Gauss–Lusztig decomposition for positive quantum groups and representation by q-tori

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Journal of Pure and Applied Algebra

www.elsevier.com/locate/jpaa

Gauss–Lusztig decomposition for positive quantum groups and representation by q-tori

IvanC.H. Ip^∗

KavliIPMU(WPI),TheUniversityofTokyo,Kashiwa,Chiba277-8583,Japan

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

Article history:

Received2December2014 Receivedinrevisedform24April 2015

Availableonline3June2015 CommunicatedbyD.Nakano

MSC:

Primary:20G42;secondary:81R50

We found an explicit construction of a representation of the positive quantum groupGL⁺_q(N,R) anditsmodulardoubleGL⁺_qq(N,R) by positiveessentiallyself- adjointoperators.Generalizing Lusztig’sparametrization,wefounda Gausstype decompositionforthetotallypositivequantumgroupGL⁺_q(N,R) parametrizedby thestandarddecompositionofthelongest elementw0 ∈ W =S_N−1.Underthis parametrization,wefound explicitlytherelationsbetweenthestandardquantum variables,therelationsbetweenthequantumclustervariables,andrealizingthem usingnon-compactgenerators oftheq-toriuv=q²vuby positiveessentiallyself- adjoint operators.The modular double arises naturally from the transcendental relations,andan L²(GL⁺_qq(N,R)) spaceinthevon Neumannsetting canalso be defined.

© 2015ElsevierB.V.All rights reserved.

1. Introduction

The goal of the present work is to give an explicit construction of a representation of the positive Hopf algebraofquantized functionGL⁺_q(N,R) andits modulardouble GL⁺_qq(N,R) bypositiveessentially self-adjoint operators actingon acertain Hilbertspace H. This isdone byfindingaquantum analogueof the Gauss–Lusztigdecompositionfor GLq(N).Byanabuseofnotation, throughoutthepaper wewillcall GL⁺_q(N,R) a“positivequantumgroup”.

The Gauss–Lusztig decomposition of the positive quantum group provides the foundation of the con- struction ofpositiverepresentations Pλ ofsplitrealquantum groupsUq(g_R)[11,14,15],whichareacertain continuousanalogueofthestandardfinitedimensionalrepresentationsoftheDrinfeld–Jimbotypequantum groups Uq(g). Such decomposition also gives the preliminaries required for the generalization of the har- monicanalysisofthequantumplaneanditsquantumdoublestudiedin[13]tohigherrank.TheL² setting described inthelastsectionmotivates theuse ofmultiplierHopfalgebrafrom thetheoryofC^∗-algebrain

* Tel.:+818021639608.

E-mailaddress:ivan.ip@ipmu.jp.

http://dx.doi.org/10.1016/j.jpaa.2015.05.038 0022-4049/© 2015ElsevierB.V.All rights reserved.

thecontextofDrinfeld–Jimbotypequantumgroup[16,17],whichprovidesanewlinkbetweenthequantized algebraoffunctionsGq(R) anditsquantum envelopingalgebraUq(g_R) inthefunctionalanalyticsetting.

Moreover, the harmonic analysis in this sense for the positivequantum group SL⁺_q(2,R) is closely re- latedto quantum Liouville theory,a certainnon-compact quantum integrable system[3,25] of interestto mathematical physicists. Itsgeneralization to higher rankwill be a veryinteresting connection to theso- calledquantumTodafieldtheory[8,31].Suchconnectionshouldbesomesortofacontinuousversionofthe Kazhdan–Lusztig equivalenceof categories [18,19], which is yetto be establishedmathematically. Finally the combinatorics of the quantum tori generators developed in this paper also provides new insights to the quantummutations appearinginthe theoryof quantum cluster algebras as well as (higher)quantum Teichmüllertheory [9,10].

1.1. Lusztig’s totalpositivity

Let Gbe a semi-simplegroup ofsimply-laced type,T itsR-split maximal torus ofrank r, and U^± its maximalunipotentsubgroupwith dimU⁺=m.TheGaussdecompositionofthemaxcellofGisgivenby

G=U⁻T U⁺. (1.1)

In type A_r (N = r+ 1), this amounts to the decomposition into lower triangular, diagonal and upper triangularmatrices.

On theother hand, given atotally positive matrix G_>0, where all entries of thematrix and itsminors (i.e.determinantsofsubmatrices)arestrictly positive,itcanbedecomposedas

G>0=U_>0⁻T>0U_>0⁺, (1.2)

wherealltheentriesandtheminorsofU^±andT arestrictlypositiveiftheyarenotidenticallyzero.Lusztig in[21]discoveredaremarkableparametrization ofG>0 usingadecompositionof themaximalWeylgroup elementw₀∈W.Letw₀=s_i1. . . s_im beareducedexpressionforw₀,thenthereisanisomorphismbetween R^m>0−→U_>0⁺ givenby

(a₁, a₂, . . . , a_m)→x_i1(a₁)x_i2(a₂). . . x_im(a_m), (1.3) wherexik(ak)=IN+akEik,ik+1andEi,j isthematrixwith1attheentry(i,j) and0otherwise.Asimilar resultalsoholdsforU_>0⁻.WiththisisomorphismLusztigwentontogeneralizethenotionoftotalpositivity toLie groupsofarbitrarytype.

