Algebraic properties of Manin matrices II: q-analogues and integrable systems

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Advances in Applied Mathematics

www.elsevier.com/locate/yaama

Algebraic properties of Manin matrices II:

q-analogues and integrable systems

A. Chervov^a, G. Falqui^b, V. Rubtsov^a,c,∗, A. Silantyev^d

aInstituteforTheoreticalandExperimentalPhysics,25,Bol.Cheremushkinskaya, 117218,Moscow,Russia

bDipartimentodiMatematicaeApplicazioni,UniversitàdiMilano-Bicocca, via R. Cozzi53,20125Milano,Italy

cLAREMA,UMR6093duCNRS,UniversitéD’Angers,Angers,France

dGraduateSchoolof MathematicalSciences,UniversityofTokyo,Komaba,Tokyo, 123-8914,Japan

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

Articlehistory:

Received10December2012 Accepted3June2014

Availableonline28August2014

MSC:

15B33 05A19 05A30 15A15 16T20 17B37 20G42 81R50

Keywords:

Non-commutativedeterminant Quasideterminant

Maninmatrix Jacobiratiotheorem Newtonidentities Cayley–Hamiltontheorem Schurcomplement Dodgsoncondensation

We study a natural q-analogue of a class of matrices with non-commutative entries, which were first considered by Yu.I. Maninin1988inrelation withquantumgrouptheory, (called Manin matrices in [5]). We call these q-analogues q-Maninmatrices.Thesematricesare defined,inthe 2×2 casebythefollowingrelationsamongtheirmatrixentries:

M21M12=qM12M21, M22M12=qM12M22, [M11, M22] =q⁻¹M21M12−qM12M21.

They were already considered in the literature, especially in connection with the q-MacMahon master theorem [10], and the q-Sylvester identities [22]. The main aim of the present paper is to give a full list and detailed proofs of the algebraic properties of q-Manin matrices known up to the moment and, in particular, to show that most of the basictheoremsoflinearalgebras(e.g.,Jacobiratiotheorems, Schurcomplement,theCayley–Hamilton theoremandsoon andsoforth) havea straightforward counterpartforsuch a class of matrices. We also show how q-Manin matrices fit

* Correspondingauthorat:LAREMA,UMR6093duCNRS,UniversitéD’Angers,Angers,France.

E-mailaddresses:chervov@itep.ru(A. Chervov),gregorio.falqui@unimib.it(G. Falqui), volodya@univ-angers.fr(V. Rubtsov),aleksejsilantjev@gmail.com(A. Silantyev).

http://dx.doi.org/10.1016/j.aam.2014.06.001 0196-8858/© 2014ElsevierInc.All rights reserved.

Laxmatrix R-matrix

within the theory of quasideterminants of Gelfand–Retakh and collaborators(see, e.g., [11]). Weframe our definitions within the tensorialapproachto non-commutativematrices oftheLeningradschoolinthelastsections.Wefinallydiscuss how the notion of q-Manin matrix is related to theory of QuantumIntegrableSystems.

© 2014ElsevierInc.All rights reserved.

1. Introduction

It iswell-knownthatmatriceswithgenericallynon-commutativeelements playaba- sic role inthe theory of quantum integrability as well as other fields of Mathematical Physics. Inparticular the best knownoccurrencesof suchmatrices (see, e.g.,[3,9] and the referencesquotedtherein) arethe theoryof quantum groups,thetheory of critical andoff-criticalphenomenainstatisticalmechanicsmodels,andvariousinstancesofcom- binatorial problems. More recently,applications of such structures inthe realm of the theory ofPainlevéequations wereconsidered[30]. Thepresentpaper builds,at leastin itsbulk,ontheresultsof[5].Inthatpaper,aspecificclassofnon-commutativematrices (that is,matriceswithentriesinanon-commutative algebra–orring)wereconsidered, namelytheso-calledManinmatrices.Thisclassofmatrices,whicharenothingbutma- tricesrepresentinglineartransformationsofgeneratorsofpolynomialalgebras,werefirst introduced intheseminalpaper[26] byYu.I.Manin, buttheyattractedsizableatten- tion onlyrecently,especiallyintheproblem of theso-calledquantum spectral curvein the theory of Gaudin models [7]. The defining relations for a (column) Manin matrix M_ij are:

1. Elementsinthesamecolumncommute;

2. Commutatorsofthecrosstermsareequal:[Mij,Mkl]= [Mkj,Mil] (e.g.[M11,M22]= [M21,M12]).

The basic claim(fully proven in[5]) is thatmost theorems of linearalgebra hold true for Manin matrices ina form identical to thatof thecommutative case, although the set of Manin matrices is fairly different from thatof ordinary matrices – for instance they donotformaringover thebase field.Moreoverinsomeexamplestheconverseis also true, thatis, the class of Manin matrices is the most general class of matrices in which“ordinary”linearalgebraholdstrue.Moreoveritisimportanttoremark thatthe structureofManinmatrixappears,asitwaspointedoutin[4],inthetheoryofquantum integrablesystems(inparticular inthetheoryof integrablespinchainsassociatedwith rational R matrices). The basic aim of the present work is to extend our analysis to another classof matrices,which wecall q-Maninmatrices, whosedefiningrelationsare insomesorttheq-analoguesoftheones forManinmatrices.Namely,theyreadas:

1. Entriesofthesamecolumnq-commute,i.e.,MijMkj=q⁻¹MkjMij

2. The q-cross commutation relations: MijMkl −MklMij = q⁻¹MkjMil−qMilMkj, (i< k,j < l) hold.

Thesedefiningrelations,astheexpertreaderhassurelyremarked,canbeconsideredas a“half” ofthe definingrelationsfor quantum matrices, thatis, fortheelements of the quantummatrixgroupGLq(N).

