Canonical bases and quantum coordinate algebras Journal of Algebra

(1)

Contents lists available atScienceDirect

Journal of Algebra

www.elsevier.com/locate/jalgebra

Canonical bases and quantum coordinate algebras

Bin Li^a,∗, Hechun Zhang^b

aSchoolofMathematicsandStatistics,WuhanUniversity,PRChina

bDepartmentofMathematicalScience,TsinghuaUniversity,PRChina

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received30January2012 Availableonline13June2014 CommunicatedbyGusI.Lehrer

Keywords:

Canonicalbasis Crystalbasis

Quantumcoordinatealgebra

Some filtrations of the tensor product of a highest weight moduleandalowestweightmoduleoverquantumgroupUq(g) areconstructedin[1]andonecanusethemtodefineatwo- sided idealof themodifiedquantized enveloping algebra.It isshownthatthequotientalgebra inheritsacanonicalbasis fromthemodifiedquantizedenveloping algebra andisdual tothequantumcoordinatealgebradefinedbyKashiwarafor asymmetrizableKac–Moodyalgebrag.

1. Introduction

A quantum coordinate algebra, or a quantum function algebra, is the q-deformed version of the coordinate algebra associated with a Lie group G. It canbe viewed, in somesense,asanalgebradualtothequantized envelopingalgebraU =Uq(g) andthus itisnaturalto studyitsstructureand representationsaswellas itsintegralform.

There are various ways to deﬁne the quantum coordinate algebraC. For any Kac–

MoodyalgebragwithasymmetrizablegeneralizedCartanmatrix,M.Kashiwaradeﬁned in[2] C as the algebragenerated by all coordinate functions of the U-modulesin the categoryOint andmoreover,there isananalogueofPeter–Weyltheorem

* Correspondingauthor.

E-mailaddresses:[email protected](B. Li),[email protected](H. Zhang).

(2)

C∼=

λ∈P⁺

V(λ)⊗V(λ)^◦

where V(λ) istheirreducibleintegrablehighestweightU-modulewithhighestweightλ and V(λ)^◦ isitsgradeddual.Inparticular, forgoffinitetype,Lusztig[3]gaveanother equivalent definition ofthe quantumcoordinate algebra.In[3], canonicalbases for the tensor productsV(λ)⊗V(−μ) aswell asforthemodifiedquantizedenvelopingalgebra U areconstructed,whereV(λ) andV(−μ) areirreducibleintegrablehighestandlowest weight modules over U with highest weight vector uλ and lowest weight vector u₋μ

respectively.Itis knownthatthereisasurjectivelinearmap ϕ_λ,−μ:U −→V(λ)⊗V(−μ)

whichtakes u∈U tou(u_λ⊗u_−μ).BytakingtheimageIm(ϕ^∗_λ,−μ) ofthedualmap ϕ^∗_λ,₋_μ:

V(λ)⊗V(−μ)∗

−→U^∗, Lusztigdeﬁnedthequantumcoordinatealgebraas

λ,μIm(ϕ^∗_λ,−μ),inwhichthemulti- plicationisdefinedthroughthecomultiplicationinU.Apartfromthis,anotherapproach to define the quantum coordinate algebra for g of finite type was also presented by Lusztig [4]. HeconsideredthesubspaceU^◦ ofthedual spaceofU spanned bythedual basisof thecanonicalbasisofU in[4]. ThemultiplicationinU^◦ isdefinedthroughthe comultiplication inU tomake itbecomeanassociativealgebrawhichisprovedlaterto be isomorphictothequantum coordinatealgebra.Inthis way,theintegralformof this algebraisnaturallydefined[4].Inthepresentpaper,wewill followLusztig’sapproachs to definethequantumcoordinatealgebraforanysymmetrizableKac–Moodyalgebrag.

In [3] Lusztig conjectured thatfor gof finite type, there is a composition series of V(λ)⊗V(−μ) compatible with the canonical basis. In [5], Lusztig gave an inductive method toconstruct thecomposition seriesof any integrablemoduleincategory Oint. A different approach to construct the composition series is given in [1] based on the theory of crystal basis. With this method we can also construct a nice filtration of V(λ)⊗V(−μ) for gof any type suchthat thequotient of any two neighborsis either zerooranirreducibleintegrablehighestweightmodule.Usingthesefiltrations,wedefine a subspace U of U spanned by all canonical base elements G(b) such that b ∈ B is contained in aconnected component not isomorphic to a highest weight crystal. It is provedthatUisatwo-sidedidealofU.ThequotientUU / Uisanassociativealgebra which inheritsfrom U acanonicalbasis.Let Utaketheplace ofU andthen wedefine, similar to whatLusztig did in[4], analgebra which is provedto be isomorphic to the quantum coordinatealgebra.Besides,following[3],aquotientmoduleofV(λ)⊗V(−μ) isconstructedandusedtodefine thequantumcoordinatealgebra.

Thequantumcoordinatealgebraconsideredinthispaperinvolvesonlyintegrablerep- resentationsincategoryOint.Thusitisexactlythealgebraofstronglyregularfunctions

(3)

onsymmetrizableKac–Moodygroup[6]whenq= 1.ItwouldbeinterestingifOintisre- placedbysomelargercategories.Namely,iftherearemoregeneratorsbesidescoordinate functionsofhighestweightmodules,say,thoseoflowestweightmodules,thegenerated subalgebra of U^∗ will be much more interesting. For g of aﬃne type, the structure of levelzeropartof U wasstudiedbyBeck andNakajima [7].Theauthors ofthe present paper believethattheirwork ishelpful to understandthis coordinatealgebraof aﬃne typethoughthis isnotincludedinthepresentpaper.

