zonotopes, braids, and quantum groups

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243 ��57698��:��;��$<=;��>��?�@ 3 ��;��A@��!�!�!BDCECGF�H IKJ1LNMOJQPSR�T�CGF�UVJEW�T'XYIZonotopes,Braids, and Quantum Groups

MarceloAguiar

Departmentof Mathematics,NorthwesternUniversity, Evanston,IL 60208,[email protected]

ReceivedSeptember16,1998

AMSSubjectClassification: 05A30,17B37

Abstract. Variousresultsaboutthe actionof the binomial braids andotherbraid analogs [2]on someparticularhigherdimensionalrepresentationsof thebraidgroupsarepresented.Theserepresentationsareconstructedfrom a fixed integer squarematrix A. The commonnullspaceof the binomial braidsis studiedin somedetail. This spaceis gradedover Z r , wherer is thesizeof A. Our mainresultsstatethat thenon-trivial componentsoccuronly on thelatticepointsof a certainhypersurfacein r-spacethat is canonicallyassociatedto A, andthat whenA is thesymmetrizationof a CartanmatrixC of finite type, theselatticepointsarecloselyrelatedto theverticesof thezonotopeassociatedto C (thepreciserelationshipis given in Theorem5.3). Thesameactionis usedto constructa quantumgroupU0

q [ A\ from anarbitraryintegersquarematrixA. Thesimplestchoicesof A yield theusualpolynomialandEulerianHopf algebrasof JoniandRota(in the correspondingrepresentations,the binomial braidsbecomethe usualbinomial orq-binomialcoefficients).Theotherchoicewe consideris thatwhenA is thesymmetrizationof asymmetrizableCartanmatrixC. Someof thepreviousresultsareusedto prove that in this caseU0

q [ A\ coincideswith the usualquantumgroupof Drinfeld andJimbo. The quantumgroupisactuallydefinedin a moregeneralsettinginvolving Hopf algebrasandcrossedbimodules.Thispaperis a continuationof [2].

Keywords: Braids,Cartanmatrices,zonotopes,quantumgroups

1. Intr oduction

Thispaperstudiesthecombinatoricsof acertainactionof thebraidgroupsonthetensorpowersof avectorspace,definedin termsof agivenmatrix,andits relationto quantumgroups.

Let Bn denotethebraidgroupin n strandsandkBn thegroupalgebraover anarbi-

trary field k. In [2], elementsb ] ni _ kBn weredefinedandshown to satisfypropertiesanalogousto thoseof the ordinaryor q-binomial coefficients ` ni a q. Braid analogs ofclassicalidentitiesof Pascal,Vandermonde,Cauchy, andseveralotherswherepresentedthere.

Theidentitiesamongbraidsin factspecializeto theq-identitiesafterpassingto theone-dimensionalrepresentationof Bn, whereeverycanonicalgeneratoractsbymultipli-cationby a fixedscalarq _ k. In particular, thecaseof theusualidentitiescorrespondsto thechoiceof the trivial representation(q b 1). Higher-dimensionalrepresentations

433

434 M. Aguiar

yieldnew realizationsof theseidentities,wherenumbersorq-numbersarenow replacedby matrices.

Thestudyof theactionsof thebinomialbraidsb ] ni _ kBn onsomeparticularhigherdimensionalrepresentationsis relevantto thedefinitionof thequantumgroupsof Drin-feld [3] andJimbo[8] (thesearecertainq-analogsof theuniversalenvelopingalgebrasof simpleLie algebras),asexplainedin [1] and[2]. A similar observation hadbeenmadebeforeby Schauenburg in [15].

In this paperwe derive someresultson thesehigher-dimensionalrepresentationsandusethemto definea quantumgroupU cq d Ae from any integersquarematrix A, thatcoincideswith thequantumgroupof Drinfeld andJimbowhenA is asymmetricCartanmatrix (or, moregenerally, whenA is the symmetrizationof a symmetrizableCartanmatrixC).

Thecontentsof thepaperareasfollows.Westartin Section2 by recallingsomedef-initionsandbraididentitiesfrom [2], aswell asthelanguageof monoidalcategoriesandsomebasicHopf algebraicnotionsthat areuseful in dealingwith higher-dimensionalrepresentationsof thebraidgroups.

In Section3.2 we defineanalgebraU0q d Ae associatedto any integersquarematrix

A and scalarq _ k, and also a companionHopf algebraU cq d Ae . We show that thesimplestchoicesof A (A bf` 0a andA bf` 1a ) yield, respectively, theusualbinomialandEulerianHopf algebrasof JoniandRota.Thiscomesasnosurprisesinceit is thebasicobservation of [2] that in the correspondingrepresentationsof the braid groups,thebinomialbraidsactastheusualbinomialor q-binomialcoefficients,andthesearethesectioncoefficientsfor theseHopf algebras(in thesenseof JoniandRota).

It will beshown in Section5.1 thatwhenA is a symmetricCartanmatrix (or, moregenerally, thesymmetrizationof a symmetrizableCartanmatrixC) U0

q d Ae andU cq d Aecoincide,respectively, with the q-analogsof the universalenvelopingalgebrasof thenilpotentandBorelsubalgebraof thesimpleLie algebracorrespondingtoC, asdefinedby Drinfeld andJimbo.

If thesizeof A is r, U0q d Ae hasr generatorsandtherelationsamongthemaredefined

in termsof thebinomialbraidsb ] ni andarepresentationof thebraidgroupsaredefinedby meansof A. The constructionof the quantumgroupis carriedout in Section3.1in a moregeneralsettingwheretheinitial dataarea Hopf algebraH anda crossedH-bimoduleinsteadof a matrix A anda scalarq, althoughthelatter is theonly caseto beconsideredelsewherein thepaper.

In Section4 we studythe higher-dimensionalrepresentationsof the braid groupsdefinedby A, with the goal of describingthe relationsof the algebraU0

q d Ae in somedetail (Corollaries4.12and4.13). By definition, the idealof relationsis generatedbythe commonnullspaceof the binomial braidson theserepresentations,andthis spaceis gradedover g r . We prove that the only non-trivial componentsoccuron the locusof a certainhypersurfacein kr that is canonicallyassociatedto A (Proposition4.4). Inparticularwededucethat,if A is symmetricandpositive-definite,thenU0

q d Ae is finitely-related(Corollary4.7).Theproofsof this typeof resultsarecombinatorial,in thesensethatthey arebasedon someof thecombinatorialidentitiesobtainedin [2].

The specialcasewhenA is the symmetrizationof a symmetrizableCartanmatrixC is treatedin Section5. The basicfactson Cartanmatrices,root systems,andtheirzonotopesarereviewedasthey areneededfor theexposition.

Zonotopes,Braids,andQuantumGroups 435

In Section5.1 we prove that, for themostgeneralcaseof a symmetrizableCartanmatrix C, the ideal of relationsis generatedby the so-calledquantumSerrerelations.Thismeansthatourdefinitionof thequantumgroupsassociatedtoC coincideswith theusualoneof Drinfeld andJimbo. Part of the proof of this result is combinatorial,butanotherpart relieson a somewhatdeepresultof Lusztigaboutthedefinitionof U0

q d C e(which is obtainedthroughrepresentationtheory)andSchauenburg’sobservationabouttheoccurrenceof thefactorialbraidin Lusztig’sdefinition.A purecombinatorialproofwouldbedesirable.

If the Cartanmatrix is of finite type (i.e., positive definite), we candescribethenullspaceof thebinomialbraids(i.e., therelationsof thequantumgroup)moreexplic-itly in termsof thetheroot systemof C andthecorrespondingzonotope.This is donein Section5.2, independentlyof the resultsin Section5.1. In this case,the commonnullspaceof the binomialbraidsis naturallygradedover the root latticeof C, andtheonly non-trivial componentsoccuronthespherethroughtheorigin, centeredat thelow-estweightof therootsystem.Theorem5.3statesthatamongtheselatticepoints,thosethatdo not lie on thewallsof theroot systemarepreciselytheverticesof thezonotopeassociatedto C. This is oneof themainresultsof thepaper.

In Section6 we list someadditionalresultsanda few openquestionsthataremoti-vatedby thepreviouswork.

Thispaperis acontinuationof [2]. Someof theresultsin Section4 alreadyappearedin [1, Section9.8.5].Therelationto quantumgroupswasstudiedfrom adifferentpointof view in [1, Section9.8].

TheauthorthanksSteve Chase,KenBrown, andDanBarbaschfor usefulconver-sationsduringthepreparationof thispaper.

2. Braid Identities and Other Prerequisites

2.1. BraidGroupsandBraidAnalogs

The groupBn of braids in n strandshasn h 1 generatorss] n1 iVjkj+jli s] nn m 1 subjectto therelations

s] ni s] nj b s] nj s] ni if n i h j n�o 2 i (A1)

s] ni s] ni c 1s] ni b s] ni c 1s] ni s] ni c 1 if 1 p i p n h 2 j (A2)

Thegenerators] ni is representedby thefollowing picture,andtheproductst of twobraidssandt in Bn is obtainedby puttingthepictureof sontopof thatof t. Theidentityof Bn is representedby thepicturewith n verticalstrands;theinverseof s is obtainedbyreflectingits pictureacrossa horizontalline, without leaving the planeof the picture.Let kBn denotethegroupalgebraof Bn over anarbitraryfixedfield k. In [2, Sections

3 and5], elementsna , b ] ni and f ] n _ kBn weredefinedandcallednatural,binomial,andfactorialbraids,respectively. Fromtheseveral identitiesinvolving themthatwere

436 M. Aguiar

q r s tttt ttttu v w

x x x x x x x x x yz { | }~ � � � � � � �� obtainedin [2], we will only needthefollowing:

f ] n b f ] j ^�� f ] n m j ^�� b ] nj i (2.1)

1 ] i ^ � b ] n m i ^j m i

� b ] ni b b ] j ^i

� 1 ] n m j ^ � b ] nj i (2.2)

b ] mc np b p

∑k� 0

1 ] k ^ � βmm k � p m k� 1 ] n m pc k � b ] mk � b ] np m k i (2.3)

f ] n b 1 ] n m 1 � ` 1a � 1 ] n m 2 � ` 2a � j+jkj � 1 � ` n h 1a � ` na i (2.4)

n

∑k� 0

µ ] k � 1 ] n m k � b ] nk b 0 � n � 0 i (2.5)

µ] n b ] ni µ ] n � 1 b b ] nn m i j (2.6)

Here,βm� n _ Bmc n is the braiding [2, Section2.4] andµ ] k _ kBk is the Mobiusbraidof [2, Sections6.2 and2.2], whosedefinitionswill be recalledbelow. Formulas(2.1), (2.2), (2.3), (2.4), (2.5)and(2.6)are,respectively, formulas(21),(20),(14),(15) ,(25)and(12)from [2]. Eachof theseidentitiesgeneralizesawell-knownq-identity. Forinstance,(2.5) is thebraidanalogof

n

∑k� 0

d h 1e kq d k2 e�� n

i � qb 0 � n � 0 i

while (2.3) is thebraidanalogof Vandermonde’s identity� m � np � q

b p

∑k� 0

q ] mm k ] p m k � mk � q

� np h k � q

iand(2.6) is simply theanalogof � n

i � qb � n

n h i � qj

We alsoneedto recall thebraids] nσ _ Bn associatedto a permutationσ _ Sn. The

pictureof s] nσ is obtainedby drawing astraightline from 1 in thebottomto σ d 1e in thetop, thenunderthata straightline from 2 to σ d 2e , etc. For instance,if σ b�� 1 2 3 4

4 2 1 3 � ,then

Zonotopes,Braids,andQuantumGroups 437��

��

��

� � � � � � � � � � � �� ¡

¢ £ ¤ ¥¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦

¦ ¦ ¦

An explicit expressionfor s] nσ in termsof thegeneratorss] ni wasgivenin [2, Section

5.1]. The mapSn § Bn, σ ¨§ s] nσ , is a sectionof the canonicalprojectionBn § Sn.Moreover, it wasshown in [2, Lemma5.2] that

if σ b τρ and © ªV«�¬V�® d σ e$b¯© ªV«�¬V�® d τ e°�K© ªV«�¬V�® d ρ e , thens] nσ b s] nτ s] nρ . (2.7)

Herethelengthof a permutationis theminimumnumberof elementarytranspositionsrequiredto write it asaproductof such.We will makeuseof this factin Section4.3.

