orphism of to the group A ut(g)such that
[T a;b]= 0 for allbut nitely m any
2 .
(3.1)
W e callsuch (g;’) an adm issible pair. C onversely,given an adm issible pair
(g;’) we denote by R (g;’) the C [ ]-m odule g w ith C -bilinear products
a( ) b= [T a;b]
(3.2)
It is straightforward to check that R (g;’) is a -conform alalgebra.
W e have proved the follow ing result:
T heorem 3.1 -conform alalgebras are classi ed by adm issible pairs (g;’).
Fix now a hom om orphism : ! C . Let s be a Lie algebra and let R
be a fam ily ofpairw ise ( )-localform aldistributions w ith coe cients in s,
w hich span s over C .A ssum e thatR isstable underallT and all -products
fora 2 ( ).T hen s iscalled a Lie algebra of ( )-localform aldistributions.
Itfollow s from Proposition 2.1 thatR is a ( )-conform alalgebra. C onversely,given a -conform alalgebra R ,we associate to it a Lie algebra s(R )
of ( )-localform aldistributions as follow s. C onsider a vector space over C
w ith the basisan ,w here a 2 R and n 2 Z ,and denote by s(R ; )thequotient
ofthis space by the C -span ofelem ents ofthe form (n 2 Z ):
( a + b)n
(T a)n
an
bn ; w here ; 2 C ; a;b2 R ;
( ) n an ; w here
2
;a 2 R :
(3.3a)
(3.3b)
T hen it is straightforward to check that the form ula (cf.R em ark 1.2)
X
[am ;bn ]=
( )m (a( )b)m + n
(3.4)
2
givesa well-de ned structure ofa Lie algebra on s(R ; )of ( )-localform al
P
distributions a(z) = n2 Z an z n 1,a 2 R . T he associated ( )-conform al
algebra is denoted by R (s).
T he relationsbetween the constructed objectscan be sum m arized by the
follow ing diagram :
10
-
-conform alalgebra R
6
Lie algebra s(R ; )
of ( )-localform al
distributions
?
adm issible pair
(g;’)
?
( )-conform alalgebra R (s)
N ow we turn to the m ost im portant exam ples of -conform alalgebras
and related constructions.
E xam ple 3.1 Let (g;Z =N Z ) be a nite-dim ensional(sim ple) Lie algebra g
w ith the action ofthe group = Z =N Z by autom orphism s ofg. D ue to this
action T , 2 ,we have the eigenspace decom position (^ is the group of
characters of ):
g=
gj :
j2 ^
T he -conform alalgebra R (g;’) is de ned as C [ ]-m odule w ith underlying
space g and C -bilinear products a( ) b = [T a;b],a;b 2 g, 2 Z =N Z . Fix a
hom om orphism
: ! C such that (n) = n ,w here 2 C is an N -th
root of1. T he corresponding Lie algebra of ( )-localform aldistributions
s(R ; ) is nothing but the tw isted a ne algebra [K ]w ith the com m utation
relations:
X
km
[am ;bn ]=
[Tk a;b]m + n
k2 Z =N Z
Equivalently:
X
[a(z);b(w )]=
[T a;b](w ) (z
w ):
2 Z =N Z
Suppose that a 2 gj,then Tka =
kj
a and
(
[am ;bn ]=
0
N [a;b]m + n
T hus we put am = 0 for m 6
=
quotient by relations (3.3b).
ifm 6
=
ifm =
j m od N
j m od N
j m od N , w hich is the sam e as to take a
11
E xam ple 3.2 C onsider = Z and the adm issible pair(g;’),w here g = g‘1
is the Lie algebra ofall Z Z m atrices over C w ith nitely m any non-zero
entries. Let E ij (i;j 2 Z ) be its standard basis,w ith the usualrelations:
[E ij;E k‘]=
j;k E i‘
‘;iE kj :
LetT be the im age of1 2 Z under ’.C onsider two di erent actionsofT on
g‘1 .
a) T :E ij ! E i+ 1;j+ 1.
In this case g‘1 is a free C [T;T 1]-m odule w ith the generators A m =
E 0;m ,m 2 Z . A s follow s from D e nition 3.1 ofa -conform alalgebra,
it is su cient to de ne the -products only on generators ofthe C [ ]m odule. So,for the case under consideration we have:
A m(r)A n = [T rA m ;A n ]= [E r;m + r;E o;n ]
=
r; m
T
m
Am +n
r;n A
m +n
; r2 Z
(3.5)
Take a hom om orphism
= q : Z ! C de ned by q(1) = q 2 C .