Furthermore,in[1],Berensteinet al.studiedthisdecompositionfortypeA_r,inthecontext nowknown as cluster algebra.They showedvarious relations and parametrizations using thecluster variables, inthis case corresponding to the different minors. Corresponding to the standard decomposition of w₀ is the parametrization using initial minors, which are the determinants of those square sub-matrices that start from eitherthetoprowor theleftmostcolumn.

Using this parametrization,we found in[11] afamily of positiveprincipalseries representations of the modulardoubleUqq(sl(N,R)),wherethenotionofthemodulardoublewasfirstintroducedbyFaddeev[6,7]

for N = 2. These positiverepresentations generalize theself-dual representations of Uqq(sl(2,R)) studied forexamplein[3,13,25],andlaterfurthergeneralizedto allothersimply-lacedand non-simply-lacedtypes in[14,15].

1.2. Gaussdecomposition

Ontheotherhand,inordertostudythepositivequantumgroupGL⁺_q(2,R) intheC^∗-algebraicandvon Neumann setting, in[13,26] aquantum version of the Gauss decomposition for GLq(2) is studied, where roughlyspeakinganymatricesaredecomposedinto productoftheform

z11 z12

z21 z22

=

u1 0 v1 1

1 u2

0 v2

, (1.4)

where {ui,vi}with uivi =q²viui are mutually commuting Weyl pairthat generatesthe algebraC[Tq] of q-tori.

Things become moreinteresting inthe split real case, where we specialize the quantum parameter to

|q|= 1,withb²∈R\Q,0< b<1 anddefine

q:=e^πib2, q:=e^πib−2. (1.5)

Then there exists acanonical representationof theWeyl pairas positiveessentially self-adjoint operators acting onL²(R)

u=e^2πbx, v=e^2πbp, (1.6)

andtheabovedecompositiongivesarealizationofthepositivequantumgroupGL⁺_q(2,R) whereallentries andthequantumdeterminantarerepresentedbypositiveessentiallyself-adjointoperatorsactingonL²(R²).

Moreover, byreplacing b−→b⁻¹ weobtaintherepresentationsforthemodulardouble GL⁺_qq(2,R).

Itisfurthershownin[13]thattheGaussdecompositionofGL⁺_q(2,R) aboveisequivalenttotheDrinfeld–

Woronowicz’s quantumdouble construction [24]over thequantumax+bgroup,and itsharmonicanalysis is studied indetail.A new Haar functional is discovered, and anL²-space of “functions”over GL⁺_qq(2,R) is defined using this Haar functional.With these set up, we provedthat L²(GL⁺_qq(2,R)) decomposesinto directintegralofthepositiveprincipalseriesrepresentationsPλ,s:

L²(GL⁺_qq(2,R)) ⊕ R

⊕ R+

Pλ,s⊗ Pλ,−sdμ(λ)dλds (1.7)

as theleftandright regularrepresentationsofthemodulardoubleUqq(gl(2,R)),wherethemeasure dμ(λ) isgivenbythequantum dilogarithmfunction.ThisisaclosequantumanalogueofthePeter–Weyltheorem in thecase ofcompact Lie group, whichcomes as a surprisesince theresult doesnot involve any kindof discrete seriesrepresentationasintheclassicalSL(2,R) case.

1.3. Gauss–Lusztig decomposition

Combiningtheapproachesabove,ouraiminthispaperistofindtheGaussdecompositionofthepositive quantum groupof higher rank, GL⁺_q(N,R), interms of Lusztig’s unipotent parameters a_i definedabove.

These parametersarenolongercommutingpositiverealnumbers,andthegoalofthis paperistodiscover theirquantumrelationswitheachother,suchthatthedecompositionrecoversthequantumgroupGLq(N), and furthermore in the split real case the generators are represented by positive essentially self-adjoint operators.

Let uscall twovariables quasi-commuting iftheycommuteupto apowerof q².Inthis paperweprove thefollowing theorem:

Theorem1.1(Gauss–Lusztigdecomposition).ThegeneratorsofthepositivequantumgroupGL⁺_q(N,R)can berepresented byN² operators

{bm,n, Uk, am,n}

with1≤n≤m≤N−1,1≤k≤N,whereeach variableispositiveself-adjoint operatorthat commutes or q²-commuteswith each other,sothat

(1) Thevariables {Uk,am,n}generatetheuppertriangular quantumBorelsubgroup T>0U_>0⁺ , (2) Thevariables {bm,n,Uk} generate thelowertriangularquantum BorelsubgroupU_>0⁻T>0, (3) Thevariables a_m,ncommute with b_m,n.

Furthermore,theGauss–Lusztig decompositionfortheotherparts ofthemodulardouble GLq(N,R)canbe obtainedby replacing allvariables{bm,n,Uk,am,n}by their tildeversion

x→x:=x^b12. (1.8)

Asacorollary,weobtainthefollowing resultsofGL⁺_q(N,R) afterspecializationto thesplit realcase:

Theorem1.2.

(1) There is an embedding of GL⁺_q(N,R) into the algebra of ^N₂² q-tori generated by {u_i,v_i} satisfying uivi=q²viui,which arerealizedby

ui=e^2πbxi, vi=e^2πbpi. (1.9)

(2) Thequantumclustervariablesx_ij,definedbythequantizedinitialminors,canberepresentedasproducts of thevariables {b_m,n,U_k,a_m,n},andhencethey quasi-commutewith eachother.