ThebraidingandMobiusbraidsaredefinedasfollows. If

σ b²± 1 2 �k�+� m m � 1 m � 2 �+�+� m � nn � 1 n � 2 �k�+� n � m 1 2 �+�+� n ³ i

thenβm� n b s] mc nσ . If σ b � 1 2 ´ ´ ´ n

n n m 1 ´ ´ ´ 1 � (thepermutationwith thelongestlength),then

µ] n b d h 1e ns] nσ j2.2. TheBraidCategoryandIts Representations

Thecollection µ¶b¸· n ¹ 0Bn of all braidgroupsformsacategory, wheretheobjectsarethenaturalnumbers,Bn is thesetof endomorphismsof n, andthereareno morphismsbetweendistinctobjects.Thiscategory is monoidal,in thesensethatthereis a functor

µ»º¼µ § µ i d si t e$¨§ s � t

that is associative andunital. Explicitly, this multiplication consistsof morphismsof

groupsBn º Bm § Bnc m thataredefinedonthegeneratorsby s] ni � s] mj b s] nc mi s] nc m

nc j .In termsof pictures,s � t is obtainedby putting t to the right of s. (For moredetailson the basicson monoidalcategoriesandthe category of braidsthe readeris referredto [12, X.6, XI.2 andXIII.2] or [10].)

We areinterestedin monoidalrepresentationsof thecategory µ . Explicitly, sucha representationconsistsof a vectorspaceX, suchthat,for every n, thebraidgroupBn

actson thetensorpowerX ½ n, with thepropertythat

s � t � x � y b d s � xe � d t � ye � s _ Bn i t _ Bm i x _ X ½ n i y _ X ½ m jSinces] ni b 1 ] i m 1 � s] 21 � 1 ] n m i c 1 , this propertyimplies that theactionof µ on X ½ n

is uniquely determinedby the action of s] 21 on X � X. Moreover, a linear operator

438 M. Aguiar

R : X � X § X � X definesa monoidalrepresentationof µ if andonly if it is invertibleandsatisfiestheYang–Baxterequation:d R � idX e°¾ d idX

� Re¿¾ d R � idX e�b d idX� Re¿¾ d R � idX e°¾ d idX

� Re jThis is a consequenceof (A2).

If X is one-dimensional,thenany invertibleoperatorR : X § X satisfiesthis equa-tion. R is necessarilygiven by multiplication by somenon-zeroscalarq _ k. In this

case,s] ni actsby multiplicationby q for every n o 2 i 1 p i p n h 1 and,aswasshown

in [2], b ] ni and f ] n acton X ½ n by multiplicationby theq-binomialcoefficient ` ni a q andtheq-factorial ` na !q, respectively (soit is thissimplestchoicethatproducestheclassicalq-identitiesfrom theidentitiesfor braids).

An equivalentway to describemonoidalrepresentationsof the braid category isby meansof the following fact: µ is thefreebraidedmonoidalstrict category on oneobject (the object 1 _ g ). This saysthat given any objectX of a braidedmonoidalcategory À , thereis a uniquefunctor F : µ § À that preserves the monoidalstruc-turesandthebraidingsandsuchthatF d 1e�b X [12, LemmaXIII.3.5]. This highlightsthe fundamentalrole of the braidingβm� n. If À carriesin additiona k-linearstructure(compatiblewith the restof the structure),thenF extendsto F : kµ § À . Usually Àconsistsof vectork-spaceswith someadditionalstructure,andthenthe vectorspaceX becomesa monoidalrepresentationof µ , asdefinedabove. An exampleof suchacategory is ÀNbÂÁ G, the category of crossedG-bimodules,for any groupG. An ob-ject of Á G is a k-spaceX equippedwith a linearactionof G anda linearG- grading,i.e., a decompositionX bfÃ g Ä GXg into subspaces,suchthat the actionof h _ G car-ries Xg to Xhgh� 1. In this context, oneusuallywrites n x nVb g whenx _ Xg, so that theconditionjust mentionedbecomesn h � x nÅb h n x n hm 1. This category is braidedmonoidalundertheusualtensorproductof k-spaces,whereX � Y is equippedwith theG-actiong � d x � yeYb d g � xe � d g � ye andtheG-grading n x � y n#bSn x n n y n , andthebraidingis

βX � Y : X � Y § Y � X i βX � Y d x � ye�b d n x n � ye � x jThis constructioncan in fact be carriedout for any Hopf algebraH in placeof

G: Thereis a category Á H of crossedH-bimodules[12, Definition IX.5.1], which isbraidedmonoidal. CrossedG-bimodulesas definedabove correspondto the choiceH b kG, the groupalgebraof G. CrossedbimodulesarealsocalledYetter–Drinfeldmodulesin theliterature.

The following statementsummarizesthe partof theseresultswe aremostly inter-estedin.

Proposition2.1. LetH bea HopfalgebraandX a crossedH-bimodule. ThenthebraidgroupBn actson X ½ n for everyn o 0, by morphismsof crossedH-bimodulesandwiththepropertythat

s � t � x � y b d s � xe � d t � ye � s _ Bn i t _ Bm i x _ X ½ n i y _ X ½ m jProof.


2.3. Hopf Algebrasin Categories

Thereis anotionof Hopf algebrain anarbitrarybraidedmonoidalcategory[14,Section10.5],thatwe review next.

Let À be a monoidalcategory and I its unit object. An algebra in À is a tripled A i µ i ue whereA is an objectof À andµ : A � A § A andu : I § A aremapsin À ,subjectto theobviousassociativity andunitality axioms. Coalgebrasin À aredefinedsimilarly. If À is alsobraided,then the notion of bialgebrasandHopf algebrasin Àarealsodefined. First, onedefinesan algebrastructureon the tensorproductof twoalgebrasA andB via

µA½ B : d A � Be � d A � Be idA ½ βB Æ A ½ idBhÇhÈhÉhÈhÉhÈhÈh § d A � Ae � d B � Be µA ½ µBh7hÉhÈh § A � B jA bialgebrain À is an objectA that is an algebraanda coalgebrain sucha way that∆ : A § A � A andε : A § I aremorphismsof algebrasin À . A Hopf algebrain À isa bialgebrain À for which id : A § A hasa convolution inverseS : A § A, calledtheantipode.When À is thecategory of vectorspaces,equippedwith the trivial braidingx � y ¨§ y � x, theabovenotionsboil down to theusualnotionsof algebras,coalgebras,bialgebras,andHopf algebras.

We aremainly interestedin Hopf algebrasin thecategory Á H of crossedH-bimod-ules. The basicexampleof suchan objectis providedby the ordinarytensoralgebraT d X e of a crossedH-bimoduleX (Proposition3.1).

If A is a Hopf algebrain Á H , thenthe tensorproductA � H carriesa structureofordinaryHopf algebra,calledthebiproductof A andH anddenotedA Ê H [14,Theorem10.6.5].Thisconstructionof aHopf algebrafrom aHopf algebrain abraidedmonoidalcategory is sometimescalledbosonization.

3. The Quantum Group Associatedto a CrossedBimodule

3.1. Definitionof theQuantumGroup

Let H beaHopf k-algebra,X acrossedH-bimodule,and

T d X e�b ∞Ën� 0

X ½ n ithetensoralgebraof thevectorspaceX. Schauenburg provedthatT d X e is a bialgebrain Á H [15, Corollary2.4 andTheorem2.7]. We presenta differentproof next, basedon thecombinatorialidentitiesfor thebraidanalogs,andshow thatT d X e is actuallyaHopf algebrain Á H . Recallthatby Proposition2.1,Bn actson X ½ n � n o 0.

Proposition3.1. Let X be a crossedH-bimoduleand T d X e the tensoralgebra of theunderlyingvectorspaceX. ThenT d X e is a Hopf algebra in Á H with comultiplication,counit,andantipodedefinedon x _ X ½ n by

∆ d xe$b n

∑i � 0

d b ] ni xe ] i ^ � d b ] ni xe ] n m i ^ i ε d xeGbÍÌ 1 if x b 1

0 if n � 0 i Sd xe�b µ ] n x j

440 M. Aguiar

Thenotationabove standsfor the natural identificationsX ½ n Î�§ X ½ i � X ½ ] n m i ^ , y ¨§y ] i ^ � y ] n m i ^ .Proof. We compute

d id � ∆ e ∆ d xe�b n

∑i � 0

n m i

∑h� 0 Ï b ] ni xÐ ] i ^ � Ï b ] n m i ^

h d b ] ni xe ] n m i ^ Ð ] h � Ï b ] n m i ^h d b ] ni xe ] n m i ^ Ð ] n m i m hb n

∑i � 0

n m i

∑h� 0 Ï 1 ] i ^ � b ] n m i ^

h� b ] ni xÐ ] i ^ � Ï 1 ] i ^ � b ] n m i ^

h� b ] ni xÐ ] h� Ï 1 ] i ^ � b ] n m i ^

h� b ] ni xÐ ] n m i m h i

and

d ∆ � id e ∆ d xeÑb n

∑j � 0

j

∑i � 0 Ï b ] j î d b ] nj xe ] j ^ Ð ] i ^ � Ï b ] j ^

i d b ] nj xe ] j ^ Ð ] j m i ^ � Ï b ] nj xÐ ] n m j ^b n

∑j � 0

j

∑i � 0 Ï b ] j î

� 1 ] n m j ^ � b ] nj xÐ ] i ^ � Ï b ] j î� 1 ] n m j ^ � b ] nj xÐ ] j m i ^� Ï b ] j î

� 1 ] n m j ^V� b ] nj xÐ ] n m j ^ jCoassociativity thusfollows from formula(2.2): 1 ] i ^ � b ] n m i ^

h� b ] ni b b ] j î

� 1 ] n m j ^ � b ] njif h b j h i.

Counitalityboils down to thefactthatb ] n0 b b ] nn b 1, which holdsby definitionofthebinomialbraids.

Takex _ X ½ n andy _ X ½ m. Then

∆ d xye�b nc m

∑p� 0

d b ] nc mp x � ye ] p � d b ] nc m

p x � ye ] nc mm p jOntheotherhand,by definitionof multiplicationin T d X e � T d X e , wehave

∆ d xe ∆ d ye$b n

∑i � 0

m

∑j � 0

d 1 ] i ^ � βn m i � j� 1 ] mm j ^ � b ] ni x � b ] mj ye ] i c j ^� d sameelemente ] nc mm i m j ^ j

Multiplicativity for ∆ thusfollows from Vandermonde’s formula (2.3). It is clearthenthat∆ andε aremorphismsof algebrasin Á H .