Thepaper is organizedas follows. InSection2, werecall somedefinitionsand facts aboutthecrystalandcanonicalbasesofU. Inparticular,theactionofCartaninvolution oncanonicalbasisofU isstudiedthroughthebilinearform onU.InSections 3and4, somenicefiltrationsofthetensorproductV(λ)⊗V(−μ) areconstructed.Wethendefine UtobethequotientofU andinvestigateitscellmodulesasin[5].Inthelastsection,an algebradualtoUisdefinedandprovedtobeisomorphictoquantumcoordinatealgebra.

2. Preliminaries

2.1. Modiﬁedquantizedenveloping algebra U

Wedenote byg =g(A) anysymmetrizable Kac–Moody algebra associated with an n×ngeneralizedCartanmatrixA.ThesetofsimplerootsisindexedbyI={1,· · ·,n}. LetQbe itsroot lattice,i.e. Q=

i∈IZαi ⊂hwhere his theCartansubalgebra and αi are the simple roots. Let Π^∨ = {hi ∈ h | i ∈ I} be the set of simple coroots. We choosed_j ∈h, 1jn-rank(A) suchthatΠ^∨

{d_j ∈h|1jn-rank(A)}formsa basisofh.Set

P^∨=

i∈I

Zh_i

⊕

1jn-rank(A)

Zd_j

⊂h.

TheweightlatticeP isdeﬁnedasP ={λ∈h^∗|λ(h)∈Z∀h∈P^∨}.LetQ⁺andP⁺be thepositive rootlattice and the set ofdominant weights respectively. Wedeﬁne P0 to bethesubsetof P⁺ consistingofweights μsuchthatμ(hi)= 0for alli∈I.LetW be theWeylgroupassociated withg.There isaW-invariantsymmetricbilinearform (,) onP×P suchthat

2(λ, α_i)

(αi, αi) =λ(h_i).

LetUq(g) bethequantizedenvelopingalgebrageneratedoverk=Q(q) byEi,Fiandq^h fori∈I,h∈P^∨[8],whichisdenotedalsobyU forsimplicity.ThesubalgebrasU⁺,U⁻ andtheintegralformU_Zare deﬁnedinthesamewayas in[8].Forξ=

i∈In_iα_i∈Q, deﬁne theheight ofξ to be

i∈I|n_i|, denoted byht(ξ).Set U_ξ ={u∈ U | q^huq⁻^h = q^ξ(h)u}.TheﬁltrationF = (Fn)n∈Z+ ofU^± isdeﬁnedby

(4)

Fn

U^±

=

ht(ξ)n

U_ξ^±.

Denote by Uq(g) or simply U the modiﬁed quantized enveloping algebragenerated by U_q(g)a_λforallλ∈P subjecttotherelations:

q^ha_λ=q^λ(h)a_λ, a_λa_μ=δ_λ,μa_λ, ua_λ=a_λ+ξu foru∈U_ξ. Note thatU =

λ∈PU aλ.TheintegralformofU isdeﬁnedas U_Z

λ∈P

U_Zaλ.

There is ananti-automorphism (resp. automorphism) of U, denotedby ∗(resp. ω), suchthat

E^∗_i =E_i, F_i^∗=F_i, q^h_∗

=q⁻^h resp.ω(Ei) =Fi, ω(Fi) =Ei, ω

q^h

=q^−h .

One can see that ∗and ω can be extended to involutions onU, denoted by the same symbols, with(aλ)^∗=ω(aλ)=a₋λ.

2.2. Crystalbasisandcanonicalbasisof U

Forλ,μ∈P⁺,letV(λ)⊗V(−μ) bethetensorproductofirreducibleintegrablehighest weight U-module V(λ) of highest weight λ with irreducible integrable lowest weight U-moduleV(−μ) oflowestweight−μ.Notethatitis,bythecomultiplicationinU,also aU-moduleandwedenoteitalsobyV(λ,−μ).Letuλ(resp.u₋μ)be thehighest(resp.

lowest)weightvectorofV(λ) (resp.V(−μ))andsetuλ,−μ=uλ⊗u₋μ∈V(λ,−μ).Itis knownin[3,8]thatV(λ,−μ) isacyclicU-modulegeneratedbyu_λ,−μ andthatitadmits acrystalbasis

B(λ,−μ)B(λ)⊗B(−μ)

where B(λ) and B(−μ) are highest and lowestweight crystalsrespectively.The corresponding globalbasisof V(λ,−μ) is constructedin[3]andfollowing Lusztig,we callit canonicalbasis,whichisdenotedby{G(b)|b∈B(λ,−μ)}.

Deﬁnition 2.1.

(i) For a U-module M with a canonical basis, a subspace N of M is called nice or compatible with the canonical basis if N is spanned over k by a subset of the canonicalbasisofM.

(5)

(ii) For U-modules M and N with canonical bases, a homomorphism of U-modules φ : M −→ N is called nice or compatible with the canonical bases if it maps a canonical base element of M to that of N or to zero and if the images of two distinctcanonicalbaseelementsaredistinct whentheyarebothnonzero.

(iii) ForaU-moduleM with acanonicalbasis, aﬁltrationor compositionseries of M is called nice or compatiblewith thecanonicalbasis ifany submoduleinvolvedin theﬁltrationorcompositionseriesisnice.

In [3], the following stability property plays a key role in the construction of the canonicalbasisofU.

Proposition 2.2. (See [3].) For λ,μ,θ ∈ P⁺, the map πλ,μ,θ : V(λ+θ,−θ−μ) −→

V(λ,−μ)whichtakesxuλ+θ,−θ−μ toxuλ,−μforallx∈U isanicesurjectivehomomor- phism ofU-modules.

We see from the proposition that there is an embedding of crystals B(λ,−μ) → B(λ+θ,−θ−μ) andnotethatitisstrict[8].Forλ,μ∈P⁺,letΦ:U a_λ−μ−→V(λ,−μ) be the U-module homomorphism taking aλ−μ to uλ,−μ. It is known thatU as well as each U a_λ have canonical bases and Φ is a nice surjective U-module homomorphism.