From the main theorem, wecan extendthe positivequantum group GL⁺_qq(N,R) into theC^∗-algebraic settingbygivinganoperatornormtoeachelementwhichisrepresentedbyintegralsofcontinuouscomplex powers of the generators, completely analogous to the N = 2 case. We can also give an L² completion anddefine theHilbertspaceL²(GL⁺_qq(N,R)).Thenitisnaturaltoconjectureitsdecompositionunderthe regularrepresentationofthemodulardoubleUqq(sl(N,R)) intothedirectintegralofpositiveprincipalseries representationsconstructedin[11],inanalogytothedecompositionsofL²(GL⁺_qq(2,R)) givenin(1.7).

1.4. Remarks

The Gauss decompositionfor ageneral quantum group isdefinitely notnew [5,30]. Howevertheusual notioninthecontextofGLq(N) isjustdecomposingthequantumgroupintoaproductoflowerandupper triangular matrices,and thequantum Plücker relationsbetween thecoordinatesare studied.Thoughthis approachisanaturalconsideration,therelationsinvolvedarequiteadhoc,andfurthermoreithasnoway to be generalized to the positive setting, its representation being rather unclear. Therefore we name our decomposition the Gauss–Lusztig decomposition to distinguish it from the standard approach, where we decompose our quantum groupinto products ofelementary matricesbearing aquantum variable,so that thepositivityandtheirrepresentationsaremanifest.

Finally we also remark that in [2,10], the notion of quantum cluster algebra is studied, where quasi- commutingclustervariablesareconsidered,andtheq-commutingrelationsarecompatiblewiththealgebraic

framework. Howeveritsrelation to theparametrizationof GL_q(N) isnotveryexplicit,and itsrepresenta- tionbythecanonicalq-tori{e^2πbx,e^2πbp}isnotshown.Inthispaper,startingfromtheverydefinitionofa quantum group, we foundusing new combinatoricsmethodthatthese cluster variables, quasi-commuting in some complicated powers of q², are actually decomposed into simpler variables {bm,n,Uk,am,n} that commuteonlyuptoafactorofq²,andexplicitformulaisgivenforthecaseGL_q(N).Theq-commutations wefoundexplicitlyarecloselyrelatedtothePoissonstructureoftheclusterX-varietyconsideredin[9].We note thatinthispaper weonlyuse asinglechoiceof clustervariables givenby theinitialminors. A more thorough understandingofthetheoryofquantumcluster algebrainthecontextofquantumgroupsshould bepossiblebyalsoconsideringexplicitlythequantummutationstootherclusters,correspondingtodifferent parametrization ofthemaximalelementw₀ explainedinTheorem 5.8(seealsoRemark 5.9).

Thepaperisorganizedasfollows.InSection2wedescribeindetailtheGaussdecompositionforGLq(2) studied in [13]. In Section 3 we describe the Lusztig parametrization of the totally positive matrix in GL⁺(N,R),and thedescriptionofthe clustervariables definedin[1]. Thenwe introducethedefinitionof GL_q(N) inSection4,andusingcertaincombinatoricsmethods,wefindinSection5thequantumrelations betweenthevariablesof theGauss–Lusztigdecomposition.InSection6weconstruct therepresentationof thesequantumvariablesusingN²−2 quantumtori,andalsopresentanexampledemonstratingtheminimal representation usingonly ^N₂² tori.Finallyusing thequantum torirealization,inSection7wedefine the positive quantum group GL⁺_q(N,R), and describe its relation to the modular double, and inSection 8a possible constructionofanL²(GL⁺_qq(N,R)) space.

2. GaussdecompositionforGL_q(2)

ThequantumgroupGLq(2) isoneofthesimplestmatrixquantumgroup.Itsrepresentationtheoryand generalpropertiesasaHopfalgebracanbefoundforexamplein[4,22,29].Inthispaperwewillusearescaled versionofGL_q(2).Thisversionisconsiderede.g.in[12,13],andhastheadvantageofactingnaturallyonthe standardL²(R) space,duetotherescaledquantumdeterminant(2.6)whichresemblestheclassicalformula withoutany qfactors.Thisalsosimplifiessomecomputationsinvolvedinlatersections.

Definition2.1.WedefineM_q(2) tobethebi-algebraoverC[q,q⁻¹] generatedbyz₁₁,z₁₂,z₂₁andz₂₂subjected to thefollowingcommutationrelations:

z11z12=z12z11, (2.1)

z21z22=z22z21, (2.2)

z11z21=q²z21z11, (2.3)

z₁₂z₂₂=q²z₂₂z₁₂, (2.4)

z₁₂z₂₁=q²z₂₁z₁₂, (2.5)

det_q:=z₁₁z₂₂−z₁₂z₂₁=z₂₂z₁₁−z₂₁z₁₂, (2.6) with co-productΔ givenby

Δ(z_ij) =

k=1,2

z_ik⊗z_kj, (2.7)

and co-unitgivenby

(z_ij) =δ_ij. (2.8)

Definition2.2. Wedefine theHopfalgebra

GL_q(2) :=M_q(2)[det⁻¹_q ] (2.9)

byadjoiningtheinverseelementdet⁻_q¹.

Remark 2.3. The antipode γ of the Hopf algebra GL_q(2) is defined through the inverse element det⁻¹_q . Howeverwewillnotusetheantipodeinthispaper.