Finally,

md S � id e ∆ d xe�b n

∑i � 0

µ] i ^ d b ] ni xe ] i ^ � d b ] ni xe ] n m i ^ b n

∑i � 0

µ] i ^�� 1 ] n m i ^l� b ] ni x b ε d xe 1preciselyby formula (2.5). The otheraxiom for the antipodefollows from a similaridentity (Equation(26) in [2]).


Let K ] 0 b K ] 1 b 0 and,for eachn o 2,

K ] n b n m 1Òi � 1

ker d b ] ni : X ½ n § X ½ n e i K b ∞Ën� 0

K ] n�Ó T d X eandlet K denotetheidealgeneratedby K in T d X e . Defineanalgebra

U0H d X e : b T d X ekÔ K j

Thus,U0H d X e is generatedby X subjectto therelationsK.

SinceBn actson X ½ n by morphismsof crossedH-bimodules(Proposition2.1), Kis acrossedsubbimoduleof T d X e . Therefore,sois K, andhencethequotientU0

H d X e isalsoanalgebrain Á H . Moreover,

Proposition3.2. In the Hopf algebra T d X e in Á H , K is a coideal stableunder theantipode. Therefore, U0

H d X e is a Hopf algebra in Á H .

Proof. If x _ K ] n , then

∆ d xe$b x � 1 � 1 � x _ K � T d X e°� T d X e � K iand,for 1 p i p n h 1,

b ] ni Sd xe$b b ] ni µ ] n d xe ] 2 ´ 6b µ] n b ] nn m i d xe$b 0 isoK is a coidealstableundertheantipode.Hence,K is a Hopf idealof T d X e andthequotientis aHopf algebrain Á H .

Remark3.1. Onecouldalsoderive this resultfrom thefactthatK consistspreciselyofthoseprimitiveelementsof T d X e of degreeatleast2 (with respectto thenaturalgradingof T d X e ).

As explainedin Section2.3,theaboveresultallowsusto constructanordinaryHopfalgebraasthebiproduct

U cH d X e : b U0H d X e�Ê H j

U0H d X e andU cH d X e arethe quantumgroupswhoseconstructionwasannouncedin

theintroduction.

Example3.1. Take H b kÕ m, thegroupalgebraof thecyclic groupof orderm � 1, qa root of unity of orderm, andX b k Ö x × , the one-dimensionalcrossedÕ m bimoduledefinedby n x nÅb 1 and n � x b qnx � n _ Õ m jTheactionof thebinomialbraidsis givenin this caseby theq-binomialcoefficients:

b ] ni xn b � ni � q

xn j

442 M. Aguiar

If m doesnot divide n, then ` n1 a q b d qn h 1e+Ô d q h 1e�Øb 0, so K ] n b 0. If m dividesn,then,for 1 p i p n h 1,� n

i � qb d qn h 1e d qn m 1 h 1e �k�+� d qn m i c 1 h 1ed qi h 1e d qi m 1 h 1e �k�+� d q h 1e b 0 i

so K ] n b k Ö xn × . Therefore,K b d xm e , the ideal generatedby xm, andU0kÙ m d X e�b

k ` xa Ô d xm e is anm-dimensionalHopf algebrain Á Ù m with structure

∆ d xn e�b n

∑i � 0

� ni � q

xi � xn m i andSd xn e$b d h 1e nq d n2e xn j

NotethatSk d xn e$b d h 1e knqk d n2 e xn, sotheantipodehasorder2m. Thebiproduct

U ckÙ m d X eGb k ` xa Ô d xmeVÊ kÕ m

is an ordinaryHopf algebraof dimensionm2 with antipodeof order2m. This Hopfalgebrawasoriginally constructedby Taft [17] (andby Sweedlerfor q bÚh 1).

Weclosethissectionwith two simplebut importantobservationsaboutthecommonnullspaceK of thebinomialbraidsK. A relatedspaceis thenullspaceF of thefactorialbraids,definedasfollows:

F b ∞Ën� 0

F ] n Ó T d X e whereF ] n b ker d f ] n : X ½ n § X ½ n e jLemma 3.3. F is an ideal of T d X e , andK Ó F.

Proof. Formula(2.1)

f ] i ^ � f ] n m i ^ � b ] ni b f ] nshows that kerb ] ni Ó ker f ] n � i; in particular, K ] n Ó F ] n , so K Ó F . On the otherhand,applyingthehorizontalsymmetryoperatorÛ of [2, Section2.3] to formula(2.1),oneobtains

b ] nAÜi

� d f ] i ^ � f ] n m i ^ eGb f ] n isincethis operatorpreservestensorproducts,reversescompositionsandfixesthe fac-torial braids(formula(17) in [2]). This immediatelyimpliesthatF is anidealof T d X e .Sinceit containsK, it mustalsocontainK.

Lemma 3.4. K ] n Ó ker d µ] n � 1e .Proof. This is animmediateconsequenceof identity (2.5), sinceb ] n0 b b ] nn b 1.


3.2. TheQuantumGroupAssociatedto aMatrix

Let A bÝ` ahka _ Mr d ÕÞe be an integersquarematrix of sizer andq _ kß a fixedscalar.We will alwaysassumethatq is not a rootof unity, q1à 2 _ k andchk b 0.

Let G bÚÕ r, the free abeliangroupof rank r, andX the vectorspacewith basisÖ x1 iVj+jkj�i xr × , beviewedascrossedG-bimodulewithn xh nÅb d a1h i�jkj+jVi arh e _ Õ r i d n1 i�j+j+jli nr e � xk b qnkxk � d n1 iVj+jkj�i nr e _ Õ r jThealgebrasU0

kG d X e andU ckG d X e will bedenotedby U0q d Ae andU cq d Ae , respectively.

They will be studiedin Sections4 and5, where,in particular, it will be shown thatwhenA is thesymmetrizationof a symmetrizableCartanmatrixC, thequantumgroupU cq d Ae definedabovecoincideswith theusualonedefinedby Drinfeld andJimbo(thusjustifying thechoiceof notation).

Accordingto thediscussionin Section2.2,theactionof s] 21 _ B2 onX � X is givenin this caseby

xh� xk ¨§ qakhxk

� xh jIt follows thattheactionof s] ni b 1 ] i m 1 � s] 21 � 1 ] n m 1 _ Bn on X ½ n is givenby

xh1�K�+�k�+� xhn ¨§ qahi á 1 Æ hi xh1

�â�+�+�+� xhi á 1� xhi

�K�k�+�+� xhn j (3.1)

We closethis sectionby consideringthe simplestinstancesof this construction,namely, thosecorrespondingto A bf` 0a andA bf` 1a .Examples3.2.

(1) ConsiderthecaseA bÚ` 0a . X ½ n is thenone-dimensionaland,asrecalledin Section

2.2,b ] ni actsby multiplicationby theordinarybinomialcoefficient � ni � . It followsthatK ] n b 0, sincechk b 0. Therefore,U0

q d Ae is just thepolynomialalgebrak ` xawith its usualHopf algebrastructure

∆ d xn e$b n

∑i � 0

± ni ³ xi � xn m i andSd xn eGb d h 1e nxn

(thebinomialHopf algebra).(2) WhenA bÝ` 1a , X ½ n is one-dimensionalandb ] ni actsby multiplication by the q-

binomialcoefficient ` ni a q. Sincechk b 0 andq is notarootof unity, thisoperatoris

injective. HenceU0q d Ae is still thepolyonomialalgebrak ` xa , but theHopf algebra

structureis

∆ d xn e$b n

∑i � 0

� ni � q

xi � xn m i andSd xn e$b d h 1e nq d n2 e xn j

This is theEulerianHopf algebraof [9].

444 M. Aguiar

4. The Braid Action Definedby a Matrix

Let A, q, andX be asabove. In this sectionwe studythe nullspacesK andF of thebinomial and factorial braidsacting on the tensorpowersof X by meansof A as inSection3.2. It will be shown that thesespacesarenaturallygradedover g r andthatthe non-trivial componentsof K occuronly on the locusof a certainhypersurfaceinkr that is canonicallyassociatedto A. Theseresultswill beappliedto decidewhenthequantumgroupU cq d Ae is finitely related.

4.1. TheGradingon K andF

For eachn andr _ g , let ã d n i r e denotethesetof all functionsÖ 1 i�j+j+jli n × § Ö 1 i�jkj+jli r × ,and ä d n i r eåbæÖ d η1 iVj+jkj�i ηr e _ g r n η1 � jkj+j � ηr b n × . For any η b d η1 i+jkj+j9iηr e _ ä d n i r e , letç d η e$bèÖ f _ ã d n i r e1Ô # f m 1 d 1eGb η1 i # f m 1 d 2e�b η2 i�j+j+jli # f m 1 d r e$b ηr × j

For eachf _ ã d n i r e , considerthetensor

x f : b xf ] 1 �K�k�+�+� xf ] n _ X ½ n jTheset Ö x f n f _ ã d n i r e�×is a k-basisof the vectorspaceX ½ n. It follows from (3.1) that, for eachη _ ä d n i r e ,thesubspaceX ] η ^ of X ½ n spannedby Ö x f n f _ ç d η e�× is invariantundertheactionofBn. In otherwords,thedistinguishedbasisof X determinesa gradingof T d X e over themonoid g r , andthis gradingis preservedby theactionof Bn. Therefore,thenullspacesK andF inherit a gradingover g r asfollows:

K b ∞Ëη Ä�é r

K ] η ^ where K ] η ^ b n m 1Òi � 1

ker d b ] ni : X ] η ^ § X ] η ^ e iand

F b ∞Ëη ÄÅé r

F ] η ^ where F ] η ^ b ker d f ] n : X ] η ^ § X ] η ^ e d and n b η1 � �k�+� � ηr e j4.2. TheAction of theBraidss] nσ

Theactionof thebraidss] nσ _ Bn on theabovebasisof X ½ n canbeeasilydescribedintermsof thenaturalactionof Sn on ã d n i r e , σ � f b f ¾ σ m 1.

Recallthatthesetof inversionsof apermutationσ _ Sn isê «Åë d σ e$b¯Ö d i i j e�n 1 p i ì j p n andσ d i e1� σ d j e�× iandthe inversionindex of σ is íî«�ë d σ e$b #

ê «Åë d σ e . We will make useof thewell-knownfactthat íî«Åëïb¯© ªV«�¬V�® .


Proposition4.1. Letσ _ Sn and f _ ã d n i r e . Then

s] nσ� x f b q

∑] i � j ^,ÄVð ñóò ] σ âf ] j ^ f ] i ^xσ ô f j

Proof. We proceedby inductionon © ª�«�¬��® d σ e . If © ª�«�¬��® d σ eGb 1, thatis, σ is thetrans-

position d i i i � 1e for somei, thens] nσ b s] ni andê «Åë d σ e�bõÖ d i i i � 1e�× , so the result

holdspreciselyby Equation(3.1).Thus,it sufficesto show thatif σ b τ � d i i i � 1e , © ªV«�¬V�® d σ e$b¯© ªV«�¬V�® d τ e°� 1, andthe

resultholdsfor τ, thenit alsoholdsfor σ.In this case,σ bÍ� 1 ô ô ô i m 1 i i c 1 i c 2 ô ô ô n

τ ] 1=ô ô ô τ ] i m 1 τ ] i c 1 τ ] i ^ τ ] i c 2=ô ô ô τ ] n � . We claim thatê «�ë d σ e$b ê «Åë d τ e9ö÷Ö d i i i � 1e�× jThis is clearly equivalent to τ d i eøì τ d i � 1e . If we hadτ d i eù� τ d i � 1e , it would bed i i i � 1e _ ê «�ë d τ e and d i i i � 1e�Ô_ ê «�ë d σ e , from whereit would follow that © ªV«�¬V�® d σ eGbíî«�ë d σ e$bÂíî«Åë d τ eÉh 1 bè© ª�«�¬��® d τ eÉh 1, againstour hypothesis.