Wedenote thecrystalbasisof U (resp.U aλ)byB (resp.B(U aλ)). Hencewe havethe embeddingofcrystals

B(λ,−μ)→B(U a_λ−μ).

Itcanbeviewedas B(λ,−μ)⊆B(λ+θ,−θ−μ)⊆B(U a_λ−μ)⊆B. Note thatB(U a_λ) canbewrittenasB(∞)⊗Tλ⊗B(−∞) whereB(±∞) isthecrystalbasisofU^∓andTλ

is acrystalconsisting of asingle element t_λ. Forb ∈B(λ,−μ)⊆B, wedenote by the sameG(b) thecorrespondingcanonicalbase elementinV(λ,−μ) orU ifthis causesno confusion.Itisknownthattheanti-automorphism∗induces abijectiononB suchthat (b₁⊗t_λ⊗b₂)^∗=b^∗₁⊗t₋_λ₋_wt(b₁₎₋_wt(b₂₎⊗b^∗₂ and G(b)^∗=G(b^∗)[8].

For any λ ∈ P, there is an extremal weight U-module V^max(λ) which admits a crystal basis B^max(λ) consisting of all b ∈ B(U a_λ) such that b^∗ is extremal [8]. We have V^max(λ) ∼= V^max(wλ) for any w ∈ W and V^max(λ) ∼= V(λ) if λ ∈ ±P⁺. It is also known that for any connected component B of B, there is an l > 0 such that (wt(b),wt(b)) l for all b ∈ B. Moreover, B contains an extremal vector and canbe embeddedintoB^max(μ) forsomeμ∈P [8].

Forgof aﬃnetype,letc∈hbethecanonicalcentralelement ofg. Givenλ∈P, we deﬁnethelevelofλtobetheintegerλ(c),denotedbylevel(λ).Sinceanintegralweight λofpositive(resp.negative) levelis W-conjugateto adominant(resp.anti-dominant) weight,itfollowsfrom thepreviousparagraphthatB(U a_λ) isaunionofhighest(resp.

lowest)weightcrystals.

(6)

Denote by S^# the cardinality of the set S. Given two crystalsB1 and B2 with B1

connected, denoteby[B2:B1] thecardinalityofthesetwhichconsists ofallconnected componentsofB2isomorphicto B1,i.e.

[B2:B1] ={B⊂B2|B∼=B1}^#. Thefollowing resultwasprovedin[1].

Proposition 2.3. Forλ∈P⁺ andμ∈P,[B(U a_μ):B(λ)]= dimV(λ)_μ. 2.3. Bilinearformon U

Weintroduceanotheranti-automorphismofU,denotedbyΨ [8],suchthat Ψ(Ei) =q_i⁻¹t⁻_i ¹Fi, Ψ(Fi) =q_i⁻¹tiEi, Ψ

q^h

=q^h, Ψ(q) =q,

whereq_i=q⁽^αi,αi² ⁾ andt_i=q⁽^αi,αi² ⁾^hⁱ.OnecaneasilycheckthatΨ²=idandΨcommutes with theautomorphism ω introducedabove.

Forλ∈ ±P⁺,thereisauniquenon-degeneratesymmetricbilinearform(,) onV(λ) suchthat

(uλ, uλ) = 1 and (su, v) =

u, Ψ(s)v

for alls∈U, u, v∈V(λ).

Thebilinear formonV(λ) and thatonV(−λ) arerelatedbythefollowingequation (su_λ, tu_λ) =

ω(s)u_−λ, ω(t)u_−λ

. (2.1)

Given λ,μ∈P⁺, wedeﬁneasymmetric bilinearform(,) onV(λ,−μ) by (u₁⊗v₁, u₂⊗v₂) = (u₁, u₂)(v₁, v₂).

Since Ψ commutes with the comultiplication Δ, i.e. (Ψ ⊗Ψ)Δ = ΔΨ, it implies that (su,v)= (u,Ψ(s)v) foralls∈U,u,v∈V(λ,−μ).

Lemma 2.4.(See[8].)Fors,t∈U andθ∈P,thereexistsauniquepolynomial f_s,t,θ(x) in x= (x_i)_i∈I suchthat for any λ,μ∈P⁺ with λ−μ=θ,(su_λ,−μ,tu_λ,−μ)=f_s,t,θ(x) with xi=q_i^λ(hⁱ⁾.

Thebilinearform onU aθ isthendeﬁnedby(saθ,taθ)=fs,t,θ(0) andthisextendsto a bilinear form on U such that(U aθ1,U aθ2)= 0 for θ1 =θ2. It was shownin[8] that (,) onU issymmetricanditsatisﬁes

(u, v) = u^∗, v^∗

and (su, v) =

u, Ψ(s)v

for alls∈U, u, v ∈U .

(7)

LetA0bethesubringofkconsistingofallrationalfunctionsregularatq= 0.Thecrystal latticeL(U) overA0andthecanonicalbasisofU arecharacterizedbythebilinearform.

Proposition2.5. (See[8].)

(i) L(U)={u∈U |(u,u)∈A0}.

(ii) Ifu∈U_Zand(u,u)∈1+qA0,thenu≡G(b)or−G(b)modqL(U)forsomeb∈B.

Wedeﬁneanotherbilinearform ((,)) onU by (u, v)

=

ω(u), ω(v)

for allu, v∈U . Proposition2.6. ((u,v))= (u,v)forallu,v∈U.