T he corresponding Lie algebra s(R ; )isthe Lie algebra w ith the basis
A mk ,m ;k 2 Z ,and the com m utation relations:
[A mk ;A n‘ ]=
X
qrk(A m(r)A n )k+ ‘ = (qm ‘
qkn )A mk++‘n :
(3.6)
r2 Z
T his is the Lie algebra ofq-pseudodi erentialoperators on the circle
(sin-algebra).
b) N ow de ne the action of’(1) = T~ by T~(E i;j) = E j+ 1;i+ 1. W e have
T~ = T ,w here isan order2 autom orphism ofg‘
1 de ned by (Eij)=
1
~
~
E ji.In thiscaseg‘1 isa free C [T ;T ]-m odulew ith generatorsA m =
E 0;m and B m = (Am ) = E m ;0, m 2 Z + . Let us com pute the rproduct,r 2 Z ,forthe elem entsofthe basisofC [ ] m odule.Suppose
n m
0 (for n m < 0 we can use axiom (C 2)):
A m(r)A n = [T~rA m ;A n ]
8
>
>
>
<
T m Am +n
m m +n
A
+
r; m T
=
m
+n
>
A
+
r;n
r;n
>
>
:
0
r; m
Am +n ;
m ;n even;
n m
; m even,n odd;
r;n m A
n m
A
;
m odd,n even
m
m ;n odd:
r;m
12
[
T~rA m ;A n ]
8
~m n m + r;n m A n m ;
>
r;0T A
>
>
>
m +n
~m n m
>
;
>
r;0T A
r;n A
<
m
n
m
m
~
~
=
+ r; m T B n+ m ;
r;0T B
>
>
>
~n n m + r;n m A n+ m
>
r;0T B
>
>
:
m +n
+ r; m T~ m B n+ m
;
r;n A
m
B (r)
Bn =
m
B (r)
An
m ;n even;
m even,n odd;
m odd,n even;
m ;n odd:
T hecorresponding Liealgebra s(R ; q)isa Liealgebra w ith basisfA mk g
and fB ‘n g, m ;n 2 Z + , k;‘ 2 Z . From (3.4) and (3.5) we have for
n m
0:
8
>
>
>
<
(qm ‘ qnk)A mk++‘n ;
m ;n even;
m ‘ m +n
k(n m ) n m
q A k+ ‘ + q
A k+ ‘ ; m even,n odd;
[A mk ;A n‘ ] =
nk m + n
k(n m ) n m
>
q
A
+
q
A k+ ‘ ; m odd,n even;
>
k+ ‘
>
:
0
m ;n odd,
m
n
m
n
[B k ;B ‘ ] = the sam e as for [A n ;A ‘ ]w ith change A ! B :
8
n m
>
q m k(qkn q m ‘)A k+
m ;n even;
>
‘ ;
>
>
m (k+ ‘) n m
kn m + n
>
>
q
A
q
A
;
m
even,n odd;
<
k+ ‘
k+ ‘
m
n
m (k+ ‘) n m
m ‘ n+ m
q
B k+ ‘ + q B k+ ‘ ;
m odd,n even;
[B k ;A ‘ ] =
>
>
m (k+ ‘) n m
k(n m ) n m
>
q
B k+ ‘ + q
A k+ ‘
>
>
>
:
n+ m
nk n+ m
+ qm ‘B k+
q
A
m ;n odd:
‘
k+ ‘ ;
T his Lie algebra is apparently new . It contains as a subalgebra the
sin-algebra fA~kn g,w here A~nk = A nk ,n > 0,n even;A nk = qnk B k n ,n < 0,
n even; and the subalgebra isom orphic to B -series ofLie algebras of
quantum torus (see [G -K -L]) for n odd.
E xam ple 3.3 C onsider the group = Z 2 Z and adm issible pair (g‘1 ;’),
w here hom om orphism ’ : ! A utg‘1 isdeterm ined by two elem ents and
T as in Exam ple 3.2 (b). W ith respect to the -action g‘1 is a free m odule
w ith the generators B m = E 0;m , m
0. T he corresponding conform al
algebra R (g‘1 ;’) is de ned by the -products:
m
B (r)
B n = [T rB m ;B n ] =
r; m
(
B (m r)B n = [ TrB m ;B n ] =
T
m
B n+ m
m
n;rB
n m
r;0T B
n n m
r;0 T B
+
B
m
B
m
r;n m
r;n
13
m +r
;
n m
n
; n
; n
(3.7)
m
0
m < 0:
Fix the hom om orphism
: ! C such that ( ) = 1, (1) = q 2 C .