Actually,this class ofmatrices wasalreadyintroduced intheliterature;indeedsuch matrices were consideredin thepapers [10,25] (under the name of right quantum ma- trices see also [8,31,32]), where the q-generalization of theMacMahon master theorem wasproven inconnection with the boson–fermion correspondenceof quantum physics.

These results (as well as the analysis of the q-Cartier Foata type of matrices) were generalizedin[22]. Moreover,amulti-parameter right-quantumanalogueof Sylvester’s identity was found in[19–21], also in connection with [23]. It is worthwhileto remark thatasuper-versionoftheMacMahon mastertheorem aswell asotherinterestingcon- sequencesthereofwasdiscussed in[27].

Wewouldlikealsoto remarkthatinvarious worksof Gelfand,Retakh andtheirco- authors(see,e.g.,[11,12,14])acomprehensivelinearalgebratheoryforgenericmatrices withnon-commutativeentrieswasextensivelydeveloped.Theirapproachisbasedonthe fundamentalnotionofquasideterminant.Indeed,inthe“generalnon-commutativecase”

there is no naturaldefinition of a “determinant” (one has to consider n² “quasideter- minants” instead)and, quite often, theanalogues of the propositionof ordinarylinear algebraaresometimesformulatedinacompletely differentway. Nevertheless itisclear thattheirresultscanbefruitfullyspecializedandappliedtosomequestionshere;indeed, inSection3wemakecontactwiththis theory,and,inAppendix Bwegiveanexample of howthis formalism canbe appliedto q-Manin matrices. However,inthe rest ofthe paperweshallnotusetheformalismofGelfand andRetakh,sinceweprefertostressthe similaritiesbetweenourcaseofq-Maninmatricesandthecaseofordinarylinearalgebra.

Hereweshallgiveasystematicanalysisofq-Maninmatrices,startingfromtheirbasic definitions,keepingintoaccount possibleapplicationtothetheoryofq-deformed quan- tumintegrablesystems(thatis,intheMathematicalPhysicsparlance,thoseassociated withtrigonometricR-matrices).Thedetailedlayoutofthepresentpaperisthefollowing:

Weshallstartwith thedescriptionofsomeelementarypropertiesdirectlystemming fromthedefinitionofq-ManinmatrixinSection2.Inparticular,weshallpointoutthat q-Maninmatricescanbeconsideredbothasdefininga(left)actiononthegeneratorsof aq-algebra,andasdefininga(right)actiononthegeneratorsofaq-Grassmannalgebra.

Wewouldliketostress thatthese properties,albeitelementary,will beinstrumentalin thedirect extension of the “ordinary”proofs of someproperties of thedeterminant to ourcase. InSection3, after having consideredtherelations/differencesof the q-Manin matriceswithquantumgrouptheory,weshallrecallthedefinitionofq-determinantand q-minorofaq-Maninmatrix.Inparticular,weshallstateandprovetheanalogueofthe Cauchy–Binettheoremaboutthemultiplicativepropertyoftheq-determinant.Weshall

limitourselvestoconsiderthesimplest case(oftwoq-Maninmatriceswiththeelements ofthefirstcommutingwiththoseofthesecond).Wewouldliketopointoutthatamore ample discussion of (suitable variants of) the Cauchy–Binet formula and the Capelli identity inthenon-commutativecasecanbefound,besides[5],in[2].StillinSection3, we shall discuss how Cramer’s rule canbe generalizedin ourcase and we preliminary discuss how thenotion ofq-characteristicpolynomial(whose propertieswill be further discussed in Section5) for a q-Manin matrixM shouldbe defined. It is worthwhileto note thatthisis aspecificcaseinwhichtheq-generalizationof some“ordinary”object requires asuitablechoiceamongthe(classicallyequivalent)possibledefinitions.Indeed, ameaningfulnotionofq-characteristicpolynomialcanbeobtainedbyitsdescriptionas theweightedsumoftheprincipalq-minors ofM.

InSection4weprovetheJacobiratiotheorem andtheLagrange–Desnanot–Jacobi–

Lewis Carrollformula (hereafterLDJC formula), alsoknownas Dodgson condensation formula.Thenweshowafundamentalpropertyofq-Maninmatrices,thatis,closenessun- dermatrix(q)-inversion,anddiscusstheanalogueofSchur’scomplementtheoremaswell as theSylvesteridentities.Further,aglancetotheq-Plückerrelationswillbeaddressed (see,forafulltreatmentinthecaseofgenericnon-commutativematrices,thepaper[24]).