We calculate

s] nσ� x f ] 2 ´ 7b s] nτ s] ni � x f b qaf ú i á 1û f ú i û s] nτ

� x ] i � i c 1,ô f jLet g b d i i i � 1e � f . Since d i i i � 1eùÔ_ ê «Åë d τ e , we have d g d k e i g d he+eGb d f d ke i f d heke forall d h i ke _ ê «Åë d τ e . Therefore,by induction,

s] nσ� x f b qaf ú i á 1 û f ú i û q ∑] h � k,ÄVð ñóò ] τ âg ] k g ] h

xτ ô g b q∑] h � k,ÄVð ñ�ò ] σ âf ] k f ] h

xσ ô f j4.3. TheAction of theMobiusBraid

It turns out that the action of µ ] n on X ½ n diagonalizesand thus can be completelydescribed.It is in this descriptionthata certainhypersurfacein kr naturallyarises.Todefineit, considerfirst thequadraticform associatedto A, namely,

QA d xe$b r

∑h� 1

ahhx2h � ∑

1 ü h ý k ü rd ahk � akh e xhxk i

wherex b d x1 i x2 i+jkj+j�i xr e is avectorin kr , andalsothelinearform

DA d xe$b r

∑h� 1

ahhxh jWe areinterestedin thehypersurfaced H e : QA d xe$b DA d xe

in particular, in pointswith non-negativecoordinatesx _ g r .

446 M. Aguiar

Lemma 4.2. Let f _ ã d n i r e andη b d # f m 1 d 1e i # f m 1 d 2e iVj+jkj�i # f m 1 d r eke _ ä d n i r e . Then

∑1 ü i þ� j ü n

af ] i ^ f ] j ^ b QA d η e=h DA d η e jProof. For eachh andk _ Ö 1 i 2 iVj+jkj�i r × , let Ahk bÿÖ d i i j e Ô 1 p i Øb j p n and f d i eEbh i f d j e�b k × . If h Øb k, thenAhk b¸Ö d i i j e�Ô f d i e�b h i f d j e�b k × , so#Ahk b ηhηk. Ontheotherhand,Ahh bâÖ d i i j e�Ô i Øb j and f d i eYb f d j eÑb h × , so#Ahh b ηh d ηh h 1e . Therefore,

∑1 ü i þ� j ü n

af ] i ^ f ] j ^ b r

∑h� 1

∑] i � j ^,Ä Ahh

af ] i ^ f ] j ^ � ∑1 ü h þ� k ü r

∑] i � j ^'Ä Ahk

af ] i ^ f ] j ^b r

∑h� 1

ahhηh d ηh h 1e°� ∑1 ü h þ� k ü r

ahkηhηk

b r

∑h� 1

ahhη2h � ∑

1 ü h ý k ü rd ahk � akh e ηhηk h r

∑h� 1

ahhηhb QA d η e=h DA d η e jThediagonalizationof µ ] n onX ½ n will bedescribedin detailin theproofof thefol-

lowing proposition.For simplicity, its statementonly providestheessentialinformationaboutits eigenvalues.

Proposition4.3. Letη _ ä d n i r e . Thenµ] n : X ] η ^ § X ] η ^ diagonalizes.Moreover, µ ] nonlyhastwo possibleeigenvalues:

q12�QA ] η ^,m DA ] η ^�� and h q

12�QA ] η ^,m DA ] η ^�� j

Proof. We needto establishsomenotation. For each f _ ã d n i r e , let f _ ã d n i r e bef d i eGb f d n � 1 h i e . If f _ ç d η e , then f _ ç d η e , too. Let

ç0 b¯Ö f _ ç d η e�Ô f b f × and

fix a disjoint decomposition ç d η e$b ç0 ö��

f Ä�� 1

Ö f i f × jCorrespondingly, X ] η ^ splitsasa directsum

X ] η ^ b Ëf Ä�� 0

k Ö x f ×GÃ Ëf Ä�� 1

k Ö x f i x f × jRecallthatµ] n b d h 1e ns] nσ whereσ bõ� 1 2 ô ô ô n

n n m 1 ô ô ô 1 � . Notethatσ � f b f andthatthesetof inversionsof this permutationis

ê «Åë d σ e$bÂÖ d i i j e1Ô 1 p i ì j p n × . Therefore,byProposition4.1,

µ ] n � x f b d h 1e nqα f x f i


whereα f b ∑1 ü i ý j ü naf ] j ^ f ] i ^ .Therefore,eachsubspacek Ö x f × with f _ ç0 or k Ö x f i x f × with f _ ç

1 is invariant

underµ] n . Onk Ö x f × with f _ ç0, µ ] n diagonalizeswith eigenvectorx f andeigenvalued h 1e nqα f . It alsofollowsreadilythat,onk Ö x f i x f × with f _ ç

1, µ] n diagonalizeswitheigenvectors

q ] 1à 2 α f x f � q ] 1à 2 α f x f and q ] 1à 2 α f x f h q ] 1à 2 α f x f

andrespectiveeigenvaluesd h 1e nq ] 1à 2 ] α f c α f ^ and h d h 1e nq ] 1à 2 ] α f c α f ^ jBut notethat

α f � α f b ∑1 ü i ý j ü n

af ] j ^ f ] i ^ � ∑1 ü i ý j ü n

af ] nc 1 m j ^ f ] nc 1 m i ^b ∑1 ü i þ� j ü n

af ] i ^ f ] j ^b QA d η e=h DA d η e iby Lemma4.2.

In particular, if f _ ç0, then

α f b d 1 Ô 2e d α f � α f eGb d 1Ô 2e�` QA d η e=h DA d η e a jThis shows thatµ] n diagonalizesandthattheeigenvaluesareasindicated.

We can now derive the main result on the nullspaceof the binomial braidsan-nouncedat thebeginningof Section4.

Proposition4.4. Theonly non-trivial componentsK ] η ^ of K occur whenη _ g r lieson thehypersurface(H). In otherwords,if K ] η ^ Øb 0, thenQA d η e$b DA d η e .Proof. By Lemma3.4, if K ] η ^ Øb 0, then h 1 is an eigenvalueof µ ] n . By Proposition4.3,this happensonly if QA d η e=h DA d η e$b 0.

4.4. TheHypersurfaceAssociatedto A

We assumethatk b for theresultsof thissection.Thefirst coupleof resultshold forarbitraryrealmatricesA bÚ` ahka of sizer (notnecessarilyinteger).

In this sectionwe obtainsomebasicinformationaboutthehypersurface d H e asso-ciatedto A by meansof theequation

QA d xe$b DA d xe j (H)

This studyis motivatedby the resultof Proposition4.4. In fact,we will obtainasa corollarythefinite generationof theidealof relationsof U0

q d Ae whenA is symmetricandpositive-definite.

448 M. Aguiar

Recallthat

QA d xe$b�� Axi x andDA d xe$b�� dA i x iwheredA b d a11 i�jkj+j�i arr e _ r and � x i y Yb ∑r

i � 1xiyi denotesthestandardinnerproductin r . Clearly,

QA b QAá At2

andDA b D Aá At2

isothereis no lossof generalityin assumingthatA is symmetric.In this case,theinnerproductandnormin r associatedto A aresimply� x i y A b�� Axi y and � x � A b QA d xe 1à 2 jProposition4.5. SupposethatA is symmetricandpositive-definite. Then,with respectto thenormdefinedbyA on r , d H e is theequationof thesphereof center d 1 Ô 2e Am 1dA

thatgoesthroughtheorigin.

Proof. We compute� x h d 1 Ô 2e Am 1dA � 2A b�� x � 2A h�� Am 1dA i x A � d 1Ô 4e�� A m 1dA � 2b QA d xeÉh DA d xe°� d 1 Ô 4e�� Am 1dA � 2 iso d H e is equivalentto � x h d 1 Ô 2e Am 1dA � A b d 1 Ô 2e�� A m 1dA � iwhich is theequationof a sphereasstated.

Corollary 4.6. Let u b d 1 i 1 iVj+jkj�i 1e _ r . If A is symmetricandpositive-definite, and

n � d 1 Ô 2e�� A m 1dA i u °� d 1Ô 2e�� Am 1u i u �� A m 1dA i dA ithenthere areno solutionsx to Equation d H e satisfyingat thesametime

x1 � x2 � jkj+j � xr b n jProof. Let Hn bethehyperplanein r with equation

x1 � x2 � jkj+j � xr b n jLet n0 _ besuchthatHn0 is tangentto thesphereof Proposition4.5(cf. Figure1).

If n � n0, Hn doesnot intersectthesphere.Sotheresultwill follow oncewe provethat

n0 b d 1Ô 2e�� Am 1dA i u °� d 1 Ô 2e�� A m 1u i u �� Am 1dA i dA j


� � �� !!!!!!!!!!!!"# $ % & ' ())))))

* + ,- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -. /0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Figure1: Thesphereandthetangentplaneof Corollary4.6.

Let P bethepointof tangency betweenHn0 andthesphere.Theradialvectorto P mustbeperpendicularto Hn0, with respectto theinnerproductdefinedby A. But thevectorAm 1u is clearlynormalto Hn0 with respectto this innerproduct,soit follows that

P b d 1 Ô 2e Am 1dA � λAm 1u

for someλ _ . We candetermineλ by imposingthefactthatP lieson thesphere:� d 1Ô 2e Am 1dA � λAm 1u h d 1 Ô 2e Am 1dA � A b d 1Ô 2e1� Am 1dA � A;

it followsthat

λ b d 1Ô 2e � A m 1dA � A� Am 1u � A b d 1Ô 2e � A m 1dA i dA 1à 2� Am 1u i u 1à 2 jNow, sincen0 is thesumof thecoordinatesof P, wehave that

n0 b2� d 1Ô 2e Am 1dA � λAm 1u i u b d 1 Ô 2e�� Am 1dA i u °� λ � Am 1u i u b d 1 Ô 2e�� Am 1dA i u °� d 1Ô 2e�� Am 1u i u 3� Am 1dA i dA andtheproof is complete.

Assumenow thatA is anintegermatrix. Considertheactionof thebraidgroupsBn

on X ½ n definedby A andtheidealK of T d X e definedby meansof thebinomialbraids(theidealof relations)asin Sections3.1and3.2.

Corollary 4.7. If A is symmetricandpositive-definite, and

n � d 1 Ô 2e4� Am 1dA i u °� d 1Ô 2e � � Am 1u i u �� A m 1dA i dA ithenK ] n b 0. In particular, theideal K is finitely generated.

450 M. Aguiar5 687�9;:< =?>�@;A B CED�F�GH IEJLK MN O8PRQ ST U8V�U�W X Y?Z;[�\

Figure2: Thesolutionsfor A b A ] 11 .

Proof. SupposeK ] n Øb 0. SinceK ] n b^] η Ä`_ ] n � r ^ K ] η ^ , theremustbesomeη _ ä d n i r esuchthat K ] η ^ Øb 0. By Proposition4.4, η lies in d H e , and it also satisfiesη1 � η2� �k�+� � ηr b n by definitionof ä d n i r e . ThiscontradictsCorollary4.6.