T he Lie algebra s(R (g‘1 ;’); )is a Lie algebra w ith the basis B km ,m 2 Z + ,
k 2 Z ,and com m utation relations:
[B km ;B ‘n ] = (qm ‘
8
>
<
m +n
qnk)B k+
‘
n m
( 1)k (qk(n m ) q m (k+ ‘))B k+
n
‘ ;
k
n(k+ ‘)
( 1) (( q)
>
:
m n
( 1) n(k+ ‘)( q)m (k+ ‘))B k+
‘ ; n
m > 0;
m < 0
m k
A fter the renorm alization of the basis B~km = q 2 B kM we get exactly the
com m utation relationsforB -seriesofsin Liealgebras,introduced in [G -K -L]:
[B~km ;B~‘n ]=
4
m ‘ nk
2
q
q
(m ‘ n k)
2
n+ m
B~k+
‘
kn + m ‘
2
( 1)k q
q
m ‘+ kn
2
n m
B~k+
‘
R epresentationsof -conform alalgebrasand
the general -conform alalgebra gc( ; ).
D e nition 4.1 A (left) m odule over a -conform al algebra R is a C [ ]m odule M with a C -linear m ap a 7
! aM ofR to E ndC M for each 2 such
that the following properties hold (here a;b 2 R , ; 2 and the action of
on M is denoted by 7
! T M ):
(M 0)
aM( ) v = 0 for v 2 M and allbut nitely m any .
(M 1)
(T a)M( ) = aM ;aM( ) T M = T M aM(
h
(M 2)
i
aM( ) ;bM( ) = a(
1
)b
M
()
1
),
.
For exam ple,a 7
! aR( ) is the adjointrepresentation ofR on itself.
It follow s from (M 1) that
(T a)M( ) (T M v)= T M (aM(
1
) v);
T his leads us to
14
v2 M :
(4.1)
T heorem 4.1 Let R be a -conform alalgebra and let (g;’) be the associated adm issible pair. T hen R -m odules M are classi ed by equivariant(g; )m odules such thatfor any a 2 g and v 2 M one has:
(T g a)v = 0 for allbut nitely m any
:
2
(4.2)
Proof: It follow s from (M 2) that a 7
! a(1) is a representation of g. It
satis es (4.1) due to (M 2),and it is equivariant,i.e. (T g a)(T M v)= T M (av)
due to (4.1). C onversely,given an equivariant (g; )-m odule M ,it becom es
an R -m odule by letting
aM = (T g a)M(1) on M :
C onsidera representation 7
! T ofthe group in a vectorspace
over
n V o
C .W ede ne a -conform alendom orphism ofV asa collection a = a( )
2
of C -endom orphism s ofV such that
a( ) T = T a(
1
);
; 2
(4.3)
;
and for each v 2 V
a( ) v = 0 for allbut nitely m any
2
:
(4.4)
W e denote the set ofall -conform alendom orphism s ofV by gc(V; ).
D e ne a C [ ]-m odule structure on the space gc(V; ) by
(T a)( ) = a(
and -product for each
2
)
;
(4.5)
;b( ) ]:
(4.6)
; ; 2
by
(a( ) b)( ) = [a(
)
It is im m ediate to check that axiom s (C 1),(C 2) and (C 3) ofa -conform al
algebra hold (though axiom (C 0) probably doesn’t hold in general).
It is clear that we have by de nition
P roposition 4.1 To give a -m odule V a structure ofa m odule over a conform alalgebra R is the sam e as to give a hom om orphism R ! gc(V; )
(i.e. a C [ ]-m odule hom om orphism preserving all -products).
15
W e w illshow now that axiom (C 0) holds, hence gc(V; ) is a -conform al
algebra,provided thatthe C [ ]-m odule V is nitely generated. Towardsthis
end we shallgive a di erent construction ofgc(V; )in the case w hen V is a
free C [ ]-m odule ofrank 1.
