Lusztig limit of quantum sl(2) at root of unity and fusion of (1,p) Virasoro logarithmic minimal models

0.23 MB·English

by  P. V. Bushlanov

LUSZTIG LIMIT OF QUANTUM SL(2) AT ROOT OF UNITY AND FUSION
OF (1,P) VIRASORO LOGARITHMIC MINIMAL MODELS
P.V.BUSHLANOV,B.L.FEIGIN,A.M.GAINUTDINOVANDI.YU.TIPUNIN
ABSTRACT. We introduce a Kazhdan–Lusztig-dualquantum group for (1,p) Virasoro
9
0 logarithmicminimalmodelsastheLusztiglimitofthequantumsℓ(2)atpthrootofunity
0 and show that this limit is a Hopf algebra. We calculate tensor products of irreducible
2
andprojectiverepresentationsofthequantumgroupandshowthatthesetensorproducts

n coincide with the fusion of irreducible and logarithmic modules in the (1,p) Virasoro
a
J logarithmicminimalmodels.

2
1
1. INTRODUCTION

]
h
t Logarithmic conformal field theories most naturally appear as a scaling limit of two-
-
p dimensional nonlocal lattice models at a critical point [1] and in quantum chains with a
e
h nondiagonalizableHamiltonian[2]. Generally speaking,aconformal field theory appear-
[
ing at the limit depends on a way of taking the limit and on chosen boundary conditions.

1
Therecentinvestigations[3,4,5]arguethatforproperchoiceofboundaryconditionsthe
v
2 latticemodels[1] givein a scalinglimitlogarithmicconformalmodels (p,q) with
0 WLM
the triplet -algebra of symmetry introduced in [6] and in [7, 8] for q = 1. The chiral
6 p,q
W
1 algebra in these conformal models is an extension of the vacuum module of the Virasoro
.
1
algebra withthecentralchargec = 13 6p/q 6q/p bythetripletoftheVirasoro
0 Vp,q p,q − −
9 primaryfields withconformal dimension∆ .
1,3
0
: The most investigated models are those with q = 1. In this case, the conformal field
v
i theories (1,p)correspondingtothelatticemodelsweredescribedintermsofsym-
X
WLM
plectic fermions in [9] (for p = 2) and were studied in numerous papers [10, 11, 12, 8,
r
a
13, 14, 15, 16, 17, 18, 27, 28]. Forthis set ofmodels, the representation categories of the
triplet algebra and of the finite-dimensional quantum group U sℓ(2) [19] at the pth
p q
W
root of unity are equivalent as braided tensor categories [20]. This is the manifestation
of theKazhdan–Lusztigduality between vertex-operatoralgebras and quantumgroups in
logarithmicmodels. This duality means that (i) there is a one-to-one correspondence be-
tween representations; (ii) fusion rules of a conformal model can be calculated by tensor
productsofaquantumgrouprepresentationsand(iii)themodulargroupactiongenerated
from chiral characters coincides withtheoneon thecenterofthecorrespondingquantum
group. In thelogarithmicmodels (1,p), theKazhdan–Lusztigdualityis presented
WLM
in its full strength (see also review [21]). In particular, the fusion calculated in [8] (see
also[29]) coincideswiththeGrothendieck ringoftheU sℓ(2).
q
1
2 P.V.BUSHLANOV,B.L.FEIGIN,A.M.GAINUTDINOVANDI.YU.TIPUNIN
For general coprime p and q, the models (p,q) also demonstrate the Kazhdan–
WLM
Lusztigdualitywithaquantumgroup[6]butrelationbetweenthequantumgroupandthe
algebra is more subtle. There is no one-to-one correspondence between representa-
p,q
W
tionsbutthemodulargroupactiononthecenter[22]coincideswiththeoneonchiralchar-
acters inthe theoryand thequantum-groupfusion[6]coincideswiththefusion[23]
p,q
W
of representationsundertheidentificationofthefusiongenerators + (1,2) and
Wp,q K1,2→ W
+ (2,1) ; we also identify + (a,b) , (∆ ) 1. This fusion was
K2,1 → W Ka,b → W Ka−,b→ a,3q−b W
alsoderived[24]from themodulargroup propertiesofthechiral characters.