Remark2.4.Itisoftenconvenientto writethegeneratorsasamatrix Z:=

z11 z12

z21 z22

,

thentheco-productcanberewritten asstandardmatrixmultiplication:

Δ

z11 z12

z₂₁ z₂₂

=

z11 z12

z₂₁ z₂₂

⊗

z11 z12

z₂₁ z₂₂

. (2.10)

Inthepapers[13,26],theGaussdecompositionofGL_q(2) isstudied.Thegeneratorsz_ijcanbedecomposed uniquelyinto

Z=

z₁₁ z₁₂ z₂₁ z₂₂

=

u₁ 0 v₁ 1

1 u₂ 0 v₂

, (2.11)

wheretheWeylpairs{ui,vi}i=1,2 arenon-commutativevariablessatisfying

uivi=q²viui, (2.12)

[ui, vj] = [ui, uj] = [vi, vj] = 0 fori=j. (2.13) Definition2.5. Wedefine C[Tq] tobethealgebraofquantumtorus:

C[Tq] :=C[q, q⁻¹]u, v, u⁻¹, v⁻¹/(uv=q²vu) (2.14) consistingofLaurentpolynomialsinthevariablesuandv.

Theninparticular,wehaveanembeddingofthealgebraGLq(2) intothealgebraofquantumtori:

GL_q(2)−→C[Tq]^⊗2, (2.15)

wheretheelements ofGL_q(2) canbe expressedasLaurentpolynomials.

Inorder to generalizethis construction to the higherrank, it turns outthatit is betterto rewrite the decomposition(2.11)intheform:

Z=

1 0 v₁ 1

u₁ 0

0 1

1 0 0 v₂

1 u₂ 0 1

=

u1 0 v₁u₁ 1

1 u2

0 v₂

(2.16) wheretheentriesofeachofthetwomatricesstillsatisfythequantumrelations(2.1)–(2.6)ofM_q(2).Finally we note that the quantum determinant det_q quasi-commutes with all other variables. It is this property thatmotivates us tostudy the Gauss decompositionfor GL_q(N) not using thestandard coordinates,but usingthe“clustervariables”whichwewillintroduceinthenextsection.

3. ParametrizationofGL⁺(N,RRR)

Inclassicalgrouptheory,thetotallypositivepartGL⁺(N,R) isthesemi-subgroupof GL(N,R) sothat all the entries are positive, and allthe minors, including the determinant, are also positive.There are in generaltwo equivalent waysto realizethetotallypositivesemi-group.In[21], aparametrization usingthe Gauss decompositionisfound:

G=U_>0⁻T_>0U_>0⁺ , (3.1)

where T_>0 is thediagonal matrixwithpositiveentries u_i, thepositiveunipotentsemi-subgroup U_>0⁺ (and similarly forU_>0⁻)isdecomposedas

U_>0⁺ = m k=1

e^akEik = m k=1

(I_N+a_kE_ik,ik+1), (3.2) where E_i,i+1 is thematrixwith1attheposition(i,i+ 1) and0otherwise,andthei_k’scorrespond tothe decompositionofthelongestelement w₀ oftheWeylgroupW =S_N−1:

w₀=s_i1s_i2. . . s_im. (3.3)

Using thestandarddecompositionforw₀:

w0=s_N−1s_N−2. . . s2s1s_N−1s_N−2. . . s2s_N−1s_N−2. . . s3. . . s_N−1, (3.4) where s_k= (k,k+ 1) arethe2-transpositions,U_>0⁺ canbeexpressed intheform:

⎛

⎜⎜

⎜⎜

⎜⎜

⎝

1 a1,1 0 0 0

0 1 a2,1 0 0

0 0 1 . .. 0

0 0 0 . .. a_N−1,1

0 0 0 0 1

⎞

⎟⎟

⎟⎟

⎟⎟

⎠

⎛

⎜⎜

⎜⎜

⎜⎜

⎝

1 0 0 0 0

0 1 a2,2 0 0

0 0 1 . .. 0

0 0 0 . .. a_N−1,2

0 0 0 0 1

⎞

⎟⎟

⎟⎟

⎟⎟

⎠

· · ·

⎛

⎜⎜

⎜⎜

⎜⎝

1 0 0 0 0

0 1 0 0 0

0 0 1 . .. 0

0 0 0 . .. a_N−1,N−1

0 0 0 0 1

⎞

⎟⎟

⎟⎟

⎟⎠ .

(3.5) The labeling is definedas follows: a_m,n is the entry at them-th row and appearsthen-th time from the left. Similarly,U_>0⁻ isgivenbythetransposeofU_>0⁺ ,i.e.

⎛

⎜⎜

⎜⎜

⎜⎝

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 . .. . .. 0

0 0 0 b_N−1,1 1

⎞

⎟⎟

⎟⎟

⎟⎠· · ·

⎛

⎜⎜

⎜⎜

⎜⎝

1 0 0 0 0

0 1 0 0 0

0 b2,1 1 0 0

0 0 . .. . .. 0

0 0 0 b_N−1,N−2 1

⎞

⎟⎟

⎟⎟

⎟⎠

⎛

⎜⎜

⎜⎜

⎜⎝

1 0 0 0 0

b1,1 1 0 0 0

0 b2,2 1 0 0

0 0 . .. . .. 0

0 0 0 b_N−1,N−1 1

⎞

⎟⎟

⎟⎟

⎟⎠ .