L
r
r
Let gc1( ) =
r2 C [ ]a be a free C [ ]-m odule w ith free generators a
labeled by elem ents of . Foreach 2 de ne the -product by
ar( ) as =
;r
1
Tr 1 ars
sr
;s a ;
(4.7)
extending to the w hole R by axiom s (C 1) and (C 1’).
T heorem 4.2 (a) T he C ( ]-m odule gc1( ) with products (4.7) is a conform alalgebra.
(b) Let V = C [ ]v be a free C [ ]-m odule of rank 1. D e ne the action of
gc1( ) on V by letting
(as)V( ) v =
;s
1
(4.8)
T v
and extending to V by (M 1). T his gives V a structure of a gc1( )m odule.
(c) T he gc1( )-m odule V = C [ ]v containsall -conform alendom orphism s
ofV . In particular,the -conform alalgebras gc(V; ) and gc 1( ) are
isom orphic.
Proof: To prove (a) we need to check that the axiom s (C 0){(C 3) are satised. (C 0)isobvious. (C 1)and (C 2)com e from the de nition (4.7)ofgc1( ).
C heck the skew sym m etry (C 2):
(ar( ) as)( ) =
T (as( 1 )ar) =
sr
(Tr 1 ars)( )
;s a( ) ;
rs
sr
;s T s 1 a
;r 1 Tr 1 a =
;r
1
(
;r
1
Tr 1 ars
sr
;s a ):
For (C 3) we need to check that both sides of
am(
r
s
) (a( ) a )=
(am( ) ar)( ) as + ar( ) (am(
s
)a )
are equal.
LH S: am(
)
sr
(ar( ) as) = am( ) ( ;r 1 Tr 1 Tr 1
;s a )
m
rs
m
sr
=
;s a( ) a
;r 1 Tr 1 a( ) a
m rs
=
;r 1
;r 1 ;m 1 T(m r) 1 a
m sr
+ ;s
;s
;m 1 Tm 1 a
16
;rs Tr
;sr
1
arsm
asrm
R H S: (am( ) ar)( ) as + ar( ) (am( ) as)
m rs
=
;(m r) 1 T(m r) 1 a
;m 1
rm s
+ ;r
;r ;(rm ) 1 a
+ ;m 1 ;r T(rm ) 1 arm s
rsm
+ ;s
;s ;r 1 a
1
1m
;s
asm r
srm
;s a
1 ;m
;m 1
sm r
;sm
a
sTm
1
am sr
;
and we have LH S = R H S.
To prove (b) we need to check only axiom (M 2) on V :
(ar( ) as)( ) v = [ar(
LH S: (ar( ) as)( ) v = (
= (
R H S: [ar(
)
;r
1
ars
;r
1
;s
;as( ) ]v = (
=
)
;as( ) ]v:
sr
;s a( ) )v
1
T(rs)
1
;s
Ts 1 ar( )
;s
1
;s
1
;r
1
;r
T(rs)
1
1
(s2)
1
Tr 1 as(
;r
1
T )v
)v
1)
;s T(sr)
1
v
so LH S = R H S.Finally,(c) is clear.
C orollary 4.1 If V is a
conform alalgebra.
nitely generated C [ ]-m odule, then gc(V; ) is a
Proof: W e m ay assum e that V = C [ ] N is a free C [ ]-m odule of rank
N . T hen gc(V; ) m ay be viewed as a C [ ]-subm odule of gc 1( N ) using
the diagonal hom om orphism C [ ] ! C [ ] N . C orollary now follow s from
T heorem 4.2a.
IfV is a free C [ ]-m odule ofrank N ,we use notation gcN ( )= gc(V; ).
E xam ple 4.1 C onsider = Z N . For the gc1(Z N ) de ned by (4.7) the corresponding Lie algebra (g;’) is a Lie algebra w ith generators a = T a ,
; 2 ZN and com m utation relations:
[a ;a ]=
;
1
a
;
1
a :
Fix q = (q1;:::;qN ) 2 (C )N and de ne the hom om orphism q :Z N !
N
C , such that q( ) = q , w here
2 Z N and q = q1 1
. T he Lie
N q
17
algebra s(R ; q) ofform aldistributions is a Lie algebra w ith generators am ,
P
2 Z N , m 2 Z . D e ne the generating functions a (z) = m 2 Z am z m 1.
From (3.4) we have:
h
i
X
(a( )a )(w ) (z
a (z);a (w ) =
( )w );
2Z N
and for m odes:
h
i
X
=
am ;an
( )m (a( )a )m + n
2Z N
= q m (T a + )m + n
= (q n q m )am++ n
q m (a + )m + n
T his is a vector generalization ofsin-algebra,considered in [G -K -L].