Otherchoiceof boundary conditionsin thelatticemodels [1]leads to logarithmiccon-
formal field models (p,q) with the Virasoro symmetry . Fusion rules for these
p,q
LM V
models were calculated in [25] using a lattice approach and for some cases in [26, 36]
usingtheNahmalgorithmandin[2]usingquantum-groupsymmetriesinXXZmodelsat
aroot ofunity.
In the present paper, we propose using the Kazhdan–Lusztig duality in calculating the
fusion rules for the simplest subset (1,p) of the (p,q) models. We construct
LM LM
a quantum group dual to the Virasoro algebra as an extension of the quantum group
p
V
U sℓ(2) dual to from the (1,p) models. This quantum group is the Lusztig
q p
W WLM
limitLU sℓ(2)oftheusualquantumsℓ(2)asq eıπ/p andhasthesetofirreduciblerep-
q
→
resentations Xα , where s = 1,2,...,p and α = are U sℓ(2) highest weight parame-
s,r ± q
tersand r 1,r N,isthesℓ(2)spin(seeprecisedefinitionsinSec.3.1). ThemoduleXα
−2 ∈ s,r
is a tensor product of s-dimensional irreducible U sℓ(2)- and r-dimensional irreducible
q
sℓ(2)-modules. ToeachXα ,aprojectivecoverPα correspondsandPα = Xα . Theset
s,r s,r p,r p,r
ofirreducibleand projectivemodulesisclosed undertensorproducts.
We show that the fusion [25] of irreducible and logarithmic -representations coin-
p
V
cides withtensor products ofLU sℓ(2)irreducibleand projectivemodules. To formulate
q
themainresultofthepaper, weintroducethefollowingsumnotations
b b
1 1
f(r) = (1 δ δ )f(r),
′ r,a r,b
− 2 − 2
r=a r=a
M M
b b
3 1 1 3
f(r) = 1 δ δ (1+δ ) δ δ f(r).
′′ r,a r,a+2 a, 1 r,b 2 r,b
− 4 − 4 − − 4 − − 4
r=a r=a
M M(cid:0) (cid:1)
1 Indecomposable rank-2 and rank-3 representations appearing in the fusion [23] are identified
with the corresponding quantum-group modules [22] in the following way: Pa+,b,+ → (Rpp−,ba,0)W,
Pa+,b,−→(R0a,,qq−b)W andP+a,b →(Rpp−,qa,q−b)W,P−a,b →(Rp2p−,aq,q−b)W.
LUSZTIGLIMITANDFUSION 3
1.1.Theorem. The tensorproductsbetween irreducibleLU sℓ(2)-modulesare
q
min(s1+s2 1,
−
r1+r2−1 2p−s1−s2−1) p−γ2
Xα Xβ = Xαβ + Pαβ
s1,r1 ⊗ s2,r2 s,r s,r
r=|srtM1e−p=r22|+1(cid:16) s=|sstM1e−p=s22|+1 s=2ps−tMesp1=−2s2+1 (cid:17)
between theirreducibleandprojectivemodulesare
min(s1+s2 1,
−
r1+r2−1 2p−s1−s2−1) p−γ2 r1+r2 p−γ1
Xα Pβ = Pαβ +2 Pαβ +2 P αβ,
s1,r1 ⊗ s2,r2 s,r s,r ′ −s,r
r=|srtM1e−p=r22|+1(cid:16)s=|sstM1e−p=s22|+1 s=2ps−tMesp1=−2s2+1 (cid:17) r=sMt|er1p−=r22|s=ps−tMes1p+=s22+1
andbetween theprojectivemodulesare
min(s1+s2 1,
−
r1+r2−1 2p−s1−s2−1) p−γ2
Pα Pβ = 2 Pαβ +2 Pαβ
s1,r1 ⊗ s2,r2 s,r s,r
r=|srMt1e−p=r22|+1(cid:16)s=|ssMt1e−p=s22|+1 s=2ps−Mtesp1=−2s2+1 (cid:17)
min(p s1+s2 1,
− −
r1+r2 p+s1−s2−1) p−γ1 r1+r2+1 p−γ2
+2 P αβ +2 P αβ +4 Pαβ,
′ −s,r −s,r ′′ s,r
r=sMt|erp1=−2r2|(cid:16)s=|p−sMtesp1=−2s2|+1 s=mpi+ns(pM1−ss12++s12)+1, (cid:17) r=|sMrt1e−p=r22|−1s=ssMt1e+p=s22+1
−
where weset γ = (s +s +1)mod 2, γ = (s +s +p+1)mod 2.