Remark4.1. If thematrixA is notpositivedefinite,then d H e doesnot representacom-pact hypersurfaceanymore, and theremay be infinitely many solutionsin g r . Forinstance,considerthematrix

A bba 2 h 2h 2 2 c j(A is theCartanmatrix of therootsystemA ] 11 .) In this case,Equation d H e becomes

x2 � y2 h 2xy b x � y jThesolutionsd x i ye _ g 2 to this equationareprecisely

x b d n2 � ne�Ô 2 i y b d n2 h ne+Ô 2 and x b d n2 h ne+Ô 2 i y b d n2 � ne+Ô 2 for everyn o 0 jThisequationrepresentsa parabolain 2 (cf. Figure2).

4.5. TheQuantumSerreRelations

In thissectionwe studythenullspacesK ] η ^ for somesimplechoicesof η _ g r .Let Ö ε1 i�j+jkj�i εr × bethecanonicalbasisof g r . Thesimplestchoiceis

η b nεh b d 0 iVjkj+j�i 0 i n i 0 iVjkj+jli 0e _ g r iwheren _ g andh _ Ö 1 i 2 i�jkj+jli r × arearbitrary. In this casethereis only onefunctionf _ ç d η e , namely, f d h, sothespaceX ] η ^ is one-dimensional,spannedby x f b x½ n

h .

Now, from Equation(3.1), every generators] ni actson this spaceby multiplicationby


qahh. Therefore,asrecalledin Section2, b ] ni actsby multiplication by the binomialcoefficient ` ni a qahh. But sinceq is not a root of unity, this elementis non-zero,so we

concludethatK ] η ^ b 0 in this case.Thenext simplestchoiceis

η b nεh � εk b d 0 iVj+jkj�i 0 i n i 0 iVjkj+jli 0 i 1 i 0 iVj+jkj�i 0e _ g r iwheren _ g andh andk _ Ö 1 i 2 iVj+jkj�i r × arearbitrarybut distinct. In this casetherearen � 1 functionsin

ç d η e , namely

fi b ± 1 �+�k� i i � 1 i � 2 �+�k� n � 1h �+�k� h k h �+�k� h ³ i for i b 0 iVjkj+j�i n j

Therefore,the spaceX ] η ^ is d n � 1e -dimensional,spannedby x fi : b x½ ih

� xk�

x½ ] n m i ^h _ X ½ ] nc 1 .

Wewill show that,for theseη, thenullspacesK ] η ^ areeithertrivial or one-dimensio-nal,spannedby someparticularvectorsSn

hk _ X ½ ] nc 1 to bedefinedin Proposition4.11,calledthequantumSerre relations. Thefirst resultin thisdirectionis

Lemma 4.8. Letη b nεh � εk beasabove. If d n h 1e ahh � ahk � akh Øb 0, thenK ] η ^ b 0.

Proof. By Lemma3.4, K ] η ^ b 0 if h 1 is not an eigenvalue of µ] nc 1 in X ] η ^ . ByProposition4.3, if h 1 is aneigenvalueof µ ] nc 1 thenQA d η e$b DA d η e . But for this η,

QA d η e$b ahhn2 � akk � d ahk � akh e n and DA d η eGb ahhn � akk j

Therefore,QA d η eGb DA d η e if andonly if d n h 1e ahh � ahk � akh b 0.

ThespacesK ] η ^ weredefinedin Section4.1astheintersectionof thenullspacesof

the binomial braidsb ] nc 1j actingon X ] η ^ Ó X ½ ] nc 1 . The following resultsaysthat,

undercertainhypothesis,all thesenullspacescoincide.

Lemma 4.9. If for each p b 1 iVjkj+jli n thebraid ` pa _ kBp is injectiveonX ]A] p m 1 εh c εk ^ ÓX ½ p, then

ker d b ] nc 1j n X ú nεh á εk û e$b K ] nεh c εk ^ � j b 1 iVj+jkj�i n j

Proof. Recallthefactorialformulas(2.1)and(2.4)

f ] j ^ � f ] nc 1 m j ^ � b ] nc 1j b f ] nc 1 b 1 ] n � ` 1a � 1 ] n m 1 � ` 2a �V�k�+�9� 1 � ` na � ` n � 1a j

We claim that1 � ` na is injective on thespaceX ] nεh c εk ^ . In fact,this spacesplitsasthedirectsumof theone-dimensionalspacespannedby x f0 b xk

� x½ nh _ X ½ ] nc 1 andthe

spacek Ö xh × � X ]A] n m 1 εh c εk ^ , andbothof theseareinvariantunder1 � ` na . On thefirst,1 � ` na actsby multiplicationby theq-analog ` na qahh, which is non-zerosinceq is not aroot of unity, while on thesecondit is injective by hypothesis.Similarly, all the lowerfactors1 ] nc 1 m p � ` pa areinjective on X ] nεh c εk ^ , for p b 1 iVjkj+jli n. It follows that f ] j ^and f ] nc 1 m j ^ areinjectiveon X ] nεh c εk ^ , for j b 1 iVj+jkjVi n, and,from here,that

ker d b ] nc 1j n X ú nεh á εk û e$b ker d ` n � 1a n X ú nεh á εk û e for j b 1 iVjkj+j�i n j

SinceK ] nεh c εk ^ is theintersectionof thesekernels,theresultfollows.

452 M. Aguiar

In theproofsbelow, qahh will beabbreviatedby qh andqahk by qhk. Wewill assume,for simplicity, that ahh Øb 0 (a hypothesisthat is alwayssatisfiedby Cartanmatrices).Also, ` na h will denotetheqh-analogof thenaturalnumbern and ` ni a h theqh-analogofthebinomialcoefficient � nj � . Thesubindicesh Øb k remainfixed.

Lemma 4.10. Assumethatahh Øb 0 andlet a b d ahk � akh e�Ô ahh. Then

n

∑i � 0

d h 1e i � ni � qahh

qi ] akh c ahk ^ c d i2e ahh b 0 if andonly if h a _ Ö 0 i 1 i 2 iVj+jkj�i n h 1 × j

Proof. First notethat

qhkqkh b qah j ( Û )

Considerthepolynomial

f d xeGb n

∑i � 0

d h 1e iq d i2 e

h� n

i � hxn m i _ k ` xa j

By oneof Cauchy’s identitiesfor q-binomials[2, Section6.2; 4, Corollary 2], f d xefactorsasfollows:

f d xeGb d x h 1e d x h qh e �k�+� d x h qn m 1h e j ( ÛlÛ )

Now,

n

∑i � 0


qi ] akh c ahk ^ c d i2 e ahh b n

∑i � 0

d h 1e iqihkq

ikhq d i

2 eh

� ni � h] ß ^b n

∑i � 0

d h 1e iqaih q d i

2eh

� ni � h

b f d xexn n x� q � a

h] ß�ß ^b d q m ah h 1e d qm a

h h qh e �+�+� d q m ah h qn m 1

h eqm an

hi

which is zeroif andonly if h a _ Ö 0 i 1 i 2 iVj+jkj�i n h 1 × .

Proposition4.11. For each i b 0 iVjkj+jli n, let

λi b d h 1e i � ni � qahh

qiakh c d i2 e ahh _ k and Sn

hk b n

∑i � 0

λix fi _ X ] nεh c εk ^ jAssumethatahh Øb 0. Then

ker d bnc 11 n X ú nεh á εk û e�bfeg h k Ö Sn

hk × if h d ahk � akh ekÔ ahh _ Ö 0 i 1 i 2 iVjkj+jli n h 1 ×0 otherwise.


Proof. By definition [2, Sections3 and7.1], b ] nc 11 b ∑nc 1

j � 1 s] nc 1σ j , whereσ j _ Snc 1 is

the j-cycle σ j b d 1 i 2 i�j+j+jli j h 1 i j e . Thesetof inversionsof this permutationisê «Åë d σ j e�b¯Ö d 1 i j e i d 2 i j e ikj+jkjli d j h 1 i j e�×and

σ j� fi b fi ¾ σ m 1

j b eiiig iiih fi c 1 if 0 p i p j h 2 if0 if i b j h 1 ifi if j p i p n j

Hence,by Proposition4.1,theactionof s] nc 1σ j on thebasiselementsx fi of X ] nεh c εk ^ is

x fi ¨§ eiiig iiih q ] j m 2 ahhc akhx fi á 1 if 0 p i p j h 2 iqiahkx f0 if i b j h 1 iq ] j m 1 ahhx fi if j p i p n j

It followsthat,for eachi b 0 iVjkj+jli n,

b ] nc 11

� x fi b Ï q ] 1 m 1 ahh � q ] 2 m 1 ahh � �k�+� � q ] i m 1 ahh Ð x fi � qiahkx f0 �� Ï q ] i c 2 m 2 ahhc akh � q ] i c 3 m 2 ahhc akh � �k�+� � q ] nc 1 m 2 ahhc akh Ð x fi á 1b ` i a hx fi � qihkx f0 � qkhqi

h ` n h i a hx fi á 1 jLet x b ∑n

i � 0µix fi bea generalelementof X ] nεh c εk ^ , whereµi _ k arearbitraryscalars.Then

b ] nc 11

� x b n

∑i � 0

µi ` i a hx fi � n

∑i � 0

µiqihkx f0 � n

∑i � 0

µiqkhqih ` n h i a hx fi á 1

b n

∑i � 1 Ï µi ` i a h � µi m 1qkhqi m 1

h ` n h i � 1a h Ð x fi � Ï n

∑i � 0

µiqihk Ð x f0 j

It follows thatx _ ker d b ] nc 11 n X ú nεh á εk û e if andonly if

0 b µi ` i a h � µi m 1qkhqi m 1h ` n h i � 1a h for eachi b 1 i�j+j+j�i n and (a)

0 b n

∑i � 0

µiqihk j (b)

454 M. Aguiar

Equation(a)determinesµi in termsof µ0, for i b 1 iVj+jkj�i n:

µ1 b h µ0qkh ` na hµ2 b h µ1qkhqh

` n h 1a h` 2a h b µ0q2khqh

` na h ` n h 1a h` 2a h b µ0q2khqh

� n2 � h

µ3 b h µ2qkhq2h

` n h 2a h` 3a h b¯h µ0q3khq3

h

� n2 � h

` n h 2a h` 3a h bÚh µ0q3khq3

h� n3 � h

andin general,for i b 1 iVjkj+jVi nµi b d h 1e iµ0qi

khq d i2 e

i� n

i � hb µ0λi i

whereλi is asdefinedin thestatementof theproposition.This meansthatx b µ0Snhk.

Thus,therearetwo possibilitiesfor thekernel.If theλi satisfyEquation(b) in placeoftheµi, thenthekernelis one-dimensionalspannedby Sn

hk, if not, thekernelis zero.Butwhenwe substituteµi for λi in theright-handsideof (b), we get

n

∑i � 0

λiqihk b n

∑i � 0


qi ] akh c ahk ^ c d i2e ahh i

which, by Lemma4.10, is zeroif andonly if h d ahk � akh e�Ô ahh _ Ö 0 i 1 i 2 i+jkj+jli n h 1 × .Thiscompletestheproof.

ThevectorsSnhk _ X ½ ] nc 1 definedin Proposition4.11arecalledthequantumSerre

relations(we will show in Section5.1 that they boil down to theusualquantumSerrerelationsfor the caseof Cartanmatrices).We cannow derive the main resultof thissection,which describesthenullspacesK ] η ^ , for η asabove, in termsof the quantumSerrerelations.

Corollary 4.12. Assumethatahh Øb 0 andlet Snhk beasbefore. Then

K ] nεh c εk ^ bjeg h k Ö Snhk × if h d ahk � akh e�Ô ahh b n h 1 i

0 otherwise.

Proof. If h d ahk � akh ekÔ ahh Øb n h 1, thenK ] nεh c εk ^ b 0 by Lemma4.8. Supposethath d ahk � akh ekÔ ahh b n h 1. Then,in particular,h d ahk � akh ekÔ ahh Ô_ Ö 0 i 1 i�j+j+jli p h 2 × � p b 1 iVj+jkj�i n jTherefore,by Proposition4.11,b ] p1 is injective on X ] ] p m 1 εh c εk ^ for p b 1 iVjkj+jli n. ButthenLemma4.9appliesto conclude,in particular, that

K ] nεh c εk ^ b ker d bnc 11 n X ú nεh á εk û e$b k Ö Sn

hk × ;

thelastequalityby Proposition4.11again.


Finally, wesummarizetheinformationwehaveobtainedonthenullspacesK andFfor arbitrarymatricesA. For thecaseof Cartanmatrices,morepreciseinformationwillbe obtainedin Section5.1, whereit will be shown that our definition of the quantumgroupboilsdown to thatof Drinfeld andJimbo.

Corollary 4.13. Let Sbetheideal of T d X e generatedby thequantumSerre relations

Snhk b n

∑i � 0


qiakh c d i2 e ahhx fi _ X ½ ] nc 1

for thoseh andk for which ahh Øb 0 and d n h 1e ahh � ahk � akh b 0. Then

S Ó K Ó F jProof. By Corollary4.12,Sn

hk _ K for thoseh andk, soS Ó K. Therestis Lemma3.3.

5. The Caseof Cartan Matrices

In this sectionwe specializethe constructionsandresultsof Sections3 and4 to thecasewhenA is a symmetricCartanmatrix, or, moregenerally, the symmetrizationofa symmetrizableCartanmatrix C. In Section5.1, it will be shown that the quantumgroupU0

q d Ae introducedin Section3.2 is theusualonedefinedby Drinfeld andJimbo.In Section5.2, we prove that, whenC is of finite type, the pointswith non-negativeintegercoordinateson thehypersurface(H) from Section4.3includetheverticesof thezonotopecorrespondingto C, andtheremainingintegerpointson (H) lie on thewallsof the root systemof C. This refinesthe resultsof Section4.4 andis oneof the mainresultsof thepaper. Therelevantnotionson Cartanmatricesandroot systemswill bereviewedasthey areneeded.

5.1. GeneralizedCartanMatrices:QuantumGroups

A matrixC bÚ` chka _ Mr d ÕÞe is calleda (generalized)Cartanmatrix if

chh b 2 � h b 1 iVj+jkj�i r ichk p 0 for h Øb k and

if chk b 0 thenckh b 0 jWe will alsoassume,without furthernotice,thatall Cartanmatricesare indecompos-able; thismeansthatthereis nopermutationσ _ Sr suchthat ` cσ ] h σ ] k a splitsasadirectsum.

An arbitrarymatrixC _ Mr d ÕÞe is calledsymmetrizableif thereis an invertibledi-agonalmatrixD _ Mr d ÕÞe suchthatDC is symmetric.

If C is a symmetrizableCartanmatrix, thenthediagonalmatrix D is uniqueup to aconstantfactor, andall its entrieshavethesamesign.ThecanonicalsymmetrizationofC,

A : b DC i

456 M. Aguiar

is theonecorrespondingto thechoiceof D with minimumpositive integerentries.(FormoredetailsonCartanmatrices,thereaderis referredto [11, Chapters1, 2, and4].)

Associatedto any symmetrizablegeneralizedCartanmatrixC, thereis Lie algebrag d C e , calledaKac–MoodyLie algebra,andaquantumgroup(Hopf algebra)Uq d g d C eke ,definedby meansof generatorsandrelations[7, 4.3]. The latterwerefirst definedbyDrinfeld [3] andJimbo[8]. We shallconcentrateon thesubalgebraU0

q d g d C eke , which isdefinedby generatorsxh for h b 1 iVjkj+j�i r (usuallydenotedby Eh instead),subjectto theso-calledquantumSerrerelations:

n

∑i � 0

d h 1e i ± ni ³ qdh

xihxkx

n m ih b 0 wheneverchk b 1 h n j

Here,chk are the entriesof C, dh are the (diagonal)entriesof D and the q-binomialcoefficient � ni � q is that of [12, VI.1.6], not the q-binomial ` ni a q of previous sections.

Theseq-binomialsarerelatedby theformula[12, VI.1.8]1± ni ³ q

b qm i ] n m i ^ � ni � q2

jLet uscomparethis definitionwith our definitionof U0

q d Ae$b T d X ekÔ K. Theentriesof A, C, andD arerelatedby ahk b dhchk; chh b 2 andahk b akh. It follows thatd n h 1e ahh � ahk � akh b 0 k chk b 1 h n iand d h 1e i � n

i � qahh

qiakh c d i2 e ahh b d h 1e i ± n

i ³ qdhi

sothat

Snhk b n

∑i � 0


qiakh c d i2 e ahhx fi b n

∑i � 0

d h 1e i ± ni ³ qdh

xihxkx

n m ih j

In other words, the quantumSerrerelationsthat were introducedin Section4.5 forarbitrarymatricesA, boil down to the usualoneswhenA is the symmetrizationof aCartanmatrixC.

To provethatourquantumgroupU0q d Ae coincideswith U0

q d g d C eke asdefinedabove,we mustshow that the ideal K is generatedby the quantumSerrerelations,i.e., thatS b K. We know from Corollary4.13thatS Ó K Ó F.

To provethatequalityholds,weneedto recallyetanotherdefinitionof thequantumgroup,namelythatof Lusztig,which is closerto ours. In Lusztig’s book,U0

q d g d C e+e isdefinedasthequotientof T d X e by theradicalof a certainbilinear form on T d X e [13,1.2.5].Lusztigproves,afterdevelopingtherepresentationtheoryof hisquantumgroup,thatthis idealis generatedby thequantumSerrerelations[13, 33.1.5].

On the other hand,Schauenburg [15, Example3.1, Theorem2.9] hasnotedthatLusztig’s ideal coincideswith F , the nullspaceof the factorial braidsas definedin

1 Warning:our l ni m q is Kassel’s n ni o q andvice versa.


Section4 (thefactthatthebraidsappearingin Schauenburg’spaper[15,Definition2.6]coincidewith our factorialbraidswasalreadynotedin [2, Section7.1]).

Therefore,S b K b F , andtheproof thatour definitionboils down to thatof Drin-feld andJimbo(or Lusztig’s) is complete.

5.2. CartanMatricesof FiniteType.Zonotopes

A Cartanmatrix is of finite type if it is positive-definite. SuchCartanmatricesarealwayssymmetrizable.Finite-dimensionalsimpleLie algebrasover p arein one-to-onecorrespondencewith symmetrizableCartanmatricesof finite type.GoodreferencesforCartanmatrices,simpleLie algebras,androot systemsare[5, ChapterIII] and[16,Chapters2, Section3.1]. We will only needthebasicinformationoutlinedbelow.

Let C be a Cartanmatrix of finite type andA b DC its canonicalsymmetrization.Let E be the root spaceof C. By definition, E is an Euclideanspaceof dimensionr(thesizeof C) with a distinguishedbasis Ö α1 ilj+jkj�i αr × (whoseelementsarecalledthesimpleroots) andinnerproduct� αk i αh E b ahk b dhchk jNotethattheentriesof C andD aregivenby

chk b 2� αk i αh E� αh i αh E anddh b d 1Ô 2e4� αh i αh E

(this shows thatour notationfor the Cartanintegerschk is the transposeof thatof [5]and[16]).

Considerthe linear transformationT : E § E whosematrix in thebasisof simplerootsisC. SinceC is invertible,sois T, andasecondbasisÖ ω1 ilj+j+j�i ωr × of E is definedby T d ωh eGb αh. Thevectorsωh arecalledthe fundamentalweights. Equivalently, wehave

αh b r

∑k � 1

chkωk j (5.1)

The lowestweightis thevector

δ b r

∑h� 1

ωh _ E j (5.2)

Let usreformulatetheresultsof Section4.4in thiscontext.

Proposition5.1. Let C be a Cartan matrix of finite type and A b DC its canonicalsymmetrization.

(1) With respectto the canonical inner product on the root spaceof C, d H e is theequationof thesphereof centerδ (thelowestweight)thatgoesthroughtheorigin.

(2) Supposethat C is symmetric. Then,if n � 2 � δ � 2E, there are no solutionsx toequationd H e satisfyingat thesametimex1 � x2 � �+�+� � xr b n. In particular, K ] n b0 for thesen.

458 M. Aguiar

Proof. By assumption,thediagonalof A is dA b 2Du, whereu b d 1 i 1 iVjkj+jli 1e _ r , soequationd H e for thematrix A becomes� DCx i x �b 2 � Du i x jSinceC is positive definite,so is A. As explainedabove, if we identify the root spacewith r in sucha way that the simplerootscorrespondto the canonicalbasisof r ,thenthecanonicalinnerproductof theroot spacebecomestheinnerproduct � Axi y of r . Therefore,Proposition4.5applies,and d H e is theequationof thesphereof centerd 1 Ô 2e Am 1dA thatgoesthroughtheorigin, with respectto thecanonicalinnerproductontherootspace.To completetheproof of 1, we mustshow that d 1Ô 2e A m 1dA b δ. Now,d 1Ô 2e Am 1dA b d 1Ô 2e C m 1D m 12Du b C m 1u jOn the otherhand,considerthe linear transformationT asabove. Underthe identifi-cationin questionbetween r andE, u correspondsto ∑r

h� 1 αh, andC to T; therefore,C m 1u correspondsto

r

∑h� 1

T m 1 d αh e$b r

∑h� 1

ωh(5.2)b δ j

Now assumethatC is symmetric,i.e., A b C. In this casedA b 2u andthe righthandsideof theinequalityin Corollary4.6becomessimply 2 � C m 1u i u . But� C m 1u i u �b�� C m 1u i C m 1u E b�� C m 1u � 2E b�� δ � 2E iso2 is justa reformulationof Corollary4.6.

Examples5.1. For the root systemAr , correspondingto the Lie algebraslr c 1 d p e , theCartanmatrix is

C bqrrrrrrrrs 2 h 1 0 jkj+j 0h 1 2 h 1 jkj+j 0

0 h 1 2 jkj+j 0

......

0 j+j+j 0 h 1 2

tvuuuuuuuuw(asquarematrix of sizer).

Fromtheproof abovewe know thatδ b C m 1u. Let

xk b d 1 Ô 2e k d r � 1 h ke _ for k b 1 i�j+j+jli r andx b d x1 i�jkj+j�i xr e _ r jOneeasilycomputesCx b u, from wherex b δ. Therefore,� δ � 2E b�� C m 1u i u �b�� x i u �b r

∑k� 1

xk b d 1Ô 12e r d r � 1e d r � 2e jThusif n � d 1Ô 6e r d r � 1e d r � 2e , thenK ] n b 0. For instanceif r b 2, thenK ] n b 0 forn � 4. Thedescriptionof K ] n for n p 4 will becarriedout in Example5.2below.


LetW betheWeyl groupof C, thatis, thegroupof isometriesof E generatedby thereflectionsacrossthehyperplanesperpendicularto thesimplerootsαh _ E. SinceC isof finite type,W is finite. Thefinite set

R bèÖ w d αi e _ E Ô w _ W andi b 1 iVj+jkj�i r ×is calledtheroot systemof C, andits elementsarecalledroots.For α _ R, let Hα

Ó Ebethehyperplaneperpendicularto α. Thesehyperplanesarecalledthewallsof therootsystem.Theconnectedcomponentsof

E x �α Ä R

Hα

arecalledtheWeyl chambers.The closure ä of a Weyl chamberä is calleda closedWeyl chamber.

As before,we identify d E i � i E e with d r i � i A e by sendingÖ α1 iVjkj+jli αr × to thecanonicalbasisÖ e1 iVj+jkj�i er × of r . Thesetof integerlinearcombinationsof thesimplerootsthengetsidentifiedwith Õ r Ó r . This is calledthe root latticeof C. Thesetofnon-negative integercombinationsof thesimplerootsis calledthepositive cone;it isidentifiedwith g r . Thesetof integerlinearcombinationsof thefundamentalweightsiscalledtheweightlattice. For instancethelowestweightδ belongsto theweight lattice(Equation(5.2)). Equation(5.1) shows that the root lattice is includedin the weightlattice.

A root α _ R is calledpositive if it lies in thepositive cone g r . Let Rc denotethesetof positiveroots.Thezonotopeof C is definedastheMinkowskisumof thepositiveroots: Ö ∑

α Ä Rá λαα _ E Ô 0 p λα p 1 × Ó E jFor instance,thezonotopeof the root systemAr is calledthe permutahedron[19, Ex-amples0.10and7.15]. Figures4 and5 show thezonotopesof A2 andB2.

Theproof below will rely on thefollowing facts[5, ChapterIII]:

(a) Theset ä F b¯Ö λ _ E Ô�� αh i λ E ì 0 � h b 1 iVjkj+j�i r ×is aWeyl chamber(theoppositeof thefundamentalWeyl chamber).

(b) The chamberä F coincideswith the negative conegeneratedby the fundamentalweights,thatis, ä F bèÖ r

∑h� 1

ahωh _ E Ô ah ì 0 � h b 1 i+jkj+j�i r × j(c) If ä is aWeyl chamberandw _ W, thenw d ä e is aWeyl chamberandw d ä e�b w d ä e .(d) Giventwo Weyl chambersä and ä�y , thereis a uniquew _ W suchthatw d ä1eGbÚä�y .(e) TheclosedWeyl chamberscoverE.

460 M. Aguiarz{ | } ~ � ��

Figure3: Thetranslatedfundamentalchamberfor A2.

(f) Theroot lattice Õ r is invariantundertheactionof W.(g) Eventhoughδ maynot lie in theroot lattice Õ r, δ h w d δ e lies in thepositive coneg r � w _ W. (In fact, δ h w d δ e is equalto the sumof thosepositive roots that

becomenegativeunderw.)(h) Theset Ö δ h w d δ e$Ô w _ W × is preciselythesetof verticesof thezonotopeof C.

Thefollowing resultdescribesin somedetailthesetof integersolutionsof Equationd H e in termsof thezonotopeof C .Theideaof theproof is to translatethehyperplanesandWeyl chambersfrom theorigin to δ, andobserve that thereis only onesolutionineverytranslatedchamber. Accordingto (d) above,it sufficesto look for solutionsin the(translated)chamberδ � ä F . We will show that theonly integral solutionof this typeis λ b 0 (part1 of Lemma5.2). This will allow us to concludethat thesetof integralsolutionsthatdo not lie on thewalls (translatedby δ) is preciselythesetof verticesofthezonotope(Theorem5.3),which is themainresultof thissection.Wewill alsoshowthatλ b 0 is theonly solutionlying at thesametime in theclosedchamberδ � ä F andin thepositiveconeg r (part2 of Lemma5.2). Figure3 reflectstheseassertionsfor therootsystemA2.

Lemma 5.2. Letλ _ Õ r bea solutionof Equation d H e lying on theroot lattice.

(1) If λ _ δ �âä F , thenλ b 0.(2) If λ _ d δ � ä F e��Og r , thenλ b 0.

Proof. (1) Accordingto (b) we canwrite

λ h δ b r

∑h� 1

ahωh with ah ì 0 � h jRecallthatδ b ∑r

h� 1 ωh. Sincetheroot lattice in includedin theweight lattice,λ alsobelongsto theweightlattice. It follows that

ah pÂh 1 � h j


Sinceλ belongsto thesphered H e , we havethat � λ h δ � E b�� δ � E. Hence,� λ � 2E b2� λ h δ � δ � 2Eb2� λ h δ � 2E � 2 � λ h δ i δ E �� δ � 2Eb 2 � δ � 2E � 2 � λ h δ i δ Eb 2r

∑h � k� 1

� ωh i ωk E � 2r

∑h � k� 1

ah � ωh i ωk E p 0 iandthusλ b 0 asneeded.

(2) Let

λ b r

∑k � 1

λkαk with λk _ gbeasolutionin d δ � ä F e�� g r . Thenλ h δ _ ä F , soby (a)

0 o�� λ h δ i αh E b�� λ i αh E h�� δ i αh E b r

∑k� 1

λkdhchk h dh

from which it follows(sincedh � 0 � h) thatr

∑k� 1

λkchk p 1 � h b 1 i�j+j+jli r j (A)

On theotherhand,sinceλ lieson thesphered H e , we have� λ h δ � E b�� δ � E � � λ i δ E b d 1 Ô 2e�� λ i λ E j (B)

Let uscomputeeachsideof (B) separately:� λ i δ E b�� r

∑h� 1

λheh i C m 1u A b r

∑h� 1

� λheh i Du �b r

∑h� 1

λhdh

while � λ i λ E b r

∑h � k� 1

λkλh � αk i αh E b r

∑h � k� 1

λkλhdhchk b r

∑h� 1

λhdh Ï r

∑k� 1

λkchk Ð jCombiningthesewith (A) andthefactthatλh o 0 � h weobtain� λ i λ E b r

∑h� 1

λhdh Ï r

∑k� 1

λkchk Ð p r

∑h� 1

λhdh b�� λ i δ E jNow, using(B), we obtain� λ i δ E b d 1Ô 2e4� λ i λ E p d 1Ô 2e4� λ i δ E iwhich implies � λ i δ E b�� λ i λ E b 0 i.e. λ b 0 jThiscompletestheproofof thelemma.

462 M. Aguiar

Theorem5.3. LetC bea Cartanmatrix of finite type. Thesetof solutionsof Equationd H e that lie in theroot lattice Õ r but noton thewallsof theroot systemofC (translatedby δ) is preciselythesetof verticesof thezonotopeof C or, equivalently, thesetÖ δ h w d δ e n w _ W × ionedifferentsolutionfor each w _ W. In particular, thesesolutionslie in thepositiveconeg r .

Proof. Let Tδ : E § E bethetranslationof vectorδ,

Tδ d λ e$b λ � δ for λ _ E jFor eachWeyl chamberä , let w_ _ W betheonly elementof theWeyl groupsuchthatw_ d ä F e$bfä , andlet

λ _ b δ h w_ d δ e _ E jWe know from (g) that

λ _ _ g r j (C)

We will show that

λ _ _ Tδ d ä e (D)

and

λ _ is theonly solutionin Tδ d ä e�� Õ r j (E)

SincetheclosedWeyl chamberscoverE andtheopenWeyl chambersaredisjoint, thiswill provethatthesetof solutionsis asdescribed.

Considertheactionof W onE obtainedby conjugatingthecanonicalactionby Tδ:

w � λ : b δ � w d λ h δ e for w _ W andλ _ E. ( Û )

Notethatw_ F b��`� E, soλ _ F b 0 _ E. Hence

λ _ b δ h w_ d δ e$b δ � w_ d λ _ F h δ e�b w_ � λ _ F jOntheotherhand,since ä b w_ d ä F e ,

Tδ d ä e�b Tδ ¾ w_ d ä F e�b w_ ¾ Tδ d ä F e jTherefore,assertion(D) is equivalentto λ _ F _ Tδ d ä F e . Sinceλ _ F b 0, this is in turnequivalentto h δ _ ä F , which is obviousfrom (b) and(5.2).

The canonicalactionof W is by isometriesof E that fix theorigin, while Tδ is anisometrythat sendsthe origin to δ. Therefore,action d ÛVe is by isometriesthat fix δ.Hencethis action preserves the sphere d H e of centerδ. Also, by (f) and (g), Õ r isinvariantunderthisaction.Thus,thesetof solutionsto d H e in Õ r is invariantunder d Û�e .Therefore,assertion(E) is equivalentto

λ _ F b 0 is theonly solutionin Tδ d ä F e�� Õ r jObviouslyλ _ F b 0 is asolutionof d H e . It is theonly suchpreciselyby part1 of Lemma5.2.Thus(E) holdsandtheproof is complete.


Remark5.1. Any closedWeyl chamberä is a fundamentaldomainfor the canonicalactionof W onE (this is thecontentof assertions(c), (d), and(e)above).This implies,asin theproofof Theorem5.3,thatthesetTδ d ä1e�� Õ r � H is a fundamentaldomainfortheaction d ÛVe of W on thesetof solutionsin Õ r to d H e . Part1 of Lemma5.2saysthat

Tδ d ä F e��OÕ r � H b¯Ö 0 × ii.e. λ b 0 is theonly solutionin the interior of this fundamentaldomain. Its W- orbitis of coursethesetof verticesof thezonotope,asin Theorem5.3. For root systemsofrank2, it is possibleto proveby a geometricargumentthat

Tδ d ä F e��OÕ r � H b¯Ö 0 × ii.e., the origin is the only solution in this fundamentaldomain,and hencethe solu-tions in Õ r to d H e arepreciselytheverticesof thezonotope.In otherwords,for theseroot systems,thereareno solutionslying on thetranslatedwalls. This will beverifieddirectly in Examples5.2.

In general,however, thereareothersolutionsin this fundamentaldomain(necessar-ily on thetranslatedwalls), andhenceothersolutionsin Õ r besidestheverticesof thezonotope.Thiswaspointedout to theauthorby DanBarbasch,who foundthesolutionα1 � α2 � α3 h α5 for the root systemA5. This solutionlies in the root lattice andinoneof thewalls of Tδ d ä F e . In termsof coordinates,theequationof d H e is in thiscase

x21 � �+�k� � x2

5 h x1 h �+�+� h x5 h x1x2 h x2x3 h �+�k� h x4x5 b 0 iandtheabovesolutioncorrespondsto d 1 i 1 i 1 i 0 i h 1e=b . Notethatthesolutiondoesnotlie in thepositiveconeg r , aspredictedby part2 of Lemma5.2.However, somepointsin its W-orbit will lie in g r (sinceonecanfind a chamberä suchthatTδ d ä e Ó g r andthena w suchthatw d ä F e�bÿä ). This shows thateven if we only look for solutionstod H e lying in g r , therearesuchsolutionswhich arenot verticesof thezonotope.

A simpledescriptionof these“extra” solutionsappearsdifficult. UsingMaple,wehavecomputedthenumberof solutionsin thefundamentaldomainTδ d ä F e��åÕ r � H forthe root systemAr for r p 11. Thesenumbersaredisplayedin Table1. In eachcase,λ b 0 is the only solution that lies in g r or that doesnot lie in any of the translatedwalls,asguaranteedby Lemma5.2. Thefirst casewhena non-zerosolutionappearsisr b 5.

Table1: Thenumberof solutionsin thefundamentaldomainfor Ar .

r 1 2 3 4 5 6 7 8 9 10 11

# Ï Tδ d ä F e�� Õ r � H Ð 1 1 1 1 3 9 27 80 255 847 2774

464 M. Aguiar

Examples5.2.