(3.6) Under this parametrization, Berenstein et al. [1] studied the parametrization by the so-called cluster variables, in this case corresponds to the initial minors of the matrix. These are the determinantsof the square submatriceswhich start from eitherthe top rowor the leftmost column.More precisely, a matrix g∈GL(N,R) istotallypositiveifandonlyifallitsinitialminors(includingthedeterminantofthematrix

itself) are strictly positive. Furthermore, the initial minor can be expressed uniquelyas aproduct of the parametersaij,bij andui,hencegivinga1–1correspondencebetweentheparametrizations.

InthestudyofthequantumGaussdecomposition,itturnsoutthatitisjustenoughtolook atT>0U_>0⁺. Letusfirstconsider U_>0⁺ .

Definition 3.1. Denote by xij, 1≤ i < j ≤N, the initial minor with the lower right corner at the entry (i,j),whichuniquelydeterminesthesubmatrix.Following[1],wewill alsocallx_ij thecluster variables.

Thenthere isanexplicitrelationbetweenxij and aij: Proposition3.2. (See[1].)Wehave

ai,N−j = xj,i+1xj−1,i−1

x_j,ix_j−1,i , (3.7)

xi,i+j = i m=1

j n=1

a_m+n−1,n. (3.8)

Herewedenotexi,i=xi,0=x0,j= 1.

The aboverelationscan be expressed schematicallyby thediagram shown inFig. 1, where the cluster variablexi,i+j is expressedastheproductof theamn variablesinsidethebox:

Fig. 1.The clusterxi,i+j fori= 2, j= 4.

AsintheN = 2 case, wesplitthediagonalsubgroupT>0 intotwo halves:

T_>0=T_>0⁻T_>0⁺ :=

⎛

⎜⎜

⎜⎜

⎜⎝

u1 0 0 0 0

0 u2 0 0 0

0 0 . .. 0 0

0 0 0 u_N−1 0

0 0 0 0 1

⎞

⎟⎟

⎟⎟

⎟⎠

⎛

⎜⎜

⎜⎜

⎜⎝

1 0 0 0 0

0 v1 0 0 0

0 0 v2 0 0

0 0 0 . .. 0

0 0 0 0 v_N−1

⎞

⎟⎟

⎟⎟

⎟⎠

, (3.9)

andjustconsider thev variablesforthedecompositionoftheupper triangularpart.Thentheformulasin T_>0⁺U_>0⁺ forai,j staythesame,whilethoseforxi,j aremodifiedasfollows:

x_i,i+j= _i

m=1

j n=1

a_{m+n−1,n i−1}

k=1

v_k. (3.10)

4. Definitionof GL_q(N)

ThequantumgroupGL_q(N) isdefinedbythefollowing relationsinvolvingtherank1case.

Definition 4.1. Wedefine M_q(N) tobe thebi-algebra overC[q,q⁻¹] generated by{z_ij}^Ni,j=1, suchthatfor every1≤i< i≤N,1≤j < j≤N, thesubmatrix

z_ij z_ij

z_ij z_ij

(4.1) is acopyofMq(2), i.e.thecorrespondinggeneratorssatisfytherelations(2.1)–(2.6).

Thequantum determinantisagaindefinedusingtheclassicalformula(withnoqinvolved):

Definition 4.2.Wedefinethequantumdeterminantas detq=

σ∈SN

(−1)^σz_1,σ(1). . . z_N,σ(N), (4.2) where S_N isthepermutationgroupofN elements.

Thenitfollowsfrom(2.6)andaninductionargumentthatdetq doesnotdependontheorderoftherow index,providedthatallthemonomialshavethesameorderofrowindex.

Definition 4.3.WedefineGLq(N) tobetheHopfalgebra

GLq(N) :=Mq(N)[det⁻¹_q ]. (4.3)

TheHopfalgebrastructureisgivenbythesameclassicalformula Δ(z_ij) =

N k=1

z_ik⊗z_kj, (4.4)

(z_ij) =δ_ij. (4.5)

Theantipodeγ canbedefinedinvolvingdet⁻¹_q , butagainwewillnotuseitinthepresent paper.

As inthecaseofGLq(2),wecanconvenientlywritethegeneratorsasamatrix Z:=

z_ijN

i,j=1. (4.6)

Let us call amatrix X of non-commutative entries a“GL_q(N)-matrix” ifthe matrix entries of X satisfy thedefiningrelationsofM_q(N) inDefinition 4.1,andthedeterminantdet_q(X) ofX isnotidenticallyzero.

Inparticular someoftheentriesareallowed tobeconstants.

Then fromtheco-associativityoftheco-productΔ,

Δ(Z) =Z⊗Z, (4.7)

we havethefollowingproperty:

Proposition 4.4. If X andY are GL_q(N)-matrices such that the matrixentries of X commute with those of Y,then thematrix product

G=XY (4.8)

isagain aGL_q(N)-matrix,where thedeterminantisnon-zero andgivenby

detq(XY) =detq(X)detq(Y). (4.9)

Henceinorderto findaGauss decomposition

Z =XY (4.10)

for GL_q(N) where X is lower triangular and Y is upper triangular, it suffices to find the corresponding matrix thatsatisfies the quantum relations (that any matrix canbe expressed in this form is proved, for example, in[5]). We will dothis byemploying theconstruction using the parametrizations of the totally positivematricesGL⁺(N,R).