Proof. Assume that u = saθ, v = taθ where s,t ∈ U, θ ∈ P. Let Ψ(s)t =

jx⁺_jx⁻_j wherex^±_j ∈U^±⊗k[q^h:h∈P^∨].Wehave

(saθ, taθ)

=

ω(s)a₋θ, ω(t)a₋θ

= a₋θ, Ψ

ω(s) ω(t)a₋θ

= a₋θ, ω

Ψ(s)t a₋θ

=

j

a₋θ, ω x⁺_jx⁻_j

a₋θ

=

j

ω Ψ

x⁺_j a_−θ, ω

x⁻_j a_−θ

=

j

Ψ x⁺_j

a_θ, x⁻_ja_θ

Sinceq^haθ=q^θ(h)aθ,itissuﬃcienttoshowtheequalitywhens,t∈U⁻.Givenλ,μ∈P⁺ suchthatλ−μ=θ,

(suλ,−μ, tuλ,−μ) = (suλ⊗u₋μ, tuλ⊗u₋μ) = (suλ, tuλ) =fs,t,θ(x) Meanwhilewehave,

ω(s)u_μ,−λ, ω(t)u_μ,−λ

=

u_μ⊗ω(s)u_−λ, u_μ⊗ω(t)u_−λ

=

ω(s)u₋λ, ω(t)u₋λ

= (suλ, tuλ) =fs,t,θ(x).

Hence((sa_θ,ta_θ))=f_s,t,θ(0)= (sa_θ,ta_θ). 2 Corollary2.7.

(i) ω(L(U))=L(U).

(ii) ω(B) =B.

(iii) ω(G(b))=G(ω(b))forallb∈B.

(8)

Proof. SimilartoKashiwara’sproofof theproperty oftheanti-automorphism∗[8],we onlyshow (ii)here.Given b=b1⊗tλ⊗b2∈B withht(wt(bi))=li,i= 1,2,wehave

G(b)≡G(b₁)G(b₂)a_λ modF_l₁₋₁ U⁻

F_l₂₋₁ U⁺

a_λ. (2.2)

Since ω : U^± −→ U^∓ induces ω : B(∓∞) −→ B(±∞) and it maps canonical base elements ofU^± tothose ofU^∓,applyingω to(2.2)wehave

ω G(b)

≡ω G(b1)

ω G(b2)

a₋λ=G ω(b1)

G ω(b2)

a₋λ

≡G ω(b2)

G ω(b1)

a₋λ modFl2−1

U⁻ Fl1−1

U⁺ a₋λ.

Sinceω(G(b))=G(b) or−G(b) forsomeb∈B,weobtainthatb=ω(b2)⊗t₋λ⊗ω(b1) and ω(G(b))=G(b). 2

3. AquotientalgebraofU

Throughout thissection,apairofdominantweights(λ,μ) isﬁxed.

3.1. Filtration

Inthissubsection,werecalltheconstructionofsomeniceﬁltrationsoftheU-module V(λ,−μ) in[1]. Inorderto obtainnicesubmodules ofV(λ,−μ),we needthefollowing lemma dueto Kashiwara[9]whoproveditincaseofU =Uq(sl2).Seealso[1]formore detailsoftheproofingeneralcase.

Lemma 3.1.

(i) LetMbeanintegrableU-modulewithacanonicalbasis.IfN isaniceU⁺-submodule of M, then U N = U⁻N is a nice U-submodule of M. More precisely, U N =

b∈B(U N)⊆B(M)kG(b).

(ii) B(U N)={f˜i1· · ·f˜imb|m0,i1,· · ·,im∈I,b∈B(N)}\ {0}.

There is a total order < on the lowest weight crystal B(−μ) such that b₁ < b₂ if wt(b₁)< wt(b₂). See[1] for theexistence of thetotal order and we note thatit is not unique.Forb∈B(−μ),onecandeﬁneasubspaceV_b(−μ) ofV(−μ) as

Vb(−μ)

cb

kG(c)

whichiseasilyshowntobeaU⁺-submodule.Henceu_λ⊗V_b(−μ) isaU⁺-submoduleof V(λ,−μ) whichhasabasis{u_λ⊗G(c)|cb, c∈B(−μ)}.Sinceu_λ⊗G(c)=G(u_λ⊗c), uλ⊗Vb(−μ) is actually a nice U⁺-submodule. Deﬁne Fλ(b) to be a U-submodule of V(λ,−μ) generatedbyuλ⊗Vb(−μ),i.e.

(9)

Fλ(b) =U

uλ⊗Vb(−μ) .

Itfollowsfrom Lemma 3.1thatF_λ(b) isaniceU-submoduleofV(λ,−μ) and B

Fλ(b)

=f˜i1· · ·f˜im(uλ⊗c)i1,· · ·, im∈I, c∈B(−μ), cb

\ {0}. Moreover,bycomparingthecrystalbasis,wehavethefollowing result.

Theorem3.2. (See[1].)Fortwoneighbors b< c∈B(−μ),Fλ(b)/Fλ(c)∼=V(λ+wt(b)) ife˜i(uλ⊗b)= 0 foralli∈I,otherwise Fλ(b)=Fλ(c).

HencewegetanicedescendingﬁltrationofV(λ,−μ)

V(λ,−μ) =F_λ(b₁)⊇F_λ(b₂)⊇F_λ(b₃)⊇ · · · (3.1) whereu_−μ=b₁< b₂< b₃<· · ·isacompletelistofB(−μ).

Remark 3.3. Similarly one can also deﬁne a total order on B(λ) such that b₁ < b₂ if wt(b₁)< wt(b₂).Set

F⁻^μ(b)

cb

U

G(c)⊗u_−μ

withwhichwecanalsoconstructaniceﬁltrationofV(λ,−μ) wherethequotientoftwo neighborsis isomorphiceithertoanirreducible lowestweightmoduleorto 0.

LetW(λ,−μ) beasubspaceofV(λ,−μ) deﬁnedby W(λ,−μ)

b∈B(−μ)

F_λ(b),

andlet

M(λ,−μ)V(λ,−μ)/W(λ,−μ). (3.2) Denote by B (resp. B(λ,−μ))the subcrystal of B (resp.B(λ,−μ)) which is aunion ofallconnectedcomponentsofB (resp.B(λ,−μ))thatarenothighestweightcrystals.