TheaimofSection5istwofold.Firstweshallshow,makinguseoftheso-calledtensor lemma, howthe theoryofq-Maninmatricesof rankncanbeframed withinthetensor approach of the Leningrad School, i.e. interpreting q-minors and the q-determinantas suitableelements inthetensor algebraofCⁿ. Thisapproachwillbe helpfulinreaching the second aim of the section, thatis, establishing the Cayley–Hamilton theorem and theNewton identitiesforq-Maninmatrices.

Section 6 shows, by using the formalism introduced in Section 5, how the theory of q-Manin matrices fits withinthe scheme of the quantum Integrable spin systems of trigonometrictype.Weshallfirstshowthat,ifL(z) isaLaxmatrixsatisfyingtheYang–

Baxter RLL = LLR relations (Eq. (6.5)) with a trigonometric R-matrix R, then the matrix

M=L(z)q^−2z∂z∂,

is actually a q-Manin matrix, and, moreover, that the quantum determinant of L(z) definedbytheLeningradSchool(see,e.g.,[29])coincideswiththeq-determinantofthe associatedq-ManinmatrixM.Then,generalizingthecorrespondingconstructionof[4,5], weshallshowhowanalternativesetofquantummutuallycommutingquantitiescanbe obtainedconsidering (ordinary)tracesofsuitably defined“quantum”powersofM.

Theappendix containstwosections.Thefirst isdevotedtothedetailedproof ofthe mosttechnicallemmaofSection5,whichweextensivelyuseinSection6.Inthesecond, weshallgiveanalternativeproofoftheLDJCformulamakinguseofaformalismsimilar to thatofquasideterminants.

As a general strategy to keep the present paper within a reasonable size, we shall present detailed proofs only in the case when these proofs are substantially different

from the “non-q” (that is, q = 1) case.Otherwise, we shall referto thecorresponding proofsin[5].

2. q-Maninmatrices

2.1. Definition ofq-Maninmatrices

LetM be ann×mmatrixwith entriesM_ij inan associativealgebraR overC and letqbeanon-zerocomplexnumber.BeingthealgebraRnon-commutative(ingeneral), thematrixM hasnon-commutativeentries.

Definition1.The matrixM is calledaq-Manin matrixifthe followingconditionshold true:

1. Entriesofthesamecolumnq-commutewitheachotheraccordingtotheorderofthe rowindices:

∀j, i < k: MijMkj=q⁻¹MkjMij. (2.1) 2. Thecrosscommutationrelations:

∀i < k, j < l: MijMkl−MklMij=q⁻¹MkjMil−qMilMkj (2.2) hold.

Thedefiningrelationsforq-Manin matricescanbe compactlywrittenas follows:

MijMkl−q^sgn(i−k)q^{−sgn(j−l)}MklMij =q^sgn(i−k)MkjMil−q^{−sgn(j−l)}MilMkj, (2.3) foralli,k= 1,. . . ,n,j,l= 1,. . . ,m, whereweusethenotation

sgn(k) =

⎧⎨

⎩

+1, ifk >0;

0, ifk= 0;

−1, ifk <0.

(2.4)

Indeed,for j=l and i=k onegetsthe columnq-commutativity(2.1),while forj =l and i = k one gets the cross commutation relations (2.2) and for i = k one gets an identity.

Remark 2.1. Definition 1 can be reformulated as follows: the matrix M is q-Manin matrix if and only if each 2×2 submatrix is a q-Manin matrix. More explicitly, for any2×2 submatrix(M_(ij)(kl)),consistingofrowsiand k,andcolumns j andl(where 1i< kn,and1j < lm)

⎛

⎜⎜

⎜⎜

⎜⎝

... ... ... ... ...

... Mij ... Mil ...

... ... ... ... ...

... Mkj ... Mkl ...

... ... ... ... ...

⎞

⎟⎟

⎟⎟

⎟⎠≡

⎛

⎜⎜

⎜⎜

⎜⎝

... ... ... ... ...

... a ... b ...

... ... ... ... ...

... c ... d ...

... ... ... ... ...

⎞

⎟⎟

⎟⎟

⎟⎠

, (2.5)

thefollowing commutationrelationshold:

ca=qac (q-commutation of the entries in thej-th column), (2.6) db=qbd (q-commutation of the entries in thel-th column), (2.7) ad−da=q⁻¹cb−qbc (cross commutation relation). (2.8) Examples.

• Let R be the algebra generated by four symbols a, b, c, d over C with the rela- tions(2.6),(2.7),(2.8)(orbeanalgebracontainingsomeelementsa,b,c,dsatisfying theserelations).Thenthematrix

M= a b

c d

(2.9) isaq-Maninmatrix(overR).

• Let us consider n elements xi ∈ R, i = 1,. . . ,n thatq-commute:xixj = q⁻¹xjxi

fori< j (e.g.we canconsider thealgebraR generatedby xi with theserelations).

Thenthecolumn-matrix

M =

⎛

⎜⎜

⎜⎝ x1

x2

...

x_n

⎞

⎟⎟

⎟⎠ (2.10)

canbe consideredaq-Maninmatrix.

• Theelements ofthe samerow ofa q-Manin matrixare notrequired to satisfyany relations.Soanarbitrary1×nmatrixM= (r₁, . . . , r_n) canbeconsideredaq-Manin matrix.

• Ifsomeelementsm_i q-commute,i.e.m_im_j =q⁻¹m_jm_i,i< j,then thematrix

⎛

⎜⎜

⎜⎝

m1 m1

m₂ m₂ ... ...

mn mn

⎞

⎟⎟

⎟⎠, (2.11)

isaq-Maninmatrix.Thecross-commutationrelation(2.2)followsfromq-commuta- tivity inthis case. Moreover, ifq =−1 then thecross-commutation relations (2.2) forthismatriximpliesq-commutativityofmi.

• Considerelements xi,ri ∈ R, i= 1,. . . ,n,such thatxixj =q⁻¹xjxi for i< j and [xi,rj]= 0∀i,j.Thenthematrix

M=

⎛

⎜⎜

⎜⎝

x1r1 x1r2 ... x1rm

x2r1 x2r2 ... x2rm

... ... ... ...

x_nr₁ x_nr₂ ... x_nr_m

⎞

⎟⎟

⎟⎠=

⎛

⎜⎜

⎜⎝ x1

x2

...

x_n

⎞

⎟⎟

⎟⎠(r1 r2 ... rm) (2.12)

isa q-Maninmatrix.Thisfactcanbe easilycheckedbydirectcalculation.

• Wereferto[4–6,33]forexamplesofq-Maninmatricesforq= 1 relatedtointegrable systems,Liealgebras,etc.Intheq= 1 case theq-Maninmatricesaresimplycalled Maninmatrices.

• Let C[x,y] be the algebra generated by x, y with the relation yx = qxy. Con- sider the operators ∂_x,∂_y:C[x,y] → C[x,y] of the corresponding differentiations:

∂_x(xⁿy^m)=nxⁿ⁻¹y^m,∂_y(y^mxⁿ)=my^m−1xⁿ.Notethatwehavethefollowingrela- tionsinEnd(C[x,y]):∂_y∂_x=q∂_x∂_y,∂_xy=qy∂_x,∂_yx=q⁻¹x∂_y,[∂_x,x]= [∂_y,y]= 1, q^2y∂y∂_yq^−2y∂y = q⁻²∂_y, where q^±2y∂y = e^{±2 log(q)y∂y} are operators acting as q^±2y∂y(xⁿy^m)=q^±2mxⁿy^m.Thenthematrix

M=

x q⁻¹q^−2y∂y∂_y y q^−2y∂y∂_x

(2.13) isa q-Maninmatrix.

• Examples of q-Manin matrices related to the quantum groups Funq(GLn) and U_q(gl_n) willbeconsideredinSubsections2.4 and6.1respectively.

Remark 2.2. If an n×m matrix M is a q-Manin matrix, then the n×m matrix M with entries Mij = Mn−i+1,m−j+1 is a q⁻¹-Manin matrix. For example, if M is the matrix(2.9)then weobtainthefollowing q⁻¹-Maninmatrix:

M= d c

b a

. (2.14)

Letusnotethatmatricesobtainedbypermutationsofrowsorcolumnsfromq-Manin matricesarenot q-Maninmatrices ingeneral.

2.2. Lineartransformationsof q-(anti)-commuting variables

TheoriginaldefinitionofaManinmatrix[26],forthecaseq= 1,wasthatofamatrix definingalineartransformationofcommuting variables–generatorsofthepolynomial algebra–aswellasforalineartransformationofanti-commutingvariables–generatorsof theGrassmannalgebra.Thiswaslargelyusedin[4,5].Letusconsiderhereananalogous interpretationoftheq-Maninmatrices.

Letusfirstintroduceaq-deformationofthepolynomialalgebraandaq-deformation of theGrassmannalgebra.Definetheq-polynomial algebra C[x1,. . . ,xm] asthealgebra with generatorsxi,i= 1,. . . ,m,and relations

xixj =q⁻¹xjxi

fori< j.Theserelationscanberewrittenintheform

x_ix_j=q^sgn(i−j)x_jx_i, i, j= 1, . . . , m. (2.15) The elements xi are called q-polynomial variables. Similarly define the q-Grassmann algebraC[ψ1,. . . ,ψn] asthealgebrageneratedbyψi,i= 1,. . . ,n,withrelationsψ²_i = 0 fori= 1,. . . ,nand ψiψj =−qψjψi,fori< j;thatis

ψ_iψ_j=−q^{−sgn(i−j)}ψ_jψ_i, i, j= 1, . . . , n. (2.16) Theelements ψi arecalled q-Grassmannvariables.

LetMbearectangularn×m-matrixoverR.WecanalwayssupposethatRcontains the q-polynomialalgebra C[x1,. . . ,xm] and the q-Grassmannalgebra C[ψ1,. . . ,ψn] as sub-algebrassuchthattheelementsofthesesub-algebrascommutewiththeentriesofM:

[x_j,M_pq]= [ψ_i,M_kl]= 0 foralli,k= 1,. . . ,nandj,l= 1,. . . ,m.Indeed,ifR doesnot contain oneofthese algebras wecanregardthematrixM as amatrixoverthealgebra R⊗C[x1,. . . ,xm]⊗C[ψ1,. . . ,ψn],wheretheelementsofdifferenttensorfactorspairwise commute.

Proposition 2.1. Let the entries of a rectangular n×m matrix M commute with the variablesx₁,. . . ,x_mandψ₁,. . . ,ψ_n.Considernewq-polynomialvariablesx˜₁,. . . ,x˜_n ∈R (resp.new q-Grassmannvariablesψ˜₁,. . . ,ψ˜_m∈R)obtainedby left(resp.right)‘action’

of M ontheold variables:

⎛

⎜⎝

˜ x1

...

˜ x_n

⎞

⎟⎠=

⎛

⎜⎝

M11 ... M1m

...

M_n1 ... M_nm

⎞

⎟⎠

⎛

⎜⎝ x1

...

x_m

⎞

⎟⎠ (2.17)

( ˜ψ1, . . . ,ψ˜m) = (ψ1, . . . , ψn)

⎛

⎜⎝

M₁₁ ... M_1m ...

M_n1 ... M_nm

⎞

⎟⎠. (2.18)

Then thefollowingthreeconditions areequivalent:

• The matrixM is aq-Maninmatrix.

• The variablesx˜_i q-commute: x˜_ix˜_j=q^sgn(i−j)x˜_jx˜_i foralli,j= 1,. . . ,n.

• The variablesψ˜_i q-anti-commute: ψ˜_iψ˜_j=−q^{−sgn(i−j)}ψ˜_jψ˜_i foralli,j= 1,. . . ,m.

Example 2.1. Let x1,x2,ψ1,ψ2 ∈ R be elements such that x1x2 = q⁻¹x2x1, ψ1ψ2 =

−qψ2ψ1,ψ²₁=ψ²₂= 0. Inthecasen= 2 theformulae(2.17), (2.18)taketheform

˜

x1=ax1+bx2, ψ˜1=aψ1+cψ2, (2.19)

˜

x₂=cx₁+dx₂, ψ˜₂=bψ₁+dψ₂, (2.20) where a,b,c,d∈ R are elements commuting with x1, x2 and with ψ1, ψ2. The matrix M =_{a b}

c d

isq-Maninifandonly ifx˜1x˜2 =q⁻¹x˜2x˜1,or ifand onlyifψ˜1ψ˜2=−qψ2ψ˜1

andψ˜²₁= ˜ψ²₂= 0.

Both factors of the algebra R ⊗ C[x1,. . . ,xn] have a natural grading, and so q-Maninmatricescanbeinterpreted asmatricesofgrading-preservinghomomorphisms C[x₁,. . . ,x_n] →R⊗C[x₁,. . . ,x_n] with respect to thevariables x_i, and/or matrices of grading-preservinghomomorphismsC[ψ1,. . . ,ψn]→R⊗C[ψ1,. . . ,ψn] withrespectto thevariablesψi.

Remark2.3.Theconditionsψ˜_i²= 0 areequivalentto therelations(2.1)and thecondi- tionsψ˜iψ˜j =−qψ˜jψ˜i,i< j,areequivalenttotherelations(2.2).

2.3. Elementaryproperties

Proposition2.2. The followingpropertieshold:

1. Any submatrixof aq-Maninmatrix isaq-Maninmatrix.

2. Any diagonal matrixwithcommuting entries isaq-Maninmatrix.

3. If M andN areq-Maninmatrices,thenM+N isaq-Manin matrixifandonly if MijNkl−q^sgn(i−k)q^{−sgn(j−l)}MklNij−q^sgn(i−k)MkjNil+q^{−sgn(j−l)}MilNkj

+NijMkl−q^sgn(i−k)q^{−sgn(j−l)}NklMij−q^sgn(i−k)NkjMil

+q^{−sgn(j−l)}NilMkj= 0, (2.21)

foralli,j,k,l= 1,. . . ,n;in particular,if

MijNkl=q^sgn(i−k)q^{−sgn(j−l)}NklMij, (2.22) foralli,j,k,l= 1,. . . ,n thenM+N isaq-Maninmatrix;

4. The product cM of q-Manin matrix M and a complex constant c ∈ C is also a q-Maninmatrix.

5. The product of q-Manin matrix M and adiagonal complex matrixD from theleft DM aswellasfrom theright M Disalso aq-Maninmatrix.

6. LetM andN ben×mandm×r q-ManinmatricesoveranalgebraRsuchthattheir elementscommute,i.e.[Mij,Nkl]= 0,i= 1,. . . ,n,j,k= 1,. . . ,m,l= 1,. . . ,r,then theproductM N isaq-Maninmatrix.

Proof. Thefirst two properties(1 and 2) are obvious. Toprove property 3 oneshould writetherelations(2.3)forM+N.Takingintoaccounttherelations(2.3)forMandN onearrivestotherelation(2.21).Thecondition(2.22)impliesthecondition(2.21).

The properties 4and 5areparticular casesof theproperty 6. To provethelast one we considerq-commutingvariables¹ xi,i= 1,. . . ,r,commutingwithM andN:xixj = q^sgn(i−j)xjxi, [Mij,xl]= [Nij,xl] = 0.Due to Proposition 2.1the new variablesx^N_k = r

l=1Nklxl, k = 1,. . . ,m, q-commute: x^N_i x^N_j = q^sgn(i−j)x^N_j x^N_i . Since [Mij,x^N_k] = 0 we can apply Proposition 2.1 to the matrix M and the variables x^N_k – the variables x^{M N}_i = m

j=1M_ijx^N_j , i = 1,. . . ,n, also q-commute: x^{M N}_i x^{M N}_j = q^sgn(i−j)x^{M N}_j x^{M N}_i . Then,formulax^{M N}_i =r

l=1(M N)_ilx_landProposition 2.1implythatM N isaq-Manin matrix. 2

Remark 2.4. If M and N are matrices over thealgebras R_M and R_N respectively we canconsider M and N as matricesoverthesamealgebraR_M⊗R_N andwethen have thecondition[M_ij,N_kl]= 0 foralli,j,k,l.

2.4. Relationswith quantum groups

Onecanalsodefine q-analogues ofManinmatrices characterizingtheconnectionsto quantum grouptheory. Actually q-Manin matrices are definedby half of the relations of thecorresponding quantum group Fun_q(GL_n)² [29]. The remaining halfconsists of relationsinsuringthatM isalsoaq-Maninmatrix,whereM isthetranspose ofM.

Definition2.Ann×nmatrixT belongstothequantumgroupFun_q(GL_n) ifthefollowing conditionsholdtrue.Forany2×2 sub-matrix(T_(ij)(kl)),consistingofrowsiandk,and columns j andl (where1i< kn,and1j < ln):

⎛

⎜⎜

⎜⎜

⎜⎝

... ... ... ... ...

... Tij ... Til ...

... ... ... ... ...

... Tkj ... Tkl ...

... ... ... ... ...

⎞

⎟⎟

⎟⎟

⎟⎠≡

⎛

⎜⎜

⎜⎜

⎜⎝

... ... ... ... ...

... a ... b ...

... ... ... ... ...

... c ... d ...

... ... ... ... ...

⎞

⎟⎟

⎟⎟

⎟⎠

(2.23)

thefollowing commutationrelationshold:

ca=qac, (q-commutation of the entries in a column) (2.24)

1 Onecanjustaswelluseq-Grassmannvariablesψi.

2 MorepreciselyweshouldwriteFunq(Matn),sincewedonotlocalizetheq-determinant.

db=qbd, (q-commutation of the entries in a column) (2.25) ba=qab, (q-commutation of the entries in a row) (2.26) dc=qcd, (q-commutation of the entries in a row) (2.27) ad−da= +q⁻¹cb−qbc, (cross commutation relation 1) (2.28) bc=cb, (cross commutation relation 2). (2.29) Asquantum groupsare usually definedwithin theso-calledmatrix (Leningrad)for- malism,letus brieflyrecallit(wewillfurtherdiscussthisissueinSection5).

Lemma2.3. Thecommutation relationsforquantumgroupmatrices canbe describedin matrix(Leningrad)notationsas follows:

R(T⊗1)(1⊗T) = (1⊗T)(T⊗1)R. (2.30) The R-matrix isgiven,in thecaseweare considering,by theformula:

R=q⁻¹

i=1,..,n

Eii⊗Eii+

i,j=1,..,n;i=j

Eii⊗Ejj

+

q⁻¹−q

i,j=1,..,n;i>j

Eij⊗Eji, (2.31)

where Eij are the standard basis of End(Cⁿ),i.e. (Eij)kl = δikδjl – zeroeseverywhere except1intheintersectionof thei-th rowwith thej-thcolumn.

Forexampleinthe2×2 casetheR-matrixis:

R=

⎛

⎜⎜

⎜⎝

q⁻¹ 0 0 0

0 1 0 0

0 q⁻¹−q 1 0

0 0 0 q⁻¹

⎞

⎟⎟

⎟⎠. (2.32)

Remark2.5. ThisR-matrixdiffersbythechangeq→q⁻¹ from thatin[29],page185.

Therelationbetweenq-Maninmatricesandquantumgroupsconsistsinthefollowing simpleproposition:

Proposition2.4.AmatrixT isamatrixin thequantum groupFunq(GLn)ifandonlyif both T andT are q-Maninmatrices.

Soonegetsthatq-Maninmatrices arecharacterized bya“half”of theconditionsof thecorrespondingquantum matrixgroup.

2.5. Hopfstructure

Let us consider the algebra C[Mij] generated (over C) by Mij, 1 i,j n, with q-Maninrelations(2.3).Onecanseethatitcanbeequippedbyastructureofbialgebra with thecoproductΔ(M_ij)=

kM_ik⊗M_kj.Thisisusually denotedas follows:

Δ(M) =M⊗^. M. (2.33)

It iseasytoseethatthiscoproductiscoassociative(i.e.(Δ⊗1)⊗Δ= (1⊗Δ)⊗Δ).

Thenaturalantipode forthisbialgebra shouldbe S(M)=M⁻¹.Soitexists onlyin someextensionsofthealgebraC[Mij].

The“coaction”-Proposition 2.1impliesthatthereexist morphismsofalgebras:

φ1:C[x1, . . . , xm]→C[Mij]⊗C[x1, . . . , xm], φ1(xi) =

k

Mikxk, (2.34) φ2:C[ψ1, . . . , ψn]→C[Mij]⊗C[ψ1, . . . , ψn], φ2(ψi) =

k

Mkiψk. (2.35) Onecancheckthatbothmapssatisfythecondition:(Δ⊗1)(φi)= (1⊗φi)(φi),i= 1,2.

Soonecanconsiderthemapsφi as“coactions”ofq-ManinmatricesonthespaceCⁿq

and itsGrassmannianversion.

3. Theq-determinantandCramer’sformula

It was shown in [4,5] that the natural generalization of the usual determinant for Manin matrices(i.e., theq= 1 case)is thecolumndeterminant.This columndetermi- nantsatisfiesallthepropertiesofthedeterminantofsquarematricesoveracommuting field, and is defined as inthe usuals case, with the provisoin mind thatthe orders of the column index in then! summands of the determinantshould be always the same.

Therole ofcolumndeterminantforq-Maninmatricesisplayedbyitsq-analoguecalled q-determinant. Most of the properties of column determinant of Manin matrices pre- sented in[5]canbegeneralizedtogeneralq.

Inthissectionwerecallthedefinitionoftheq-determinant.Wewillseethatinthecase ofq-Maninmatricesitgeneralizesthenotionoftheusual determinantsforthematrices overcommutativerings.Weshallstartbyconsideringtheq-determinantforanarbitrary (i.e., notnecessarilyq-Manin)matrixwithelements inanon-commutativeringR.

3.1. Theq-determinant

Wedefinetheq-determinantofanarbitrarymatrixM withnon-commutativeentries and relate it with coaction of M on the q-Grassmann algebra. We prove here some formulaeused belowforq-determinantsandq-minors ofq-Maninmatrices.

3.1.1. Definition of theq-determinant

Definition3.Theq-determinant,ofann×nmatrixM = (Mij) isdefinedbytheformula detqM:=

τ∈Sn

(−q)^−inv(τ)M_τ(1)1M_τ(2)2· · ·M_τ(n)n, (3.1) wherethesumrangesoverthesetSnofallpermutationsof{1,. . . ,n}.Recallthatinv(τ) isnumberofinversions–thenumberofpairs1i< jnforwhichτ(i)> τ(j).Inother words,inv(τ) isthelengthofτ withrespect toadjacenttranspositionsσ_k =σ_(k,k+1). Example3.1.

det_q

a b c d

def= ad−q⁻¹cb. (3.2)

Example3.2.

detq

⎛

⎜⎝

M11 M12 M13

M21 M22 M23

M₃₁ M₃₂ M₃₃

⎞

⎟⎠

def= M11M22M33+q⁻²M21M32M13+q⁻²M31M12M23

−q⁻¹M11M32M23−q⁻¹M21M12M33−q⁻³M31M22M13 (3.3)

=M₁₁det_q

M11 M12

M21 M22

+ (−q)⁻¹M₂₁det_q

M12 M13

M32 M33

+ (−q)⁻²M₃₁det_q

M12 M13

M22 M23

= (−q)⁻²det_q

M21 M22

M31 M32

M₁₃

+ (−q)⁻¹det_q

M11 M12

M31 M32

M₂₃+ det_q

M11 M12

M21 M22

M₃₃. (3.4) Itiseasy tosee fromthedefinitionthat,ifLisanarbitrary lower-triangularmatrix with d_i on the diagonal, and R an arbitrary upper-triangular matrix with d_i on the diagonal,then

det_qL= det_qR=

i

d_i. (3.5)

Itis alsoeasyto seethat,if, S isthepermutation matrixcorrespondingto thepermu- tationσ∈Sn,thendetqS = (−q)^−inv(σ).

Remark 3.1. For q = 1 the present definition of q-determinant coincides with that of column-determinant i.e.the determinantdefinedbythe columnexpansion,first taking

elementsfromthefirstcolumn,thenfromthesecondandsoonandsoforth.Forq=−1 the q-determinantyields the(column) permanent ofthe matrixM. Justlike theusual determinant, the q-determinant is linear over C both with respect to its columns and rows.

3.1.2. The q-Grassmann algebra

Let uscollectsomeuseful relationsintheq-GrassmannalgebraC[ψ1,. . . ,ψn].Some of themholdwithoutassumingψ_i²= 0,whilesomeofthemrequirethisproperty.

Lemma 3.1.Letψi satisfy therelationsψiψj =−qψjψi forall1i< jn;thentheir monomialsof n-thorder arerelated asfollows

ψ_τ(1)· · ·ψ_τ(n)= (−q)^−inv(τ)ψ1· · ·ψn, ∀τ∈Sn, (3.6) or equivalentlyas

ψ_στ(1)· · ·ψ_στ(n)= (−q)⁻inv(στ)+inv(σ)ψ_σ(1)· · ·ψ_σ(n), ∀σ, τ ∈S_n. (3.7) Proof. Notethatiftherelation(3.7)isvalidforsomeτ₁,τ₂∈S_n (forallσ∈S_n)then itis soforτ=τ₁τ₂.Indeed

ψ_στ1τ2(1)· · ·ψ_στ1τ2(n)= (−q)^{−inv(στ1τ2)+inv(στ1)}ψ_στ1(1)· · ·ψ_στ1(n)

= (−q)^{−inv(στ1τ2)+inv(σ)}ψ_σ(1)· · ·ψ_σ(n). (3.8) Sinceeachτcanbepresentedasaproductofadjacenttranspositionsσk,k= 1,. . . ,n−1, itis sufficienttoprove(3.7)forτ =σ_k. Inthis casewecanwrite

ψσσk(1)· · ·ψσσk(n)=ψσ(1)· · ·ψσ(k+1)ψσ(k)· · ·ψσ(n)

= (−q)⁻sgn(σ(k+1)−σ(k))ψσ(1)· · ·ψσ(k)ψσ(k+1)· · ·ψσ(n). (3.9) Thus, formula(3.7)forτ=σk followsfromtheequality

inv(σσk) = inv(σ) + sgn

σ(k+ 1)−σ(k)

. 2 (3.10)

Corollary 3.1.1. Under the condition of Lemma 3.1, one can relate the monomials of m-thorder asfollows:

ψjτ(1)· · ·ψjτ(m)= (−q)^−inv(τ)ψj1· · ·ψjm, (3.11) ψjτ(1)· · ·ψjτ(m)= (−q)⁻inv(τ)+inv(σ)ψjσ(1)· · ·ψjσ(m), (3.12) where 1j1< . . . < jmn,τ,σ∈Sm.

Proof. Consider the elements ψ˜k = ψjk. They satisfy the conditions of Lemma 3.1:

ψ˜iψ˜j = −qψ˜jψ˜i for all 1 i < j m. Writing the formulae (3.6), (3.7) for ψ˜k we obtain(3.11),(3.12). 2

Corollary 3.1.2. Assumingadditionally that ψ²_i = 0, i.e. ψiψj =−q^{−sgn(i−j)}ψjψi,it is convenient towritethemore generalformula

ψi1· · ·ψin=ε^q_i1,...,inψ1· · ·ψn, (3.13) where1iln andtheq-epsilon-symbol isdefined bytheformula

ε^q...,i,...,i,...= 0, ε^qσ(1),...,σ(n)= (−q)^−inv(σ), σ∈Sn. (3.14) Letus denote byI⊕J = (i1,. . . ,im,j1,. . . ,jk) thecontraction oftwo multi-indices I = (i₁,. . . ,i_m) andJ = (j₁,. . . ,j_k).For amulti-indexK = (k₁,k₂,. . . ,k_r) denoteby

\K the multi-index (1,. . . ,kˆ1,. . . ,kˆm,. . . ,n), thatis \K is obtained from (1,2,. . . ,n) bydeletingk_i foralli= 1,. . . ,r.

Let(i1,. . . ,in) beapermutationof(1,. . . ,n) suchthati1< i2< . . . < imandim+1<

im+2< . . . < in forsomemn. LetI= (i1,i2,. . . ,im) and \I = (im+1,im+2,. . . ,in).

Itisquiteeasytosee that

ε^q_(I⊕\I)= (−q)^{−ml=1(il−l)}= (−q)^{+nl=m+1(il−l)}, (3.15) ε^q_(\I⊕I)= (−q)^{ml=1il−nl=n−m+1l}= (−q)^{n−ml=1 l−nl=m+1il}, (3.16) whereweused

l=1,...,ni_l=

l=1,...,nl.

LetusalsomentionthatforI= (n,n−1,n−2,. . . ,1) we have

ε^q_I = (−q)^{−n(n−1)/2}. (3.17) 3.1.3. The q-determinant, q-minorsandtheq-Grassmann algebra

Letusgiveamoreconceptualapproachtoq-determinantsandq-minors–q-determi- nantsofsub-matrices–viatheq-Grassmannalgebra.WeconsideranarbitrarymatrixM, butthe factthat thenotion of theq-determinantcan be reformulatedinterms of the q-Grassmannalgebraisaclearhintthatitisrelatedwiththenotionofq-Maninmatrix.

LetM beanarbitraryn×mmatrixandI= (i₁,i₂,. . . ,i_k) andI= (i₁,i₂,. . . ,i_l) be two arbitrarymulti-indices, where knand l m.Denote by MIJ the k×k matrix definedas (M_IJ)_ab = M_iajb, where a,b = 1,. . . ,k. Then the q-determinant det_q(M_IJ) canbe consideredas theq-analogueoftheminor (q-minor)ofthematrixM.