5. Gauss–Lusztigdecomposition ofGLq(N)

Let T⁺ and U⁺ be given by the same matrices as in (3.5) and (3.9), but instead with formal non- commutingvariablesv_m,a_mn for1≤n≤m≤N−1.

Definition 5.1. We define the variables xij,1≤ i < j ≤N to be the quantum determinant of the initial submatrices (with thesame parametrization given inSection 3) of the matrixproduct Z = T⁺U⁺ using thedeterminantformula(4.2). Wewill callxij thequantum clustervariables.

Thenwecanstateourmain results:

Theorem 5.2. The product Z = T⁺U⁺ is a GLq(N)-matrix if and only if we have the following q-commutation relationsbetweenthevariablesgivenby:

• a_mnv_m=q²v_ma_mn foralln,

• a_mna_mn =q²a_mna_mn forn> n,

• amnam−1,n =q²am−1,namn forn≤n,

• commuteotherwise.

Furthermore thevariablesx_ij can bewritten as

x_i,i+j= _i

m=1

j n=1

a_{m+n−1,n i−1}

k=1

v_k (5.1)

= (a11a22a33. . .)(a21a32a43. . .). . .(. . . ai+j−1,j)(v1v2. . . vi−1) (5.2) inthis particularorder.Finallyfor everyGL_q(N)-matrix,thecommutationrelationsbetweenthevariables x_ij aregivenby

xi,i+jxk,k+l=q2P(i,j;k,l)xl,k+lxj,i+j, (5.3) whereforj ≤l,

P(i, j;k, l) = #{m, n|l+ 2≤m+n≤k+l+ 1,1≤m≤i,1≤n≤j}

−#{m, n|1≤m+n≤i,1≤m≤k,1≤n≤l} (5.4)

and

P(k, l;i, j) =−P(i, j;k, l). (5.5)

Corollary5.3.LetU⁻ andT⁻ bedefinedby(3.6)and(3.9)sothatb_mn andu_mcommutewitha_mn andv_m. Then {bmn,u⁻¹_m}satisfies exactlythe samerelations as{amn,vm}. LetT =T⁻T⁺ be thediagonal matrix with entries Tk=ukvk−1 for1≤k≤N,wherewedenoteby v0=uN = 1.Thentheproduct

Z=U⁻T U⁺ (5.6)

gives theGauss–LusztigdecompositionofGLq(N).Moreprecisely,thismeansthatthegenerators{zij}and det⁻¹_q oftheHopfalgebra GLq(N)canbeexpressed intermsofN² variables{amn,bmn,um,vm}(andtheir inverses) that commute uptoafactorof q².

Theq-commutation relationsforamn(andalsobmn)canberepresentedneatly byadiagram:

(5.7)

whereu−→vmeansuv=q²vu,anddoublearrowsmeansitq²-commuteswitheverythinginthatdirection.

Inother words,thearrowsconsistofallthepossible leftdirections,andallthenorth-eastdirectionsgoing up onelevel.Furthermore, note thatthecommutation relationsfor amn, vm,um andbmn are justcopies of theGauss decomposition(2.16)forGL_q(2).

Remark5.4.It waspointedoutbyA.Goncharovthatifwemakeachangeofvariablesbytakingratiosof thegenerators:

a_m,n=

am,1 n= 1,

qam,na⁻¹_m,n−1 n >1 (5.8)

(theq factoris usedto preservepositivity,cf. Section7), thenthecommutation relationsamong thea_m,n variables takeamoresymmetricform,representedbythediagram

(5.9)

ThischoiceofgeneratorsiscloselyrelatedtothePoissonstructureoftheclusterX-varieties,studiedfor examplein[9].

Wewilluseseverallemmas toprovethetheorem.

Lemma5.5. Assumetheq-commutation relationsinTheorem 5.2 fora_mn andv_m hold. Then(5.1)holds.

Proof. Weusethefactthat,byinduction,eachentry zij oftheuppertriangularmatrixhasaclosedform expressiongiven by

zi,i+j = vi−1

1≤t1<t2<...<tj≤i+j−1

(ai,t1ai+1,t2. . . ai+j−1,tj)

:=

t

S_i,t. (5.10)

Wealsohavezi,i= 1 andzi,i−j = 0.

Hencethequantized initialminorx_i,j isgivenbysumsofproductsoftheform

Sj,t=S1,t1S2,t2. . . Sj,tj. (5.11) Now using theq-commutation relations, which saythatamn commuteswith amn when both m> m and n > n, we canarrange the order on eachmonomial Sj,t so that ithas a“maximal” ordering: If the producta_mna_mn appearsintheordering,theneitherm=m+ 1 and n> n,orm < m.Furthermore,if thelastterminS_k,tisa_m,n,thentheterma_m+1,n forn> nwillnotappearinS_k+1,t,sothatnothingcan commutetothefront,whilewecanpushallthevmtothebacksincevmcommuteswithamn form< m. This orderingis uniqueinthesense thatfor everymonomialwhere theorder inwhichap,∗ appearsfor eachfixed pis the same,thecorresponding maximal ordering is thesame. Hencetheclassical calculation works and all the terms will cancel, except the one with minimal lexicographical ordering. This term is precisely

S1,tminS2,tmin. . . Si,tmin, where

S_k,tmin =v_k−1a_k,1a_k+1,2. . . a_k+j−1,j.