(1) For therootsystemA2, wehave

C b A b a 2 h 1h 1 2 c andδ b C m 1u b d 1 i 1e jEquationd H e is simply

x2 � y2 b xy � x � yjFromexample5.1we know that thereareno solutionswith x � y � 4. In fact,it iseasyto seedirectly thattheonly solutionsin Õ 2 ared 0 i 0e i d 1 i 0e i d 0 i 1e i d 1 i 2e i d 2 i 1e i and d 2 i 2e jTheseare the verticesof the zonotopeof A2, so in this case,thereareno othersolutionsthanthosepredictedby Theorem5.3.Therefore,

K ] n b 0 � n o 5 or n p 2;

K ] 3 b K ] 2 � 1 Ã K ] 1 � 2 and

K ] 4 b K ] 2 � 2 jThefirst two non-trivial componentsarespannedby thequantumSerrerelations,asdescribedby Corollary4.12. Notethat h d a12 � a21e�Ô a11 b 1 bSh d a21 � a12ekÔ a22,therefore,

K ] 2 � 1 b k Ö S212 × whereS2

12 b x112 h d q � q m 1e x121 � x211

and

K ] 1 � 2 b k Ö S221 × whereS2

21 b x112 h d q � q m 1e x212 � x211 jTheremainingnon-trivial componentcanbecomputedby handandit turnsout tobeone-dimensionalaswell:

K ] 2 � 2 b k Ö R22 × whereR22 bÚh x1122 � d q � qm 1 e x1212 h d q � q m 1 e x2121 � x2211jFromSection5.1weknow thattherelationat d 2 i 2e mustbegeneratedby thequan-tum Serrerelations. In fact, in this caseeitherSerrerelationsufficesto generateR22, sincewehave that

R22 b¯h x1� S2

12 � S212

� x1 b¯h S221

� x2 � x2� S2

21 jThezonotopeis shown in Figure4.


(0,0) (0,1)

(1,0) (1,2)

(2,2)(2,1)

Figure4: Thezonotopeof A2.

(2) For therootsystemB2 we have

C b a 2 h 1h 2 2 c andD bba 2 0

0 1c i soA b DC b¡a 4 h 2h 2 2 c andδ b C m 1u b d 3 Ô 2 i 2e ;equationd H e is

4x2 � 2y2 h 4xy b 4x � 2yitheonly solutionsin Õ 2 ared 0 i 0e i d 1 i 0e i d 0 i 1e i d 2 i 1e i d 1 i 3e i d 3 i 3e i d 2 i 4e and d 3 i 4e iandthezonotopeis shown in Figure5.

(0,1)

(2,1)

(1,0) (1,3)

(2,4)

(3,4)(3,3)

(0,0)

(3/2,2)

Figure5: Thezonotopeof B2.

ThequantumSerrerelationsoccurat theverticesd 2 i 1e and d 1 i 3e .(3) For therootsystemG2, wehave

C bba 2 h 1h 3 2 c andD b a 3 0

0 1c i soA b DC bba 6 h 3h 3 2 c andδ b C m 1u b d 3 i 5e j

466 M. Aguiar

Equationd H e is

6x2 � 2y2 h 6xy b 6x � 2yiandtheonly solutionsin Õ 2 ared 0 i 0e i d 1 i 0e i d 0 i 1e i d 2 i 1e i d 1 i 4e i d 4 i 4e i d 2 i 6e i d 5 i 6e i d 4 i 9e i d 6 i 9e i d 5 i 10e i and d 6 i 10e iandthezonotopeis shown in Figure6.

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Figure6: Thezonotopeof G2.

6. Further Questionsand Results

Considertheactionof thebraidgroupsBn on thetensorpowersX ½ n of a vectorspaceX, definedby meansof a fixed matrix A _ Mr d ÕÞe asin Sections3.1 and3.2. In thispapertheattentionhasbeenconcentratedon thestudyof thecorrespondingnullspacesof the binomialbraids,motivatedby its relevanceto quantumgroups. In Section4.5,somecomponentsK ] η ^ of thesenullspaceswereexplicitly described.It maybepossibleto obtainsimilar descriptionsof theothernon-trivial components(andthusunderstandwhy thequantumSerrerelationssufficeto generatetheidealof relationsfor thecaseofCartanmatrices,withoutresortingto Lusztig’sresultaswedid in Section5.1.A relatedquestionthatarisesnaturallyis whetherK b F always(we know it holdsfor thecaseof Cartanmatricesfrom Section5.1.

Thereareotherinterestingquestionsandresultsaboutthis action. It is possibletoobtainexplicit expressionsandfactorizationsfor thedeterminantsof someof thebraidanalogsdiscussedin this paper. For instance,onecanshow that for any σ _ Sn andη _ ä d n i r e ,

det Ï s] nσ : X ] η ^ § X ] η ^ Ð¼b d h 1e d nη e m N ] σ � η ^ � q� ñóò ] σ ^ ] nη ^

n ú n � 1û �QA ] η ^,m DA ] η ^�� i (6.1)


where � nη � b #

ç d η e (the multinomial coefficient), N d σ i η e is the numberof orbits fortheactionof σ on

ç d η e andQA andDA areasin Section4.3. It follows from herethat

detÏ s] nσ : X ½ n § X ½ n Ð b d h 1e rn m N ] σ ^ � q� ñ�ò ] σ ^ rn � 2 ∑h Æ k ahk i (6.2)

whereN d σ e is thenumberof orbitsfor theactionof σ on ã d n i r e . Explicit expressionsfor N d σ i η e andN d σ e canbe obtainedthroughPolya’s theory; for instancefor the n-cycleσ b d 1 i 2 iVj+jkj�i ne ,

N d σ i η e$b 1n ∑

d � Dφ d d e ± nÔ d

η Ô d ³ andN d σ e$b 1n ∑

d � nφ d d e rnà d i

whereφ is Euler’s functionandD b gcdd η e is thegreatestcommondivisorof thecom-ponentsof η.

We also have obtainedfactorizationsfor the determinantof b ] n1 b ` na acting onsomeparticularcomponentsX ] η ^ :

det Ï ` n � 1a : X ] nεh c εk ^ § X ] nεh c εk ^ Ð bÚ` na !qahh

n m 1

∏i � 0

d 1 h qahkc akh c iahh e i (6.3)

detÏ ` n � 2a : X ] nεh c 2εk ^ § X ] nεh c 2εk ^ Ð b n

∏i � 1

` i a !qahh d 1 h qahkc akh c ] n m i ^ ahh e i

n

∏i � 0 Ï 1 � d h 1e iqakk c ] n m i ^ ] ahk c akh ^ c d n � i

2 e ahh Ð i (6.4)

and

det Ï ` r a : X ] ε1 c ε2 c ô ô ô c εr ^ § X ] ε1 c ε2 c ô ô ô c εr ^ Ð b r m 2

∏i � 0

∏I Ä�� i ] r ^ Ï 1 h q

�QA ] ηI ^,m DA ] ηI ^�� Ð ] r m 2 m i ^ !i! i

(6.5)

whereç

i d r e denotesthesetof all subsetsI of Ö 1 i 2 iVjkj+j�i r × of cardinalityi andηI _ g r

hascoordinatesηI � h b Ì 1 if h Ô_ I

0 if h _ I. Note that, by Lemma4.2, QA d ηI eÑh DA d ηI e1b

∑ h �� kh Æ k ��

Iahk.

FromEquation(6.5)onecandeduce(by induction,using(2.1)) that

det Ï f ] r ^ : X ] ε1 c ε2 c ô ô ô c εr ^ § X ] ε1 c ε2 c ô ô ô c εr ^ Ðb r m 2

∏i � 0

∏I Ä�� i ] r ^ Ï 1 h q

�QA ] ηI ^'m DA ] ηI ^�� Ð ] r m 2 m i ^ ! ] i c 1 ! i (6.6)

notetheslightdifferencebetween(6.5)and(6.6).Equation(6.5) generalizesa formulaof HanlonandStanley [6, Lemma3.4]; their

formulais thecaseahk d 1 of ours.Otherformulasin theirpapercanalsobegeneralizedto thecontext of braids.

468 M. Aguiar

TheparticularcasewhenA is symmetricof Equation(6.6)becomesaspecialcaseofaformulaof Varchenko[18]. Thisauthorassociatesacertainmatrixto any weightedhy-perplanearrangement(thatis, ahyperplanearrangementwhereanumberhasbeencho-senfor every hyperplane)andobtainsa formulafor its determinant.In thespecialcaseof thebraidarrangement� r m 1 bfÖ Hhk Ô 1 p h ì k p r × , whereHhk bfÖ d x1 iVj+jkj�i xr e _ r Ô xh b xk × , weightedby numbersqahk, Varchenko’smatrix turnsout to bethematrixof f ] r ^ : X ] ε1 c ε2 c ô ô ô c εr ^ § X ] ε1 c ε2 c ô ô ô c εr ^ with respectto thecanonicalbasisof Section4, andhis formulaturnsout to beprecisely(6.6).

It would be interestingto obtainfactorizationformulasof this typeon anarbitrarycomponentX ] η ^ .References

1. M. Aguiar, Internalcategoriesandquantumgroups,Ph.D.Thesis,CornellUniversity, Ithaca,1997.

2. M. Aguiar, Braids,q-binomialsandquantumgroups,Adv. PureandAppl. Math.20 (1998)323–365.

3. V.G.Drinfeld, QuantumGroups,in: Proc.Int. Cong.Math.Berkeley, 1986,pp.798-820.4. J. GoldmanandG.C. Rota, On the foundationsof combinatorialtheory IV. Finite vector

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Mathematics,Vol. 9, Springer-Verlag,1972.6. P. HanlonandR.Stanley, A q-deformationof atrivial symmetricgroupaction,Trans.Amer.

Math.Soc.350(1998)4445–4459.7. J.C.Jantzen,Lectureson QuantumGroups,GraduateTexts in Mathematics,Vol. 6, Ameri-

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61 (1979)93–139;JosephP.S.Kung,Ed.,Gian-CarloRotaon Combinatorics:IntroductoryPapersandCommentaries,Birkhauser, Boston,1995(reprint).

10. A. Joyal andR. Street,Braidedtensorcategories,Adv. Math.102(1993),20–78.11. V.G.Kac, Infinite dimensionalLie Algebras,CambridgeUniversityPress,1990.12. C. Kassel,QuantumGroups,GraduateTexts in Mathematics,Vol. 155, Springer-Verlag,

1995.13. G. Lusztig,Introductionto quantumgroups,Progr. Math.110,Birkhauser, 1993.14. S.Montgomery, Hopf AlgebrasandTheirActionson Rings,CBMS82,1993.15. P. Schauenburg, A characterizationof the Borel-like subalgebrasof quantumenveloping

algebras,Comm.in Algebra24 (1996)2811–2823.16. H. Samelson,Noteson Lie Algebras,SecondEd.,Universitext, Springer-Verlag,1990.17. E.J.Taft, Theorderof theantipodeof a finite-dimensionalHopf algebra,Proc.Nat. Acad.

Sci.USA 68 (1971)2631–2633.18. A. Varchenko, Bilinear form of a real configurationof hyperplanes,Adv. Math. 97 (1993)

110–144.19. G.M. Ziegler, Lectureson Polytopes,GraduateTexts in Mathematics,Vol. 152, Spriger-

Verlag,1995.

zonotopes, braids, and quantum groups

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