5
N on-com m utative generalization of -local
form aldistributions.
In x3 we associated to a -conform al algebra R a Lie algebra s(R ; ) of
( )-local form al distributions, by xing a hom om orphism
: ! C .
C onversely,each Lie algebra of ( )-conform alform aldistribution de nes a
right ( )-conform alalgebra.In thisparagraph wew illconsiderm oregeneral
case,w here ( )isa subgroup ofthe group ofm erom orphic transform ations
of C . In allexam ples we w illconsider : ! G L 2(C ). For 2 we w ill
denoteby (z)thecorresponding underthehom om orphism transform ation
from G L 2(C ). C onsider the generalization of the property (1.2) of the function:
P roposition 5.1 Let (z) and (z) be from G L2(C ),then:
0
(z) ( (z) w ) =
0
(w ) (z
(w )) =
1
(z
(w ))
1
( (z) w )
(5.1)
Proof: T he group G L 2(C ) is generated by the transform ations a) z 7
!
z,
1
2 C ;b) z ! z
, 2 C ;c) z 7
! z . It is su cient to prove (5.1) for (a),
18
(b) and (c) transform ations. For (a) this is (1.3). To prove (b),let
X
(z
(w
)) =
z
k 1
)k
(5.2a)
wk :
(5.2b)
)k
(5.3)
(w
k2 Z
X
((z +
) w) =
(z +
)
k 1
k2 Z
W e de ne
(z + w )
‘ 1
X1
=
(k + ‘)!
z
k!‘!
k= 0
k 1
for‘ 0.C om paring coe cients ofz
(5.3) we get (b). So,the function
X
(z
w+
)=
z
k 1
‘ k 1
(
w ‘ in right-hand sides of(5.2)and
)k =
(w
X
k
(z +
)
‘ 1
w‘
(5.4)
‘
is wellde ned. T he proofof(c) is sim ilar.
Further we w ill suppose for sim plicity that
elem ent subset S
we shalluse the notation
(z
G L 2(C ). For an N -
Y
w )NS =
(z
(w )):
2S
D e nition 5.1 T wo form aldistributions a(z) and b(z) with coe cients in
a Lie algebra g are called S-localifin g[[z;z 1;w ;w 1]]one has:
(z
w )NS [a(z);b(w )]= 0:
P roposition 5.2 Ifa(z)and b(z)are S-localform aldistributions,then there
exists a unique decom position
X
[a(z);b(w )]=
a(
b (w ) (z
1)
(w ))
2S
T he form aldistributions (a(
b)(w ) are given by the form ula:
1)
Y
a(
b (w )= R es
1)
z
2S
6
=
z
(w )
[a(z);b(w )]:
(w )
(w )
19
(5.5)
Proofis the sam e as the proofofProposition 1.1.
For 2 introduce the follow ing operator:
T
1
a(z)=
0
(z)a( (z))
(5.6)
It is clear,that T , 2 ,preserve -locality.
For -products (a( ) b)(w ) de ned as (5.5) we have allthe properties of
left -conform alalgebras (see D e nition 3.1). C onversely, if we have a conform alalgebra R and a hom om orphism : ! G L 2(C ),wecan construct
a Lie algebra s(R ; ) of ( ) localform aldistributions as follow s. C onsider
a vector space over C w ith the basis an ,a 2 R ,n 2 Z ,and denote by s(R ; )
the quotient ofthis space by the C -space ofelem ents ofthe form :
( a + b)n
an
bn ;
(T 1a)n ( 0(z)a( (z)))n ;
; 2 C ; a;b2 R
2
w here ( 0(z)a( (z)))n denotes the coe cient ofz
decom position. T hen the form ula
n 1
(5.7)
in the Fourier series
X
[a(z);b(w )]=
(a(
b)(w ) (z
(w ))
1)
(5.8)
2
givesa Lie algebra structure on s(R ; )of -localform aldistributionsa(z)=
P
n 1
.
n2 Z an z
R em ark 5.1 Ifwe leta(z)T = 0(z)a( (z)),then we have allpropertiesof
right -conform alalgebras. W e prefer (the m ore custom ary) left -m odules.