Wehavethefollowingpropositionin[1].

Proposition3.4.

(i) W(λ,−μ)isa niceU-submoduleof V(λ,−μ)andB(W(λ,−μ))=B(λ,−μ).

(ii) M(λ,−μ)admits acanonicalbasisandB(M(λ,−μ))=B(λ,−μ)\B(λ,−μ).

(10)

Remark3.5.OnecanseethatU(λ,−μ)

b∈B(λ)F⁻^μ(b) hasacrystalbasisB(λ,−μ) as well as a canonical basis where B(λ,−μ) consists of all connected components of B(λ,−μ) thatarenotlowestweightcrystals.SimilarlyN(λ,−μ)V(λ,−μ)/U(λ,−μ) admits acanonicalbasis.

Note that when gis of finite type, V(λ,−μ) is finite dimensional. Hence there are finitely manyterms inthe filtration(3.1) and furthermore, we canobtain anice composition seriesofV(λ,−μ)[1]bydeletingthesuperfluoustermsin(3.1)whichprovides a complete proof tothe conjecture raised byLusztig [3]. Moreover, W(λ,−μ)= 0 and M(λ,−μ)=V(λ,−μ) inthiscase.Butwhengisofaffineorindefinitetype,thesituation is quitedifferent. Forgofaffinetype,thefollowing resultwasshownin[1].

Proposition 3.6.

(i) W(λ,−μ) = N(λ,−μ) = 0 and M(λ,−μ) = U(λ,−μ) = V(λ,−μ) if level(λ−μ)>0.

(ii) W(λ,−μ) = N(λ,−μ) = V(λ,−μ) and M(λ,−μ) = U(λ,−μ) = 0 if level(λ−μ)<0.

(iii) M(λ,−μ)=N(λ,−μ) is a1-dimensional trivialmodule if λ−μ ∈P0,otherwise if λ−μ ∈/ P0 is of level 0, W(λ,−μ) = U(λ,−μ) = V(λ,−μ) and M(λ,−μ) = N(λ,−μ)= 0.

3.2. U

We denote by O⁺ (resp. O⁻) the completely reducible category whose objects are directsumsofirreducibleintegrablehighest(resp.lowest)weightU-modules.Notethat O⁺ hereisoftenreferredtoas Oint intheliteratures.

Theorem 3.7.Forb∈B, thefollowingconditions areequivalent.

(i) G(b)acts onV(λ)aszero forallλ∈P⁺. (ii) G(b) actsonM aszeroforany M∈ob(O⁺).

(iii) b∈B.

Proof. Theequivalence of (i)and (ii) isclear. If b satisfies (ii), weshow thatit satisfies (iii). Otherwise assume thatb ∈/ B, b is contained in ahighest weight subcrystal of B. Thereexist λ,μ∈P⁺ suchthatb∈B(λ,−μ)⊂B. Werewritethenicefiltration (3.1)ofV(λ,−μ) as

V(λ,−μ) =F₀⊇F₁⊇ · · · ⊇F_l⊇ · · ·. (3.3) There existsans0 such thatG(b)∈FsbutG(b)∈/Fs+1.Hence

(11)

0=G(b)(u_λ,−μ+Fs+1)∈V(λ,−μ)/Fs+1

where V(λ,−μ)/Fs+1 is an object in O⁺. This contradicts (ii). Finally we show that (iii)implies(i).AssumethatG(b)V(λ)= 0 forsomeλ∈P⁺,thenthereexistsanm∈ V(λ)_ξsuchthatG(b)m= 0 andb∈B(U a_ξ)⊂B. Wecanﬁndλ,μ∈P⁺withλ−μ=ξ such that b ∈ B(λ,−μ) ⊂ B(U a_ξ) and there exists a homomorphism of U-modules φ: V(λ,−μ)−→V(λ) which takes u_λ,−μ to m. Since b ∈B, G(b)∈W(λ,−μ), that is,G(b)∈Fs forany Fsintheﬁltration(3.3).Restricting φonFs,we getaU-module homomorphism

φ|Fs:F_s−→V λ

.

SincethesetofgeneratorsofFsisoftheformuλ⊗Vb(−μ) forsomeb,thecorresponding weights ofthese generators arenotlower than or equalto λ for asuﬃcientlarges by the construction. Hence φ|Fs is zero for s 0. It follows that φ(G(b)) = G(b)m = 0 whichisacontradiction. 2

Asisknownin[10,7.1.9],ifu∈U actsoneachM ∈ob(O⁺) as zero,thenu= 0 for gofany type.Buttheabovetheoremtellsusthatu∈U annihilatingallobjectsinO⁺ mightbenonzerowhengisofaﬃneorindeﬁnitetype.

Proposition 3.8. For u =

k_bG(b) ∈ U such that u acts on M as zero for all M ∈ ob(O⁺)andif k_b = 0,thenb∈B.

Proof. Weassumethatkb0 = 0 forsomeb0∈/B.There existλ,μ∈P⁺ suchthatb0∈ B(λ,−μ)⊂B. Since b0 ∈/B,there exists anssuchthatG(b0)∈Fs butG(b0)∈/Fs+1

whereFsandFs+1 areintheﬁltration(3.3)of V(λ,−μ).Hencewehave 0=u(u_λ,−μ+F_s+1)∈V(λ,−μ)/F_s+1

withV(λ,−μ)/Fs+1∈ob(O⁺).Thisisacontradiction. 2

Bythis propositionweknowthatanyu∈U annihilatingallM ∈ob(O⁺) isalinear combination of G(b) with b ∈ B. Denote by U the set of all such u. It follows from Theorem 3.7and Proposition 3.8that

Theorem3.9. U isanice two-sidedideal ofU and itadmitsacrystal basisB.

Wedefine UtobethequotientofU byU,i.e.UU / U.HenceUinheritsfrom U a canonicalbasisand we denote byBthe corresponding crystalbasis. Onecanseefrom thedefinitionofB and B=B\B thatBis aunionofallhighestweightsubcrystals of B. We know also from Theorem 3.7 thatany M ∈ ob(O⁺) is also a representation of U. Note thatwhen gis of finite type, U = U. If g is of affine type, it follows from

(12)

Proposition 3.6 thatU is isomorphic to thesubalgebra ofU generated by U aξ and aη

forallξ withapositivelevelandη∈P0.

Remark3.10.SimilarlyonecandeﬁneUtobethesetofallu∈Usuchthatuannihilates all M ∈ ob(O⁻). Then U is also anice ideal of U with a crystal basisB where B consistsofallconnectedcomponentsofBthatarenotlowestweightcrystals.Wedenote byVthequotientalgebraU / Uwhichadmitsbothacrystal basisandacanonicalbasis.

4. CellsinU

Recall thatintheprevioussection,by(3.2)wedeﬁne M(λ,−μ) whichis arepresen- tationofU aswell asU.AlsoitcanbeviewedasarepresentationofU.Toseethat,we need thefollowing lemma.

Lemma 4.1.Forb∈B ⊂B andλ,μ∈P⁺,G(b)M(λ,−μ)= 0.

Proof. WeonlyshowthatG(b)V(λ,−μ)⊆W(λ,−μ).SinceV(λ,−μ)/Fs∈ob(O⁺) for any F_s intheﬁltration(3.3),byTheorem 3.7wehave

G(b)

V(λ,−μ)/F_s

= 0 whichmeansG(b)V(λ,−μ)⊆Fs.Hencewehave

G(b)V(λ,−μ)⊆

s0

Fs=W(λ,−μ). 2

Applying this lemma wehave UM(λ,−μ)= 0 andthus we equip M(λ,−μ) with a U-action.QuotientbyW(λ,−μ),weobtainfrom(3.3)aﬁltrationofM(λ,−μ) consisting of niceU or U-submodules

M(λ,−μ) =M₀⊇M₁⊇ · · · ⊇M_l⊇ · · · (4.1) where Mi =Fi/W(λ,−μ).Denotebyvλ,−μ theimageofuλ,−μ inM(λ,−μ).Hencethe map

¯

α_λ,−μ:U−→M(λ,−μ) x−→xv_λ,−μ

takes the canonical base elements of U to those of M(λ,−μ) or to zero. For ξ ∈ P⁺, let M(λ,−μ)_[ξ] (resp. U_[ξ])be thesubspace ofM(λ,−μ) (resp. U)spanned byall G(b) such thatb iscontainedinasubcrystal ofB(M(λ,−μ)) (resp.B)isomorphicto B(ξ).

Note that M(λ,−μ)_[ξ] is usually not a U-submodule of M(λ,−μ). Set M(λ,−μ)_[_ξ], M(λ,−μ)_[>ξ],U_[ξ] andU_[>ξ] as follows

(13)

M(λ,−μ)[ξ]

ηξ

M(λ,−μ)[η], U[ξ]

ηξ

U[η],

M(λ,−μ)[>ξ]

η>ξ

M(λ,−μ)[η], U[>ξ]

η>ξ

U[η].

For a U-module M ∈ ob(O⁺) with a canonical basis, M can be written as M =

λ∈P⁺M[λ] whereM[λ] is thesumof allsubmodules ofM isomorphic toV(λ).Here M[λ] is usually not nice. But it is known in [5, Proposition 27.1.7] that M[ξ] is a nice U-submodule of M for a maximal ξ, i.e. ξ is maximal in the sense of dominant order among all λ such that M[λ] = 0. Moreover, both M[ξ]

λξM[λ] and M[>ξ]

λ>ξM[λ] are nice. In particular, when gis of ﬁnite type, M(λ,−μ)_[_ξ] = M(λ,−μ)[ξ].

Similarto [5,Lemma 29.1.3]wehavethefollowing lemma.

Lemma4.2. Forx∈Uandξ∈P⁺,thefollowingare equivalent (i) x∈U_[_ξ].

(ii) Forallλ,μ∈P⁺,xv_λ,−μ∈M(λ,−μ)_[_ξ]. (iii) Forany M∈ob(O⁺)and m∈M,xm∈M[ξ].

(iv) Ifxactson V(η)asanonzeromapforsome η∈P⁺,then ηξ.

Proof. Itisclearthattheequivalenceof(i)and(ii)followsfromdefinitionsofU[ξ] and M(λ,−μ)[ξ].(iii) and (iv) are equivalent sinceany M ∈ ob(O⁺) can be writtenas a directsumofsomeV(η).Ifxsatisfies(iii)weshowthatitsatisfies(ii).Setx=

kbG(b).

Assume that(ii) does not hold, then there exists some b0 ∈ B(M(λ,−μ)) ⊆ B with k_b₀ = 0 suchthatb₀ iscontained inasubcrystalof Bisomorphicto B(η) withη ξ.

ItfollowsthatthereexistsanssuchthatG(b₀)∈M_sbutG(b₀)∈/M_s+1where M_s and Ms+1 areintheﬁltration(4.1).Thus

M_s/M_s+1∼=V(η).

SetM M(λ,−μ)/Ms+1 andm =v_λ,−μ+Ms+1 ∈M. Then M ∈ob(O⁺) andxm∈/ M[ξ] which contradicts(iii).Converselyweshow that(ii)implies(iii). ForanyM ∈ ob(O⁺) andm∈ Mθ, there exist λ,μ∈P⁺ with λ−μ =θ suchthatxvλ,−μ = 0 and φ:V(λ,−μ)−→M, u_λ,−μ−→misanonzerohomomorphismofU-modules.Asinthe proofofTheorem 3.7,onecanseethatφ(W(λ,−μ))= 0. Hencewe have

φ¯:M(λ,−μ)−→M, vλ,−μ −→m

ahomomorphism of both U-modules and U-modules.As isproved before, there exists an s such that the weights of the generators of M_s are not lower than or equal to any weight in M. Hence φ(M¯ s) = 0 and furthermore, φ¯ factors through the U-map

(14)

φ¯ : M(λ,−μ)/Ms −→ M, v_λ,−μ +Ms −→ m. Since xv_λ,−μ ∈ M(λ,−μ)[ξ] and M(λ,−μ)/Ms∈ob(O⁺),

x(vλ,−μ+Ms)∈

M(λ,−μ)/Ms

[ξ].

It followsthatφ¯(x(v_λ,−μ+M_s))=xm∈M[ξ] whichproves(iii). 2 Similarlyonecanprovethefollowinglemmasince

U_[>ξ]=

η>ξ

U_[η], M_[>ξ] =

η>ξ

M_[η], M(λ,−μ)_[>ξ]=

η>ξ

M(λ,−μ)_[η].

Lemma 4.3.Forx∈Uand ξ∈P⁺,thefollowingareequivalent (i) x∈U_[>ξ].

(ii) Forallλ,μ∈P⁺,xvλ,−μ ∈M(λ,−μ)_[>ξ]. (iii) Forany M∈ob(O⁺)andm∈M,xm∈M[>ξ].

(iv) If xactsonV(η) asanonzeromapforsomeη∈P⁺,thenη > ξ.

Thecorollarybelowfollowsimmediately fromLemma 4.2andLemma 4.3.

Corollary 4.4.Both U_[ξ] and U_[>ξ] are nicetwo-sided idealsof Uforany ξ∈P⁺. Remark 4.5. For ξ ∈ P⁺, we can deﬁne V[−ξ] (resp. V[<−ξ]) to be the subset of V consisting of allxsuch thatη ξ (resp.η > ξ) ifxactson V(−η) asanonzero map.

Similarlyboth ofthemareniceidealsofV.

For an integrable left U-module M with ﬁnite dimensional weight spaces, let M^◦ denotethegradeddualofM,i.e.M^◦=

θ∈PM_θ^∗whereM=

θ∈PM_θ.Thenthereis arightU-actiononM^◦ as

(f·x)(v) =f(xv) forf ∈M^◦, v∈M, x∈U.

For instance, V(λ)^◦ is an irreducible integrable right U-module with highest weight λ∈P⁺.GivenarightU-moduleM,wedenoteby^∗M thesamek-vectorspaceequipped with aleftU-actionas

x◦m=m·x^∗ forx∈U, m∈^∗M,

where x^∗ is the image of x under the anti-automorphism ∗. It is clear that^∗V(λ)^◦ ∼= V(−λ) asleftU-modules forλ∈ ±P⁺. Given aleftU-moduleN, deﬁne^ωN to be the left U-modulewiththeunderlyingspace^ωN =N suchthat

x◦v=ω(x)·v forx∈U, v∈^ωN.

Seethat^ωV(λ)∼=V(−λ) forλ∈ ±P⁺.

(15)

Lemma4.6.

(i) Both∗andω on U inducebijections ∗, ω:U←→V.

(ii) There are bijections ∗, ω : U_[ξ] ←→ V_[−ξ] and ∗, ω : U_[>ξ] ←→ V_[<−ξ] for ξ∈P⁺.

Proof. Toprove(i),itissuﬃcienttoshowthat∗(U)=Uandω(U)=U.Forb∈B, G(b) annihilatesallV(λ) forλ∈P⁺.Thenwe haveG(b)◦^∗V(−λ)^◦= 0 for allλ∈P⁺ whichimpliesthatG(b)^∗=G(b^∗) annihilatesallV(−λ) forλ∈P⁺.Hence∗(U)⊆U. Similarly we have∗(U) ⊆ U. It follows from ∗² = id on U that ∗(U) =U. Given b∈B,G(b) annihilates all^ωV(−λ) forλ∈P⁺.It impliesthatω(G(b)) annihilatesall V(−λ) for λ∈ P⁺ and thus ω(U) ⊆U. The proof of the equality is similar to that for ∗. In order to prove(ii), we only show ∗(U_[ξ]) ⊆V_[−ξ] and ω(U_[ξ]) ⊆ V_[−ξ]. Given x ∈ U_[ξ], if x^∗ acts on V(−η), η ∈ P⁺, as a nonzero map, one can see that x(^∗V(−η)^◦) = xV(η) = 0 which implies η ξ. Hence x^∗ ∈ V_[₋_ξ]. Similarly if ω(x) actson V(−η) for someη ∈P⁺, as anonzeromap, thenω(x)V(−η)=x◦^ωV(−η)= xV(η)= 0.Henceη ξ whichimpliesω(x)∈V[−ξ]. 2

NotethatUcanbe writtenas adirectsumofvectorspaces

ξ∈P⁺U_[ξ].Wehavean isomorphism

U[ξ]/U[>ξ]∼=U[ξ]

ask-vectorspaces.Furthermore,U[ξ]/U_[>ξ]isanalgebraaswellasaU-bimodulewhich wecallatwo-sidedcellmoduleofUanddenotebyU(ξ) forsimplicity.Thiscellnaturally inheritsfrom Uacanonicalbasisand itscrystalbasis isafamilyofcopies ofB(ξ). We havethefollowing resultsimilar to[5,Theorem 29.3.3].

Proposition4.7. Forξ∈P⁺,

(i) U(ξ)decomposesintoadirectsumof niceirreduciblehighestweightleft U-submod- ules,each summand isisomorphic toV(ξ).

(ii) U(ξ)decomposesintoadirectsumofniceirreduciblehighestweightrightU-submod- ules,each summand isisomorphic toV(ξ)^◦.

(iii) U(ξ)∼=V(ξ)⊗V(ξ)^◦ as U orU-bimodules.

Proof. (i)isobvious.Sincewehavebijections

ω◦ ∗:U_[ξ]←→U_[ξ], U_[>ξ] ←→U_[>ξ]

by Lemma 4.6, ω ◦ ∗ induces an anti-automorphism of U(ξ). Applying ω ◦ ∗ to any summandV in(i),weobtainaniceirreduciblerightU-moduleω◦∗(V) byCorollary 2.7 andthis proves(ii).Letφbe therestrictingmap onU_[ξ] oftheU-actiononV(ξ), i.e.

(16)

φ:U[ξ] −→Endk(V(ξ)).Then φis ahomomorphismofalgebras without 1.It canbe seen from Lemma 4.3 thatthe kernel of φ is exactly U_[>ξ] and thus the induced map φ¯:U(ξ)−→Endk(V(ξ)) isinjective.WeviewV(ξ)⊗V(ξ)^◦ as asubset ofEndk(V(ξ)), i.e.

(x⊗f)(v) =f(v)x forx, v∈V(ξ), f ∈V(ξ)^◦.

It is easyto see thatV(ξ)⊗V(ξ)^◦ isaU or U-subbimodule as well as asubalgebraof End_k(V(ξ)) whereU or UactsonV(ξ)⊗V(ξ)^◦ as

x(v⊗f)y

(m) =f(ym)xv forv, m∈V(ξ), f ∈V(ξ)^◦, x, y∈U orU.

In fact the φ¯ deﬁned above maps U(ξ) injectively into V(ξ)⊗V(ξ)^◦, and moreover, φ¯ : U(ξ) −→ V(ξ)⊗V(ξ)^◦ is a homomorphism of U or U-bimodules. Fixing a right weightη∈P,U(ξ)a_η is,byProposition 2.3,adirectsumofdimV(ξ)_η copiesofV(ξ) as a left U-module,where we denote theimage of a_η inU(ξ) by thesame symbol.Hence φ¯:U(ξ)−→V(ξ)⊗V(ξ)^◦ issurjective andU(ξ)∼=V(ξ)⊗V(ξ)^◦. 2

5. Quantumcoordinatealgebra

5.1. U^◦

For U1,U2 ∈ {U , U_Z, U, U, U-modules with canonical bases}, let U1⊗ U2 be the set ofallformal(possiblyinﬁnite)linearcombinations

k_b₁_,b₂G(b₁)⊗G(b₂),

where k_b₁_,b₂ ∈ k or Z[q,q⁻¹]. For λ,λ₁,λ₂ ∈ P, thecomultiplication inU induces the map Δ_λ,λ₁_,λ₂ :U a_λ−→U a_λ₁⊗U a_λ₂ whereΔ_λ,λ₁_,λ₂ isnonzeroonlyifλ=λ₁+λ₂.Set

Δ =

λ,λ1,λ2∈P

Δλ,λ1,λ2 :U −→U⊗ U .

Fora,b,c∈B, wedeﬁne mˆ^b,c_a ∈kto satisfythat Δ

G(a)

=

b,c

ˆ

m^b,c_a G(b)⊗G(c).

Note thatrestrictingΔ onU_Z, wehavein[5,23.2.3]

Δ :U_Z−→U_Z⊗U_Z.

(17)

Since{G(b)|b∈B}formsaZ[q,q⁻¹]-basisofU_Z,thestructureconstantmˆ^b,c_a isactually inZ[q,q⁻¹].Forλ,λ1,λ2, μ,μ1,μ2∈P⁺ withλ=λ1+λ2 andμ=μ1+μ2,letτ1,τ2

betheU-modulehomomorphism

τ₁:V(λ)−→V(λ₁)⊗V(λ₂), u_λ−→u_λ₁⊗u_λ₂,

τ2:V(−μ)−→V(−μ1)⊗V(−μ2), u_−μ−→u_−μ₁⊗u_−μ₂. SetRλ2,−μ1 tobe theuniqueisomorphismofU-modules(theR-matrix)

Rλ2,−μ1 :V(λ2)⊗V(−μ1)−→ V(−μ1)⊗V(λ2)

suchthatRλ2,−μ1(uλ2⊗u_−μ₁)=u_−μ₁ ⊗uλ2. Letτ be thecomposition ofτ1⊗τ2 and 1⊗Rλ2,−μ1⊗1,i.e.

V(λ,−μ)

τ τ1⊗τ2

V(λ1)⊗V(λ2)⊗V(−μ1)⊗V(−μ2)

1⊗Rλ2,−μ1⊗1

V(λ1,−μ1)⊗V(λ2,−μ2) Letρbe themapρ:U ⊗ U −→V(λ₁,−μ₁)⊗ V(λ₂,−μ₂) suchthat

ρ

ka,bG(a)⊗G(b)

=

ka,b

G(a)uλ1,−μ1

⊗

G(b)uλ2,−μ2

.

OnecanseethatUactsonu_λ₁_,−μ₁⊗u_λ₂_,−μ₂ asamapwhichcanbeobtainedthrough Δ.

Moreprecisely,wehaveacommutativediagram

U ^Δ

γ

U⊗U

ρ

V(λ1,−μ1)⊗V(λ2,−μ2) ⁱ V(λ1,−μ1)⊗V(λ2,−μ2) whereγ(x)=x(u_λ₁_,−μ₁⊗u_λ₂_,−μ₂) andiisthecanonicalinclusion.

Proposition5.1. The followingdiagramiscommutative

U ^Δ

α_λ,−μ

Im(Δ)

ρ|_ImΔ

V(λ,−μ) ^τ V(λ1,−μ1)⊗V(λ2,−μ2)

(5.1)