Againeachv_k−1 ineachSk commuteswith allthea’s, sowe canmovethem towardstheback, andhence givingtheexpression (5.1). 2

Lemma 5.6.Assume theq-commutationrelations fora_mn andv_m hold.Then (5.3)holds.

Proof. Usingthe expression givenbyLemma 5.5,we canstudyhow x_ij andx_kl commute.Wedothis by counting how many q-commutations it takes for a fixed a_m,n appearing in x_i,i+j to travel through each variable am,n inxk,k+l.

First, notice thatam,n appears inxk,k+l only if1≤n ≤l and 0≤m−n ≤k−1.Now fix m,n and consider amn.Itq²-commuteswithamn inxk,k+lwhen:

• q²:a_m,n with n< n,hencealso1≤n ≤l and0≤m−n ≤k−1.Wecanrewritethisas A1= #{n|max(m+ 1−k,1)≤n≤min(l, n−1, m)},

• q⁻²:am,n withn> n,hencealso1≤n≤l and0≤m−n≤k−1 whichreducesto A₂= #{n|max(m+ 1−k, n+ 1)≤n≤min(l, m)},

• q²:a_m−1,n withn≥n,hence

A3= #{n|max(m−k, n)≤n ≤min(l, m−1)},

• q⁻²:am+1,n withn≤n,hence

A4= #{η|max(m−l+ 2,1)≤n ≤min(l, n, m+ 1)}.

Hencetheamountofq²powerspickedupisjustthesignedsumofthecountabove.Byacasebycasestudy, these expressionscanbesimplified:

A3−A2=

1 m+n≥l+ 2, n+m≤k+l+ 1, n≤l

0 otherwise,

A1−A₄=

⎧⎪

⎨

⎪⎩

1 n≥l+ 1, k+ 1≤m+n≤k+l

−1 m+n≤k,1≤n≤l

0 otherwise.

Hence,thetotalamountofpowerpicked upafter summingallm,nisgivenby

#{l+ 2≤m+n≤k+l+ 1, n≤l}+ #{k+ 1≤m+n≤l+k, l+ 1≤n}

−#{m+n≤k,1≤n≤l}, subjectto 1≤m≤i,1≤n≤j.

Letus assumej ≤l.Then n≤j≤l,and theexpressioncanbe simplifiedto

#{l+ 2≤m+n≤k+l+ 1} −#{m+n≤k}, subjectto 1≤m≤i,1≤n≤j.Thistakescareofamn.

Westillneedtocalculatethoseforv_m.Sincethereisonlyonev_mappearinginx_k,k+lforeach1≤m≤k, wejustneedtocounthowmanya_mn’swithindexm≤k+ 1 arethere.Henceusingtherenameda_m+n−1,n theconditionis

#{m, n|m+n≤k,1≤m≤i,1≤n≤j},

andthisis theamountofq² pickedup,hencecanceledwiththelastterminthepreviouscalculation.

Similarlyconsideringtheotherdirection,theamountof q⁻² pickedupis

#{m, n|m+n≤i,1≤m≤k,1≤n≤l}. Hencewearriveat ourformula. 2

Lemma5.7. Wehave

P(i, j;k, l) =P(i, j−1, k, l−1). (5.12) Proof. This is done by simplecounting. Assumej ≤l. Let us compare thedifference between the corre- spondingtermsoftheP function.Wehaveforthesecond term:

#{m, n|1≤m+n≤i,1≤m≤k,1≤n≤l}

−#{m, n|m+n≤i,1≤m≤k,1≤n≤l−1}

= #{m, n|1≤m+l≤i,1≤m≤k}, whileforthefirsttermwehave

#{m, n|l+ 2≤m+n≤k+l+ 1,1≤m≤i,1≤n≤j}

−#{m, n|l+ 1≤m+n≤k+l,1≤m≤i,1≤n≤j−1}

= #{m, n|l+ 2≤m+n≤k+l+ 1,1≤m≤i,1≤n≤j}

−#{m, n|l+ 2≤m+n≤k+l+ 1,1≤m≤i,2≤n≤j}

= #{m|l+ 2≤m+ 1≤k+l+ 1,1≤m≤i}

= #{m|l+ 2≤m+l+ 1≤k+l+ 1,1≤m+l≤i}

= #{m|1≤m≤k,1≤m+l≤i}. Hencetheamountscancel. 2

ProofofTheorem5.2. Wewillprovethetheorem byinduction.WhenN = 2 thedecompositionisjust 1 0

0 v1

1 a₁₁

0 1

witha₁₁v₁=q²v₁a₁₁. Hencethiscaseholdstrivially.

Assumeeverythinghold fordim =N−1.

For dim =N, firstwe notice thataN−1,N−1 commuteswith aii fori < N−1 by lookingat theentry z1,i+1=a11a22. . . aii,whichcommuteswitheachotherbytheGLq(2) relations.

Next we notice that the cluster variables for a general GL_q(N)-matrix depend only on the variables appearing in T⁺U⁺, since we assumed that the lower triangular matrix U⁻T⁻ commutes with T⁺U⁺. Hencetherelationsbetweenx_i,i+j whichhold forT⁺U⁺ will alsoholdforGL_q(N).