C onsider som e im portant exam ples.
E xam ple 5.1 Lie algebra ofpseudodi erentialoperators on the circle.
Let R be the general Z -conform alalgebra gc1(Z ) de ned by (4.7). Fix
a hom om orphism
: Z ! G L 2(C ) de ned as (n) = 10 n1 ,n 2 Z . T he
corresponding Lie algebra s(Z ; ) is a Lie algebra w ith basis amk ,m ;k 2 Z .
P
Let am (z) = k amk z k 1 be the generating function. T hen from (5.8) we
have:
[am (z);an (w )] =
X
(am(s)an )(w ) (z
w + s)
s2 Z
= (T
m +n
m
a
)(w ) (z
m)
w
20
m +n
a
(w ) (z
(5.9)
w + n)
To w rite dow n explicitly the com m utation relationsforcoe cients ofform al
distributions am (z),consider the Lie algebra PD i (S 1) ofpseudodi erential
operators on the circle (see [K h-L-R ]form ore details). T his is a Lie algebra
ofthe associative algebra w ith the basis xm @n ,m ;n 2 Z . T he com m utation
relations are de ned from the com m utation relations on x and @:
@
0
f(x) = f(x) @ + f
(x)
1
f(x) =
@
X1
( 1)n f(n)(x)@
n 1
;
n= 0
w here f(x) 2 C 1 (S 1). Introduce a new basis of PD i (S 1) w ith generators xm D k,m ;k 2 Z , w here D = x@. T hese generators have m ore sim ple
com m utation relations:
[xm D k;xn D ‘]= xm + k (D + n)k D ‘
(D + m )‘D k
(5.10)
Fork < 0 we w illunderstand (D + n)k asan expansion by the negative power
ofD (see (5.3)). Introduce the generating function:
X
am (z)=
x
m
D kz
k 1
(5.11)
:
k2 Z
U sing the property (5.4) of -function,we have
[am (z);an (w )]
X
=
x
(m + n)
(D
n)k D ‘z
k 1
w
‘ 1
(D
m )‘D kz
k 1
wk
D ‘D k (w + m )
w
‘ 1
k;l2 Z
= x
(m + n)
X
D k D ‘(z + n)
k 1
w
‘ k 1
‘ k 1
(w + m )kz
k 1
k;‘2 Z
m +n
= a
(w + m ) (z
w
m)
am + n (w ) (z
w
n):
C om paring (5.9)and (5.12),we see thatthe Lie algebra s( ; )isisom orphic
to the Lie algebra PD i (S 1).
T hus,we have show n,that the Lie algebra ofq PD i (S 1) and the Lie
algebra of PD i (S 1) correspond to the sam e general Z -conform al algebra
gc1(Z ),but to the di erent hom om orphism s Z ! G L 2(C ).
M oregenerally,wecan considerthegeneralconform alalgebra gc1(G L 2(C ))
w ith the usualaction ofG L 2(C ) on C . Let A ;B ;C 2 G L 2(C ). By (4.7) we
have:
BA
:
aA(C )aB = C ;A 1 TA 1 aA B
C ;B a
21
(5.12)
Itisclear,thatthegeneralconform alalgebra adm itreduction to thesubalgebra.T hisreduction iswellde ned attheleveloftheLiealgebra s(g‘1(G L(C ))):
[aA (z);aB (w )]= A (w )0w aA B (A (w )) (z
A (w )) aB A (w ) (z
B 1(w )):
(5.13)
m
q2
0
A s was show n, for the subgroup H 1 =
0
m
q 2
;m 2 Z
we get the
n
o
m
1
1
0
Lie algebra ofq PD i (S 1);for the subgroup H 2 =
get the Lie algebra PD i (S 1). For the subgroup H 3
; m 2 Z we
G L 2(C ),generated
1
q2
1
by two elem ents a = 10 11 and b =
we get the Lie algebra,that
q 2
containsassubalgebra theLiealgebra ofPD i (S 1)and q PD i (S 1).Taking
di erent subgroups H in G L(2;C ) (not necessarily discrete) we get a fam ily
ofin nite-dim ensionalLie algebras s(gc1(H )).
E xam ple 5.2 C onsider the group
w ith relations: 2 = 1 and T = T
A ut(g‘1 ),i= 1;2.
1
generated by two elem ents and T
. Fix two hom om orphism s ’i : !
(
’1 :
(
(Eij)= E j; i
T (E ij)= E i+ 1;j+ 1
and
’2 :
(Eij)= E i; j
T (E ij)= E i+ 1;j+ 1
In both cases g‘1 is a free C [ ]m odule w ith the basis A m = E 0;m ,m 2 Z + .
T he -products on generators A m are as follow s:
a) For ’ 1 we have:
A m( )A n = [T rA m ;A n ]=
r; m
A m( )A n = [ TrA m ;A n ]=
m
T
r;0T
Am +n
m
r;n A
A n+ m +
m +n
for
m + n; rA
m +n
= Tr ;
for
=
Tr :
b) For ’ 2 we have:
A m( )A n =
r; m
T
A m( )A n =
r; m
Tm B n
m
Am +n
m
r;n A
m +n
for
= Tr ;
r; n B
n m
for
=
Tr and n
m :
T hese give us two structures ofa -conform alalgebra,w hich we w ill
denote R 1 and R 2 respectively. C onsidertwo hom om orphism s i : !
22
G L 2(C ),i= 1;2:
!
a)
1(
0 1
1 0
) =
;
1(z)=
1
;
z
1
1(T )
0
q2
1
0 q 2
=
; T1(z)= qz;
!
b)
2(
1
0
) =
0
1
;
2(z)=
z;
!
2(T )
1 1
0 1
=
; T (z)= z + 1;
W e can de ne four Lie algebras s(R i; j),i;j = 1;2,of -localform al
distributions w ith generators A mn ,m 2 Z + ,n 2 Z ,and the generating
P
function A m (z)= n A mn z n 1.
(a) S(R ; 1):
[A m (z);A n (w )]
= qm A m + n (qm w ) (z
qm w ) A m + n (w ) (z
1
+ A m + n(w )
qm A m + n (qm w ) z +
w
For the coe cients A
[A mk ;A n‘ ]= (q
m
k
m‘
q nw )
!
qm + n
z+
w
we have the com m utation relations
q
nk
)A mk++‘n
( 1)k qm k (q
n‘
qnk)A m‘ +kn
T his is the C -series ofLie algebras ofquantum torus ([G -K -L]).
(b) s(R 2; 1):
[A m (z);A n (z)] = A m + n (qm w ) (z
qn w ) A m + n (z q n w )
qm
+ A n m (w )
q m A n m (q m w ) z +
w
for m ;n 2 Z ; n m
and
[A mk ;A n‘ ] = (q
m‘
q
k‘
)A mk++‘n
23
( 1)k (qm ‘
qnk)A ‘n
m
k
:
z+
qn
w
;
(c) S(R 1; 2):
[A m (z);A n (w )]= A m + n (w + m ) (z w m ) Am + n (w ) (z w + n)
A m + n (w + m ) (z + w )+ Am + n (w ) (z + w m n)
(5.14)
C onsider in PD i (S 1) w ith the basis (5.11) a subalgebra,stable
under the autom orphism w de ned by: w (xn D k)= xm (m D )k.
T his is a Lie algebra w ith the basis xm D k w (xm D k). Introduce
the generating function ofthe form :
C m (z)=
X
(x
m
Dk x
m
( m
D )k )z
k 1
= am (z)+ am (m
z);
k2 Z
w here am (z) are given by (5.12). It is easy to check, that the
elds C m (z) satisfy equation (5.14). W e w illcallthis subalgebra
a C -series ofPD i (S 1).
(d) S(R 2; 2) For generating functions we have:
[A m (z);A n (w )]= A m + n (w + m ) (z w
+ A n m (w m ) (z + w
m ) Am + n (z w + n)
m ) An m (z + w + n)
(5.15)
for m ;n 2 Z + ,and n m . T his subalgebra is called the B -series
ofPD i (S 1). It can be de ned as a subalgebra stable under the
second-order autom orphism
de ned by (xn D n )= x m ( D )n .
R elations (5.15) are exactly the relations on generating series
P
B m (z)= k2 Z (x m D k + (x m D k))z k 1 = am (z) a m ( z).
A cknow ledgm ents
T he research ofM .G .-K .wassupported in partby IN TA S grant942317 and
D utch N W O organization,by R FFIgrant96-0218046,and grant96-15-96455
forsupportofscienti c schools. M .G .-K .isalso very gratefulto the Institut
G irard D esarques,U R A 746,U niversite Lyon-1 for the kind hospitality and
support.
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