1 1 2 2 1 2
TheLU sℓ(2)representationcategoryC isadirectsumoftwofullsubcategoriesC =
q p p
C+ C suchthattherearenomorphismsbetweenC+ andC andthesubcategoryC+ is
p ⊕ −p p −p p
closedundertensorproducts. Thesetofirreduciblemodulesbelongingtothesubcategory
C+isexhaustedbytheirreduciblemodulesXα withα = +wheneverrisodd,andα =
p s,r −
wheneverr iseven.
The category C+ is equivalent as a tensor category to the category of Virasoro algebra
p
representations appearing in (1,p). We do not describe here this Virasoro category
LM
butnoteonlythatunderthisequivalenceirreducibleandprojectivemodulesareidentified
inthefollowingway
(1.1) X+p,2r−1 → R02r−1, X−p,2r → R02r, P+s,2r−1 → Rp2r−−s1, P−p−s,2r → Rs2r,
X+ (2r 1,s), X (2r,s), 16s6p, r>1,
s,2r 1 → − −s,2r →
−
where(r,s)aretheirreducibleVirasoromoduleswiththeheighestweights∆ = ((pr
r,s
−
s)2 (p 1)2)/4p and the s are logarithmic Virasoro modules from (1,p). Un-
− − Rr LM
der this identification, the fusion [25] for (1,p) is given by tensor products of the
LM
correspondingLU sℓ(2)representations.
q
The quantum groups dual to the logarithmic conformal models (1,p) as well as
LM
(1,p) can be constructed in the free field approach [19, 6]. In this approach, the
WLM
4 P.V.BUSHLANOV,B.L.FEIGIN,A.M.GAINUTDINOVANDI.YU.TIPUNIN
corresponding quantum groups and the chiral algebras are mutual maximal centralizers
of each other. To construct the quantum group for (1,p), we first note that the chi-
LM
ral algebras realized in the (1,p) models admit SL(2)-action by symmetries.
p
W WLM
Invariants of this action is the universal enveloping of the Virasoro algebra with the
p
V
central charge c = 13 6p 6/p. This suggests that a quantum group LU sℓ(2) dual
p q
− −
to the Virasoro algebra from (1,p) should be a combination of the quantum group
LM
U sℓ(2)dualto and ordinary sℓ(2).
q p
W
ThequantumgroupLU sℓ(2)isconstructedinthefree-fieldapproachusingthescreen-
q
ing operators of the Virasoro algebra with the central charge c . We recall there are
p p
V
two screening operators e = e√2pϕ(z)dz and F = e−qp2ϕ(z)dz commuting with p
V
and thescreeningF commutes[8]withtheextendedchiral algebra and generates the
H H Wp
lower-triangularpartoftheU sℓ(2)withtherelationFp = 0. Then,consideringthedefor-
q
mationFǫ = e(−qp2+ǫ)ϕ(z)dz,wecanconstructanoperatorf = lǫim0 Fǫǫp. Theoperatorse
and f generatethesℓ(2)from thepreviousparagraph. Toobtaina→Hopf-algebrastructure
H
on LU sℓ(2), weusehere thepurely algebraicapproach followingLusztig. Weconstruct
q
thequantumgroupLUqsℓ(2)asalimitofthequantumgroupUq(sℓ(2))asq eıpπ. There
→
is an evidentlimit in which Ep, Fp and Kp become central but we consider anotherlimit
inwhich therelationsEp = Fp = 0, K2p = 1 are imposedbut thegenerators e = Ep and
[p]!
f = Fp are kept in the limit. In the limit q eıpπ, we have [p]! = 0 and the ambiguity 0
[p]! → 0
issolvedinsuchawaythattheeandf becomegeneratorsoftheordinarysℓ(2). Wethus
obtainaHopfalgebraLU sℓ(2)thatcontainsthequantumgroupU sℓ(2)asaHopfideal
q q
and thequotientistheU(sℓ(2)), theuniversalenvelopingofthesℓ(2).
It is noteworthy to mention that a similar duality is also presented in quantum spin
chains (XXZ models) with nondiagonalizable action of the Hamiltonian. Here, there are
two commuting actions of a Temperley–Lieb algebra (or a proper its extension) and of a
quantumgroup. Inotherwords,thespaceofspinstatesisabimoduleovertheTemperley–
Lieb algebra (or its extension) and the corresponding quantum group [2]. Moreover, the
quantum group symmetries are stable with respect to increasing the number of sites and
shouldbekeptinascalinglimit;fusionrules forTemperley–Liebalgebrarepresentations
areobtainedbyaninductionprocedurejoiningtwochainsendtoendandcanbecalculated
using only the quantum group symmetries [2, 30]. Thus, these two algebraic objects are
insomedualitywhich isalatticeversionoftheKazhdan–Lusztigduality.
We also note that tensor products of LU sℓ(2) projective modules reproduce the Vira-
q
soro fusionrules proposedin [2]undertheidentifications:
p = 2 : P+ , P , r N,
1,2r−1 → R2r−1 −1,2r → R2r ∈
p = 3 : X+ , P+ , P , r N.
3,2r−1 → R3r−2 1,2r−1 → R3r−1 −2,2r → R3r ∈
LUSZTIGLIMITANDFUSION 5
Thepaperisorganizedasfollows. InSec.2,weintroducethequantumgroupLU sℓ(2)
q
dual to the Virasoro algebra and describe a Hopf algebra structure on LU sℓ(2). In
p q
V
Sec. 3, we describe irreducible representations of LU sℓ(2), calculate all possible ex-
q
tensions between them and then construct projective modules. In Sec. 4, we decompose
tensorproductsbetween irreducibleand projectiveLU sℓ(2)-modules.
q
2. QUANTUM GROUPS AS CENTRALIZERS OF VOAS.
Inthissection,weintroduceaquantumgroupthatcommuteswiththeVirasoroalgebra
onthechiral spaceofstatesinthefree masslessscalar field theory
p
V
ϕ(z)ϕ(w) = log(z w)
−
withtheenergy-momentumtensor
(2.1) T = 1∂ϕ∂ϕ+ α0∂2ϕ,
2 2
where the background charge α = α + α = √2p 2/p. This quantum group is
0 +
denotedasLU sℓ(2)andconstructedassome−extension−ofthefinite-dimensionalquantum
q p
group U sℓ(2) which is the maximal centralizer of the triplet algebra . We recall that
q p
W
isan extensionof by thesℓ(2)-tripletofthefields W ,0(z) [8]:
p p ±
W V
W (z) := e α+ϕ(z), W0(z) := [S ,W (z)], W+(z) := [S ,W0(z)],
− − + − +
where S is the “long screening” operator eα+ϕdz. These three fields are Virasoro
+
primariesandtheirconformaldimensionsequalto(2p 1). ThefieldW (z)isthelowest-
H − −
weight vector (with respect to the Cartan sℓ(2)-generator h = 1 ϕ and ϕ is the zero-
α+ 0 0
mode of ∂ϕ(z)), the field W0(z) has the h-weight equals to 0 and W+(z) is the highest-
weightvectorofthesℓ(2)-triplet.
Irreducible representations of the triplet algebra admit two commuting actions,
p
W
sℓ(2)- and -actions [6], and the Virasoro algebra is the invariant of the sℓ(2) ac-
p p
V V
tion in the vacuum representation of . This suggests a construction of the maximal
p
W
centralizer for as an extension of the centralizer U sℓ(2) for the triplet algebra by
p q p
V W
the sℓ(2) triplet: e = S , h = 1 ϕ , and a “conjugate” operator f to the long screening
+ α+ 0
S .
+
2.1. The centralizer of . We recall thedefinition ofthe quantumgroup U sℓ(2) [19]
p q
W
thatcommuteswiththetripletalgebra actiononthechiralspaceofstates. TheU sℓ(2)
p q
W
can be constructed as the Drinfeld double of the Hopf algebra generated by the “short
screening” operator F = eα−ϕ(z)dz and K = e iπα−ϕ0. The Hopf algebra structure is
−
found from the action of these operators on fields. In particular, the comultiplication is
H
calculated from the action of F and K on operator product expansions of fields. Details
ofconstructingU sℓ(2)aregivenin[19].
q
6 P.V.BUSHLANOV,B.L.FEIGIN,A.M.GAINUTDINOVANDI.YU.TIPUNIN
The quantum group U sℓ(2) is the “restricted” quantum sℓ(2) with q = eiπ/p and the
q
threegenerators E, F,and K satisfyingthestandard relationsforthequantumsℓ(2),
K K−1
(2.2) KEK−1 = q2E, KFK−1 = q−2F, [E,F] = q−q−1 ,
−
withsomeadditionalconstraints,
(2.3) Ep = Fp = 0, K2p = 1,
and theHopf-algebrastructureisgivenby
(2.4) ∆(E) = 1 E +E K, ∆(F) = K 1 F +F 1, ∆(K) = K K,
−
⊗ ⊗ ⊗ ⊗ ⊗
(2.5) S(E) = EK 1, S(F) = KF, S(K) = K 1,
− −
− −
(2.6) ǫ(E) = ǫ(F) = 0, ǫ(K) = 1.
The quantum group U sℓ(2) admits (p 1) dimensional family of inequivalent ex-
q
−
tensions by sℓ(2) algebras acting on U sℓ(2) as exterior derivatives [31]. There is an
q
extensionthat admitsaHopfalgebrastructuretobedefined in thefollowingsubsection.
2.2. The centralizer of . Here, we define a quantum group LU sℓ(2) (i.e. a Hopf
p q
V
algebra)that commuteswiththeVirasoroalgebra on thechiral spaceofstates.
p
V
2.2.1. Definition. TheHopf-algebrastructureonLU sℓ(2)isthefollowing. Thedefining
q
relationsbetweentheE,F,andK generatorsarethesameasinU sℓ(2)andgivenin(2.2)
q
and (2.3), andtheusualsℓ(2)relationsbetweenthee, f, andh:
(2.7) [h,e] = e, [h,f] = f, [e,f] = 2h,
−
and the“mixed”relations
(2.8) [h,K] = 0, [E,e] = 0, [K,e] = 0, [F,f] = 0, [K,f] = 0,
1 qK q−1K−1
(2.9) [F,e] = [p 1]!Kp q− q−1 Ep−1,
− −
( 1)p+1 qK q−1K−1
(2.10) [E,f] = [−p 1]! Fp−1 q− q−1 ,
− −
1 1
(2.11) [h,E] = EA, [h,F] = AF,
2 −2
where
p 1
(2.12) A = − (us(q−s−1)−us(qs−1))K+qs−1us(qs−1)−q−s−1us(q−s−1) u (K)e
(qs−1 q−s−1)u (q−s−1)u (qs−1) s s
s s
s=1 −
X
with us(K) = pn−=11,n=s(K − qs−1−2n), and es are the central primitive idempotents of
U sℓ(2)giveninApp. A6 .
q Q
LUSZTIGLIMITANDFUSION 7
ThecomultiplicationinLU sℓ(2)isgivenin (2.4) fortheE, F, andK generators and
q
p 1
(2.13) ∆(e) = e 1+Kp e+ 1 − qr(p−r)KpEp r ErK r,
− −
⊗ ⊗ [p 1]! [r] ⊗
− r=1
X
p 1
(2.14) ∆(f) = f 1+Kp f + (−1)p − q−s(p−s)Kp+sFs Fp−s,
⊗ ⊗ [p 1]! [s] ⊗
− s=1
X
an explicitform of∆(h) = 1[∆(e),∆(f)]isverybulkyand wedonotgiveithere.
2
TheantipodeS and thecounityǫ aregivenin(2.5)-(2.6)and
(2.15) S(e) = Kpe, S(f) = Kpf, S(h) = h,
− − −
(2.16) ǫ(e) = ǫ(f) = ǫ(h) = 0.
2.2.2. LU sℓ(2)throughdividedpowers. ThequantumgroupLU sℓ(2)canberealized
q q
astheLusztigextensionoftherestrictedquantumgroupU sℓ(2)bydividedpowersofthe
q
E and F generators. In the usual quantum U sℓ(2) with the relations (2.2), (2.4), (2.5),
q
and (2.6)andwithgeneric q,thee, f and hexistas thefollowingelements,
1 ( 1)p
e = KpEp, f = − Fp,
[p]! [p]!
and theCartan element
p 1
1 1 q2p q2p 1 ( qp)p−1K2p − ( 1)rqr(p−1)[p 1]!
h = [e,f] = − ef + − − + − − K2r +
2 2 2p(q q−1) [p] [p r]![r]!
− (cid:16) Xr=1 − (cid:17)
p 1 p n 1 n
( 1)p+1 − ( 1)n[p 1]! − − C q−(2k+1)K q(2k+1)K−1 qr−1K q−r+1K−1
+ − Kp − − − − − ,
2[p 1]! ([p n]!)2[n]! (q q−1)2 q q−1
− n=1 − k=0 − r=1 −
X Y Y
where the U sℓ(2) Casimir element C is given in App. A. The commutation relations
q
between these elements and their comultiplication, antipode, and counity satisfy (2.7)-
(2.16)whenq eiπ/p.
→
3. REPRESENTATIONS OF LU sℓ(2)
q
To describe the category C of finite-dimensional LU sℓ(2)-modules, we first study
p q
irreducibleLU sℓ(2)-modulesin3.1andthenobtainessentialinformationaboutpossible
q
extensions between them in 3.2. This let us construct all finite-dimensional projective
LU sℓ(2)-modulesin3.3. Thenin3.4,wedecomposetherepresentationcategoryC into
q p
a direct sum of two full subcategories C+ and C . The subcategory C+ is then identified
p −p p
withatensorcategoryoftheVirasoroalgebrarepresentations.
8 P.V.BUSHLANOV,B.L.FEIGIN,A.M.GAINUTDINOVANDI.YU.TIPUNIN
3.1. Irreducible LU sℓ(2)-modules. An irreducible LU sℓ(2)-module X is labeled
q q ±s,r
by ( ,s,r), with 16s6p and r N, and has the highest weights qs 1 and r 1 with
± ∈ ± − −2
respect to K and h generators, respectively. The sr-dimensional module X is spanned
±s,r
byelementsa ,06n6s 1,06m6r 1,wherea isthehighest-weightvectorand
±n,m − − ±0,0
theleftaction ofthealgebraon X is givenby
±s,r
(3.1) Ka = qs 1 2na , ha = 1(r 1 2m)a ,
±n,m ± − − ±n,m ±n,m 2 − − ±n,m
(3.2) Ea = [n][s n]a , ea = m(r m)a ,
±n,m ± − ±n−1,m ±n,m − ±n,m−1
(3.3) Fa = a , f a = a ,
±n,m ±n+1,m ±n,m ±n,m+1
whereweset a = a = a = a = 0.
±1,m ±n, 1 ±s,m ±n,r
− −
3.1.1.Remark. TheelementAdefinedin(2.12)and(2.11)isrepresentedinanirreducible
representation of LU sℓ(2) by an operator acting as the identity on the highest-weight
q
vector and zero on all other vectors. Therefore, as it follows from the relations (2.8)-
(2.11), the E, F, and K generators of the subalgebra U sℓ(2) commute on X with the
q ±s,r
e, f, andhgenerators ofthesubalgebrasℓ(2).
3.2. Extensions among irreducibles. Here, we study possible extensions between irre-
ducible LU sℓ(2)-modules to construct indecomposable modules in what follows. Let
q
A and C be left LU sℓ(2)-modules. We say that a short exact sequence of LU sℓ(2)-
q q
modules 0 A B C 0 is an extension of C by A, and we let Ext1 (C,A)
→ → → → LUq
denotetheset ofequivalenceclasses (see, e.g., [32])ofextensionsofC byA.
3.2.1. Lemma. For 16s6p 1, r N and α,α +, , there are vector-space
′
− ∈ ∈{ −}
isomorphisms
C, α = α, s = p s, r = r 1,
Ext1 (Xα ,Xα′ ) = ′ − ′ − ′ ±
LUq s,r s′,r′ ∼ (0, otherwise.
There arenonontrivialextensionsbetween X andanyirreduciblemodule.
±p,r
Proof. We first recall [20] that the space Ext1 of extensions between irreducible mod-
U
q
ules over the subalgebra U sℓ(2) is at most two-dimensionaland there exists a nontrivial
q
extension only between X and X , where 16s6p 1 and we set X =X .
±s ∓p−s − ±s ±s,1|Uqsℓ(2)
Moreover,thereisanactionofLU sℓ(2)onprojectiveresolutionsforirreducibleU sℓ(2)-
q q
modules and this generates an action of the quotient-algebra sℓ(2) on the correspond-
ing cochain complexes and their cohomologies. Therefore, for an irreducible X and an
U sℓ(2)-module M, all extension groups Ext (X,M) are sℓ(2)-modules. In particular,
q •U
q
the space Ext1 (X ,X ) is the sℓ(2)-doublet and there is a nontrivial sℓ(2) action on
U ±s ∓p s
q −
all HochschildcohomologiesofU sℓ(2)(see[31]).
q
LUSZTIGLIMITANDFUSION 9
Next,tocalculatethefirstextensiongroupsbetweentheirreducibleLU sℓ(2)-modules,
q
weusetheSerre-HochschildspectralsequencewithrespecttothesubalgebraU sℓ(2)and
q
the quotient-algebrasℓ(2). The spectral sequence is degenerate at thesecond term due to
thesemisimplicityofthequotientalgebra andwethusobtain
Ext1 (Xα ,Xα′ ) = H0(sℓ(2),Ext1 (Xα ,Xα′ )),
LU s,r s′,r′ U s,r s′,r′
q q
where the right-hand side is the vector space of the sℓ(2)-invariants in the sℓ(2)-module
Ext1 (Xα ,Xα′ ). This module is nonzero only in the case α = α, s = p s and
Uq s,r s′,r′ ′ − ′ −
isomorphic to the tensor product X X X of the sℓ(2) modules, where X is the
2 r r′ r
⊗ ⊗
r-dimensionalmodule. Obviously,thetensorproductcontainsatrivialsℓ(2)-moduleonly
inthecaser = r 1. Thiscompletestheproof. (cid:3)
′
±
3.2.2. Example. As an example, we describe an extension of X by X . This
±s,r ∓p s,r+1
can be realized as an extension of r copies X by (r +1) copies X of th−e irreducible
±s ∓p s
modulesoverthesubalgebraU sℓ(2), −
q
X± f // X± f // ... f // X±
s s s
◦ ◦ ◦
E F E F E F E F
zz ## {{ ## {{ ## {{ $$
X∓ // X∓ // X∓ // ... // X∓ // X∓
p−s p−s p−s p−s p−s
f f f f f
• • • • •
with the indicated action of the E and F generators mixing X with X modules, and
±s ∓p s
−
with the e and f generators mapping different copies with the same sign. The extension
thusconstructedcan bedepicted as
X±
s,r
◦
(cid:28)(cid:28)
X∓
p−s,r+1
•
withtheconventionthatthearrowisdirectedtothesubmoduleinthebottommarkedby ,
•
incontrast tothesubquotient in thetop.
◦
3.3. Projective LU sℓ(2)-modules. We next construct projective LU sℓ(2)-modules as
q q
projectivecoversofirreduciblemodules. Aprojectivecoverofan irreduciblemoduleisa
“maximal” indecomposable module that can be mapped onto the irreducible. Lem. 3.2.1
state that we can “glue” two irreducible modules into an indecomposablemoduleonly in
the case ifthe irreducibles have oppositesigns ofthe α-index, thedifference between the
two r-indexes equals to one and the sum of the two s-indexes is equal to p. Therefore,
to construct a projective cover for X , 16s6p 1 and r>2, we first have to obtain a
±s,r −
nontrivialextensionofX bythemaximalnumberofirreduciblemodules,that is,by
±s,r
X ⊠Ext1 (X ,X ) X ⊠Ext1 (X ,X ),
∓p−s,r−1 LUq ±s,r ∓p−s,r−1 ⊕ ∓p−s,r+1 LUq ±s,r ∓p−s,r+1
10 P.V.BUSHLANOV,B.L.FEIGIN,A.M.GAINUTDINOVANDI.YU.TIPUNIN
whichis
0 X X M X 0.
→ ∓p−s,r−1⊕ ∓p−s,r+1 → ±s,r → ±s,r →
where M is an indecomposable module. Next, to find the projective cover of X , we
±s,r ±s,r
extendthesubmoduleX X M bythemaximalnumberofirreducible
∓p s,r 1⊕ ∓p s,r+1 ⊂ ±s,r
modules,thatis,byX − −2X X− . ThecompatibilitywiththeLU sℓ(2)-algebra
±s,r−2⊕ ±s,r⊕ ±s,r+2 q
relations (with Fp=Ep=0 and (2.11) in particular) leads to an extension corresponding
tothemoduleP withthefollowingsubquotientstructure:
±s,r
(3.4) X±s,r
•
zz $$
X∓ X∓
p−s,r−1 p−s,r+1
◦ ◦
(cid:28)(cid:28) (cid:2)(cid:2)
X±
s,r
•
and the LU sℓ(2) action is explicitly described in App. B. This module being restricted
q
tothesubalgebraU sℓ(2)isadirect sumofprojectivemodulesthatcoversthedirect sum
q
r X , where we set X = X . Therefore, the P module is the projective
⊕i=1 ±s ±s ±s,1|Uqsℓ(2) ±s,r
cover of X , for 16s6p 1 and r>2. Similar procedure gives the projective cover
±s,r −
P for theirreduciblemoduleX withthefollowingsubquotientstructure:
±s,1 ±s,1
(3.5) X±s,1
•
(cid:15)(cid:15)
X∓
p−s,2
◦
(cid:15)(cid:15)
X±
s,1
•
and theLU sℓ(2)action isalsoexplicitlydescribed inApp. B.
q
A “half” of these projective modules is then identified in the fusion algebra calculated
belowinSec. 4 withsomelogarithmicVirasororepresentations.
3.3.1. Remark. We note there are no additional parameters distinguishing nonisomor-
phic indecomposable LU sℓ(2)-modules with the same subquotient structure as in (3.4)
q
and (3.5). This trivially follows from the fact that all infinitesimal deformations of ho-
momorphisms f : LU sℓ(2) End(P ) continuing infinitesimal deformations of the
q → ±s,r
algebra End(P ) are in one-to-one correspondence with elements in the cohomology
±s,r
space H1(LU sℓ(2),End(P )). Since the module End(P ) is a direct sum of projec-
q ±s,r ±s,r
tivemodulesthenthecohomologiesspaceH1 istrivial.