Nowforageneralclustervariable xk,N inthenewrank,weknowfromLemma 5.5thata_N−1,N−k isthe onlynewtermappearing.HencethecommutationrelationsbetweenaN−1,N−kandai+j−1,jisequivalentto thecommutationrelationsbetweenxk,N andxi,i+j byinductiononnewterms.Nowconsiderthe(N−1)× (N−1) submatrix correspondingto x_N−1,N. Thisby definitionsatisfiestheGL_q(N−1) relations,and in particular the commutation relationbetween x_k,N and x_i,i+j shouldbe the sameas the relation between x_k,N−1 andx_i,i+j−1.However,thisisprecisely thestatementprovedinLemma 5.7. 2

Theaboverelationscanbegeneralizedtoarbitraryreducedexpressionforw₀asfollows.Let(a,b,c) and (a,b,c) bepositiveq-commutingvariablessuchthat

b

a c

and

a c

b

(5.13)

where againu−→v meansuv=q²vu.

Then asin(3.2),theproducts

x2(a)x1(b)x2(c) =x1(a)x2(b)x1(c)

form acopyof U⁺ oftheGauss decompositionofGL_q(3) correspondingto thereducedexpressions w0=s2s1s2=s1s2s1,

where

a= (a+c)⁻¹cb=bc(a+c)⁻¹, b=a+c,

c= (a+c)⁻¹ab=ba(a+c)⁻¹, and thismap

φ: (a, b, c)→(a, b, c) (5.14)

is aninvolution between(a,b,c)←→(a,b,c).Inparticular, we seethatbyapplying this transformation to any three consecutive variables a_mn corresponding to the sub-word of the form s_is_js_i with i adjacent to j, allthe arrows inthediagram (5.13) arepreserved. Applying this transformation,we candeduce all quantumLusztig’svariablesforarbitraryreducedexpressionforw0.Hencewecanrestatethecommutation relationsinTheorem 5.2as follows:

Theorem5.8.Letain,m bethecoordinatesofU⁺ correspondingtothereducedexpressionofw0=si1. . . sin. Then the product T⁺U⁺ is a GLq(N)-matrix if and only if for any |i − j| = 1, the coordinates {v_i,v_j,a_i,m,a_j,n,a_i,k} form a copy of GL_q(3), where {a_i,m,a_j,n,a_i,k} appear in this exact order in the parametrization of U⁺.Inotherwords,wehave

(5.15)

Remark5.9.Theabovetransformationoftheamn coordinatesbetweendifferentreducedexpressionofthe longest Weyl element w0 can be rewritten in terms of the variables xi,j using (5.1). This becomes the quantized cluster mutations,or underthe limitq −→ 1,the cluster mutationsfor the parametrization of totallypositivematrices,whichishistoricallythefirstexamplesandthemainmotivationfortheintroduction of thetheory ofclusteralgebra[1].

6. Embedding intothealgebraofquantumtori

In order to deal with positivity for the split real case in the next section, we would like to find an embeddingofthealgebraM_q(N) (resp.GL_q(N))into copiesofthealgebraofquantumtori C[Tq],so that itselementsareexpressedintermsofpolynomials(resp.Laurentpolynomials)inthequantumtorivariables.

HenceduetotheGauss–Lusztigdecompositionestablishedintheprevioussection,theremainingtaskisto findanexplicitrealizationofthegeneratorsa_mn,v_m usingseveral copiesoftheWeylpair{u,v}satisfying uv=q²vu.

Theorem6.1. Thereisan embeddingofalgebra

T⁺U⁺−→C[Tq]^{⊗N2 +N2 −4} givenby

vm→vm (6.1)

amn→um

_m−1

k=n

v_m−1,k

n−1

l=1

vm,l

um,n, (6.2)

whereT⁺U⁺isnowrealizedasthealgebrageneratedby{amn,vm}satisfyingtherelationsfromTheorem 5.2, andC[Tq]^{⊗N2 +N−42} isgeneratedbytheWeylpairs{um,vm}and{umn,vmn}for1≤n≤m≤N−1,where wehaveomittedthelastsetofgenerators{uN−1,N−1,vN−1,N−1}.(WedefineuN−1,N−1:= 1intheformula.)

Similarly, forU⁻T⁻ generated by{b_mn,u_m},wehave theembedding U⁻T⁻−→C[Tq]^{⊗N2 +N−42} givenby

um→u_m (6.3)

bmn→v_{m m−1}

k=n

v_m−1,k

n−1 l=1

v_m,l

u_m,n, (6.4)

wherethegenerators{u,v}(with sameindexing above) commutewith {u,v}used above.

Together,this gives anembeddingof thealgebra GLq(N)U⁻T⁻⊗T⁺U⁺:

GLq(N)−→C[Tq]^⊗N2+N−4. (6.5)

Proof. The proof is straightforward to check, since for the uvariables only u_m and u_mn appear in a_mn. Hencewejustneedto count,at mostonce, howmanyvmn appearsinanothervariables. 2

Remark6.2.ThisembeddingresemblestheDrinfelddoubleconstruction,whichreads

D(Uq(b)) =Uq(g)⊗ Uq(h). (6.6)

WithourassignmentforU⁻T⁻⊗T⁺U⁺,wecanactuallycombinethediagonalvariables(hence“modding”

outh)as follows: