On Morita equivalence for simple Generalized Weyl algebras

On Morita equi v alence for simple Generalized W eyl algebras Lionel Ric hard ∗ and Andrea Solotar † No v em b er 15, 20 18 Abstract W e give a necessary condition for Mor ita equiv alence of simple Genera lized W eyl algebras of classical t ype. W e propose a reformulation of Ho dges’ r e- sult, which describ es Morita equiv alences in case the p olynomia l defining the Generalized W eyl alg ebra has degree 2, in ter ms of iso morphisms of quan- tum tor i, inspired by similar considera tions in nonc o mmut ative differential geometry . W e study how far this link can be g eneralized for n ≥ 3. 1 In tro duction The aim of th is p ap er is to describ e Morita equiv alence of generalized W eyl alge- bras of type k [ h ]( σ cl , a ), wh ere σ cl ( h ) = h − 1 and a ∈ k [ h ] is a p olynomial, un d er the assump tion that the algebra is simple and has fin ite global dim en sion. Gen- eralized W eyl algebras w ere introdu ced by V.Ba vula [1], and pro du ce a common framew ork for the study of some classical algebras and their quan tum counterpart. Examples of GW A are, n -th W eyl algebras, U ( sl 2 ), p rimitiv e qu otien ts of U ( sl 2 ), its qu an tized versions and also the subalgebras of inv ariants of these algebras un- der the action of finite cyclic sub groups of automorphisms. It results from the discussion in [13, § 3.1] that from the p oint of view of Morita equiv alence these t w o cases (classical . qu an tum) m ight b e treated separately . W e fo cus h ere on th e classical case, also studied by T.J.Hod ges [9] und er the name of N on c ommutative deformation s of typ e A -Kleinian singularities . These algebras are n aturally Z -graded, and they play a crucial role in a recen t pap er [15] by Su s an Sierra on r ings graded equiv alen t to the W eyl algebra. Nev- ertheless w e are d ealing here with usual Morita equiv alence, and the gradin g will not p la y an y visible role in the follo wing. ∗ Universit y of Edinburgh, S chool of Mathematics and Maxw ell Institute for Mathematical Sciences, JC MB - Ma yfield Road, Edinburgh EH9 3JZ, United Kingdom. lionel.ric hard@ed.ac.uk † Dto. de Matem´ atica, F acultad de Cs. Exactas y Naturales. Un ivers idad d e Buenos Aires. Ciudad Universitaria Pab I. (1428), Buenos Aires - Argentina. asolotar@dm.uba.ar 0 Researc h partially supp orted by UBACyT X 169 , PIP-CONICET 5099 , PICS-C N RS 3410 , and Cooperaci ´ on Inte rnacional–CONICET-CNRS . The first author is supp orted by an EP- SRC grant (EP/D034167/1), th e second auth or is a researc h member of CONICET (A rgenti na). 1 Notation. F or a ∈ k [ h ], d en ote A ( a ) = k [ h ]( σ cl , a ) the k -algebra generated o v er k [ h ] by tw o generators x, y satisfying th e relations xh = ( h − 1) x, y h = ( h + 1) y , y x = a ( h ) , xy = a ( h − 1) . (1) W e recall th e follo wing resu lt, wh ic h follo ws from [2, Prop osition 2 and Corollary 2]. Prop osition 1.1. The classic al GW A A ( a ) = k [ h ]( σ cl , a ) is simple if and only if for any two distinct r o ots α and β of the p olyno mial a , then α − β 6∈ Z . ⊓ ⊔ F urthermore, we will assume in the follo wing that th e p olynomial a has d istin ct ro ots. Thanks to [9, Theorem 4.4], this is equiv alen t to saying that the algebra A ( a ) has finite global dimension. W e write explicitly this condition for fur ther use: λ i − λ j 6∈ Z , ∀ i 6 = j. (2) In this p ap er we will make use of th e pro of giv en b y Ho d ges for B λ ’s in [8], which are exactly the GW As defin ed by a p olynomial of degree 2, using add itional results from [9]. Ho wev er, we prop ose a r eform ulation of Ho dges’ result, r elying on the link with quan tum tori, insp ired by similar considerations in noncommutat iv e differen tial geometry [14, 10]. It is natural to study , then, h o w far this link can b e generalize d for n ≥ 3. The pap er is constructed as follo ws. Next section is dedicated to our main r esult Theorem 2.6.2. Along the wa y we w ill study in d eep detail the link b etw een K 0 ( A ) and H H 0 ( A ). In S ection 3 w e explicit our result in the case n = 3, and inv estigat e how far our n ecessary condition is to b e suffi cien t. A t last, in Section 4 we present some links with qu antum tori, in spired by similar considerations in noncomm utativ e differentia l geometry [14, 10]. In all the f ollo wing k is an algebraicall y closed field of c haracteristic zero, an d in Sections 3 and 4 we will sp ecify k = C . 2 F ramew ork 2.1 Normal form and degree 2 case W e recall the follo wing result of Ba vula and J ordan. Theorem 2.1.1 ([3], Th eorem 3.28) . F or a 1 , a 2 ∈ k [ h ] , A ( a 1 ) ≃ A ( a 2 ) if and only if a 2 ( h ) = ρa 1 ( ǫh + β ) for some ǫ ∈ {− 1 , 1 } and ρ, β ∈ k with ρ 6 = 0 . ⊓ ⊔ 2 Thanks to this r esult w e will alw a ys assume our p olynomials to b e monic, i.e. we will wr ite them in the form a ( h ) = Q n i =1 ( h − λ i ) with λ 1 , . . . , λ n the ro ots of the p olynomial a ( h ). Note that w e m a y also, up to isomorphism, translate all r o ots b y the same − β and change the sign of all of them. Remark 2.1.2 . It follo ws from [9] that the p olynomials defin ing t w o Morita equiv alent GW As m ust ha v e the same degree. Before s tudying the general case w e recall the follo wing resu lt in degree 2. Theorem 2.1.3 ([8], Theorem 5) . L et a ( h ) = ( h − λ 1 )( h − λ 2 ) and b ( h ) = ( h − µ 1 )( h − µ 2 ) b e two p olynomials of de gr e e 2. Then A ( a ) and A ( b ) ar e Morita e qui valent if and only if λ 1 − λ 2 = ± ( µ 1 − µ 2 ) + m for some m ∈ Z . ⊓ ⊔ 2.2 A sufficient condition The follo wing is a direct consequence of [9, Lemma 2.4 an d Theorem 2.3]. Prop osition 2.1. Set a, b ∈ k [ h ] two p olynomials with distinct r o ots r esp e c tively { λ i , 1 ≤ i ≤ n } and { µ i , 1 ≤ i ≤ n } , satisfying c ondition (2). Supp ose that ther e exist τ ∈ S n and ( m 1 , . . . , m n ) ∈ Z n such that λ i = λ ′ τ ( i ) + m i for al l 1 ≤ i ≤ n . Then the GW A’s A ( a ) and A ( b ) ar e Morita-e quivalent. ⊓ ⊔ Note that for n = 2 th is condition is equiv alent to the one app earing in T h eorem 2.1.3. 2.3 Morita equiv alence and trace function In the r est of this Section w e study necessary conditions for Morita equiv ale nce. Assume that a and b are t w o p olynomials in k [ h ], with simp le ro ots ha ving non- in teger differences, such that A ( a ) and A ( b ) are k -linearly Morita equiv alent. Suc h an equiv alence from the category of (sa y) left A ( a )-mo d ules to left A ( b )-mo d ules is giv en by tensorin g with a bimo du le A ( b ) P A ( a ) , finitely generated and pro jectiv e as a left and a right mo dule. Th is fu nctor induces a group isomorphism K 0 ( F ) b et ween K 0 ( A ( a )) and K 0 ( A ( b )) and a k -linear isomorphism H H 0 ( F ) b etw een H H 0 ( A ( a )) and H H 0 ( A ( b )). Here as usually K 0 ( A ) d enotes the Gr othendiec k group of A , generated b y finitely generated pr o jectiv e m o dules, and H H 0 ( A ) the Ho c hsc hild homology space in degree zero, wh ic h is also the k -v ector sp ace of traces A/ [ A, A ]. Moreo v er, K 0 ( F ) m ust pr eserv e the usu al rank fu nction r k : K 0 ( A ) → Z , defined on a pro jectiv e P as th e length of the F rac A -mo dule (F rac A ) ⊗ A P . So if we denote e K 0 ( A ) = Ker( rk ), we h a v e the follo wing commuta tiv e diagram: 3 e K 0 ( A ( a )) i / / e K 0 ( F )   K 0 ( A ( a )) tr / / K 0 ( F )   H H 0 ( A ( a )) H H 0 ( F )   e K 0 ( A ( b )) i / / K 0 ( A ( b )) tr / / H H 0 ( A ( b )) (3) Here i denotes th e canonical injection and t r the usual Hattori-Stallings trace map. Remark that e K 0 ( F ) is an isomorp hism of groups to o (for more d etails see [4]). F ollo wing the ideas of [8 ], we will describ e as pr ecisely as p ossible the m aps e K 0 ( F ) and H H 0 ( F ). 2.4 A basis for e K 0 ( A ( a )) Let a ∈ k [ h ] b e a p olynomial of degree n with simp le r o ots satisfying (2). Thanks to [3 , Th eorem 3.28], we can assume that a ( h ) is monic, that is a ( h ) = Q n i =1 ( h − λ i ). Then thanks to [9, Theorem 3.5] and Quillen’s lo calization sequence [7 ] we kno w (b y an argumen t analo gous to [8, Prop osition 1]) t hat [ A ( a )] , [ P ( a ) 1 ] , . . . , [ P ( a ) n − 1 ] form a basis of K 0 ( A ( a )), with P ( a ) i = A ( a ) x + A ( a )( h − λ i ). Moreo v er, thanks to [9, Lemma 2.4], w e kno w that the P ( a ) i are progenerators, and give Mo rita equiv alences b et w een A ( a ) and A ( b i ), where b i = ( h − λ i − 1) Q j 6 = i ( h − λ j ) is th e p olynomial obtained from a by replacing λ i with λ i + 1. Then one easily v erifies that Prop osition 2.4.1. With the notations ab ove ([ P ( a ) i ] − [ A ( a )] , 1 ≤ i ≤ n − 1) is a b asis of e K 0 ( A ( a )) . ⊓ ⊔ 2.5 T r ace of [ P ( a ) i ] − [ A ( a )] W e compute here the trace of the pro jectiv e P ( a ) i . Prop osition 2.5.1. L et a ( h ) ∈ k [ h ] b e a p olynomial of de gr e e at le ast 2 satisfying the criterion of Pr op osition 1.1, and denote A = A ( a ) . F actorize a ( h ) = u ( h ) w ( h ) with u and w non-c onstant p olyno mials. Assume that u and w ar e r elatively prime p olynomials. The left A -ide al P = Ax + Aw ( h ) is pr oje ctive, and its tr ac e is the class of the p olynomia l 1 + w ( h ) B ( h ) − w ( h − 1) B ( h − 1) , wher e B ( h ) , C ( h ) ar e two p olyno mials such that B ( h ) w ( h ) + C ( h ) u ( h ) = 1 . Pro of . Consider the epimorph ism of A -mo du les G : A ⊕ A → P defined by G (1 , 0) = x, G (0 , 1) = w ( h ). T hen one ma y easily c hec k that G admits a section F : P → A ⊕ A defin ed b y F ( x ) = ( C ( h − 1) u ( h − 1) , B ( h − 1) x ) , F ( w ( h )) = ( C ( h ) y , w ( h ) B ( h )). Then tr ( P ) is nothing bu t the usual trace of the idemp otent F ◦ G ∈ M 2 ( A ), and one conclud es using the d efining relation b et w een B and C . ⊓ ⊔ 4 Notations. • Since a ( h ) = ( h − λ 1 ) . . . ( h − λ n ) has degree n , we see from 3.1.1 in [6] th at H H 0 ( A ( a )) is n aturally isomorphic to the s u bspace of k [ h ] sp anned by the classes of 1 , h, . . . , h n − 2 . F or con v enience we w ill den ote 1 a and h p a the classes of 1 and h p , so t hat H H 0 ( A ( a )) = k . 1 a ⊕ L n − 2 p =1 k .h p a . Similarly H H 0 ( A ( b )) = k . 1 b ⊕ L n − 2 p =1 k .h p b . • F or an y int eger ρ ≥ 0 denote k ρ the sp ace of p olynomials of d egree n ot greater than ρ . Giv en n distinct scalars λ 1 , . . . , λ n , denote b y ( v 1 , . . . , v n ) the basis of k n − 1 consisting of Lagrange interp olation p olynomials asso ciated to ( λ 1 , . . . , λ n ), i.e. v i = u i /r i , with u i = Q j 6 = i ( h − λ j ) and r i = Q j 6 = i ( λ i − λ j ) = u i ( λ i ). In fact, eac h u i is th e quotien t of t w o V and ermonde determinants, u i = V ( λ 1 ,...,λ i − 1 ,h,λ i +1 ,...,λ n ) V ( λ 1 ,..., ˆ λ i ,...,λ n ) .  V ( λ 1 ,...,λ n ) V ( λ 1 ,..., ˆ λ i ,...,λ n )  − 1 = V ( λ 1 ,...,λ i − 1 ,h,λ i +1 ,...,λ n ) V ( λ 1 ,...,λ n ) with the con v en tion that V ( λ 1 , . . . , λ n ) is the determinant of th e n × n matrix with ( i, j )th en try λ n − i j for all 1 ≤ i, j ≤ n . Prop osition 2.5.2. Set a ( h ) = ( h − λ 1 ) . . . ( h − λ n ) . L et P ( a ) i = A ( a ) x + A ( a )( h − λ i ) for al l 1 ≤ i ≤ n − 1 b e the left A ( a ) -mo dules c onsider e d ab ove. Then tr ( P ( a ) i ) is the class of the p olynomial 1 + ( σ − 1) u ( a ) i /r i . Pro of . W e apply the preceding P r op osition with v ( h ) = a ( h ), u ( h ) = Q j 6 = i ( h − λ j ) and w ( h ) = h − λ i . Th e euclidian division of u b y w giv es u = ( h − λ i ) Q + r i , with deg Q = n − 1. Setting B ( h ) = − Q/r i and C ( h ) = 1 /r i , one gets tr ( P ( a ) i ) as the class of the p olynomial 1 + r i − u r i − σ ( r i − u ) r i . On e concludes then using the fact that σ is an algebra morphism. ⊓ ⊔ Since the p olynomial giving the trace of P ( a ) i is of degree n − 2, it may b e iden tified with its class in H H 0 ( A ( a )). Denote p ( a ) i the image of ( σ − 1)( v i ) in H H 0 ( A ( a )), so that tr( P ( a ) i ) = 1 a + p ( a ) i . Lemma 2.5.3. The set ( p ( a ) 1 , . . . , p ( a ) n − 1 ) is a b asis of H H 0 ( A ( a )) . Pro of . First we c hec k th at replacing v n ( h ) by the constan t p olynomial 1, the set ( v 1 , . . . , v n − 1 , 1) is s till a basis of k n − 1 . L et α 0 , α 1 , . . . , α n − 1 ∈ k su c h that α 0 + P n − 1 i =1 α i v i ( h ) = 0. Replacing h = λ n w e get α 0 = 0, and w e conclude thanks to the fact that t he v i ’s are linearly indep endant. No w defin e t he linear map S : k n − 1 → k n − 2 b y S ( P ) = σ ( P ) − P . T he set ( p ( a ) 1 , . . . , p (0) n − 1 , 0) is the image of the basis ab o v e b y S . W riting the matrix of S in the canonical bases, one easily sees that it is surjectiv e. This ends the pro of. ⊓ ⊔ Clearly we ha ve the same r esults with b instead of a and the µ i ’s in stead of the λ i ’s. W e giv e no w an int erpretation of the trace p olynomials p ( a ) i in terms of Sc h ur p olynomials. 5 Prop osition 2.5.4. Set as b efor e a ( h ) = Q n i =1 ( h − λ i ) . L et P ( a ) i = A ( a ) x + A ( a )( h − λ i ) for 1 ≤ i ≤ n − 1 . Then tr ( P ( a ) i ) = 1 a + p ( a ) i with p ( a ) i = n P i =1 ( − 1) i + l (( h − 1) n − l − h n − l ) σ ( 1 , . . . , 1 | {z } l − 1 , 0 , . . . , 0 | {z } n − l ) ( λ 1 , . . . , ˆ λ i , . . . , λ n ) . V ( λ 1 ,..., ˆ λ i ,...,λ n ) V ( λ 1 ,...,λ n ) wher e σ ( 1 , . . . , 1 | {z } l − 1 , 0 , . . . , 0 | {z } n − l ) ( λ 1 , . . . , ˆ λ i , . . . , λ n ) denotes the Schur p olyno mial asso ci- ate d to the p artition (1 , . . . , 1 | {z } l − 1 , 0 , . . . , 0 | {z } n − l ) evaluate d in ( λ 1 , . . . , ˆ λ i , . . . , λ n ) . Pro of . Recall from Prop osition 2.5.2 that p ( a ) i = ( u i ( h − 1) − u i ( h )) . 1 r i =  Y j 6 = i ( h − 1 − λ j ) − Y j 6 = i ( h − λ j )  . 1 Q j 6 = i ( λ i − λ j ) =  V ( λ 1 , . . . , λ i − 1 , h − 1 , λ i +1 , . . . , λ n ) V ( λ 1 , . . . , ˆ λ i , . . . , λ n ) − V ( λ 1 , . . . , λ i − 1 , h, λ i +1 , . . . , λ n ) V ( λ 1 , . . . , ˆ λ i , . . . , λ n )  . V ( λ 1 , . . . , ˆ λ i . . . , λ n ) V ( λ 1 , . . . , λ n ) = V ( λ 1 , . . . , λ i − 1 , h − 1 , λ i +1 , . . . , λ n ) − V ( λ 1 , . . . , λ i − 1 , h, λ i +1 , . . . , λ n ) V ( λ 1 , . . . , λ n ) = det        λ n − 1 1 λ n − 1 2 · · · ( h − 1) n − 1 − h n − 1 · · · λ n − 1 n . . . . . . . . . . . . . . . . . . λ 1 λ 2 · · · h − 1 − h · · · λ n 1 1 · · · 1 − 1 | {z } i 1 1        V ( λ 1 , . . . , λ n ) . Dev eloping by th e i -th column w e obtain: n P i =1 ( − 1) i + l (( h − 1) n − l − h n − l ) . det 0 B B B B B B @ λ n − 1 1 λ n − 1 2 · · · λ n − 1 i − 1 λ n − 1 i +1 · · · λ n − 1 n . . . . . . . . . . . . . . . . . . λ 1 λ 2 · · · λ i − 1 λ i +1 · · · λ n 1 1 · · · 1 1 · · · 1 1 C C C C C C A V ( λ 1 ,...,λ n ) = 6  n P i =1 ( − 1) i + l (( h − 1) n − l − h n − l ) .σ ( 1 , . . . , 1 | {z } l − 1 , 0 , . . . , 0 | {z } n − l ) ( λ 1 , . . . , ˆ λ i , . . . , λ n ) . V ( λ 1 ,..., ˆ λ i ...,λ n ) V ( λ 1 ,...,λ n )  . ⊓ ⊔ Let us remark h er e that Sch ur p olynomials also pla y a cen tral role in the classifi- cation up to Morita equiv alence of Cher ed nik algebras in [4]. 2.6 Computing H H 0 ( F ) In this sub section w e consider t w o p olynomials a ( h ) = Q n i =1 ( h − λ i ) and b ( h ) = Q n j =1 ( h − µ j ) with all distinct ro ots with non-integ er differences. Assuming that the algebras A ( a ) and A ( b ) are Morita equiv alen t, and usin g the notations of 2.3 , w e describ e no w H H 0 ( F ) as a matrix ( α ij ) ∈ GL n ( k ), in the bases ( p ( a ) 1 , . . . , p ( a ) n − 1 ), ( p ( b ) 1 , . . . , p ( b ) n − 1 ). 2.6.1 Notations • Set P = A ( b ) P A ( a ) the progenerator such that F ≡ P ⊗ A ( a ) ( ). It must h a v e rank 1 as an A ( b )-module (b ecause b oth rings are no etherian domains), so [ P ] − [ A ( b )] has rank 0 in K 0 ( A ( b )), and there exist m 1 , . . . , m n − 1 ∈ Z s u c h that [ P ] = [ A ( b )] + m 1  [ P ( b ) 1 ] − [ A ( b )]  + . . . + m n − 1  [ P ( b ) n − 1 ] − [ A ( b )]  in K 0 ( A ( b )) and t r ( b ) ( P ) = 1 b + m 1 p ( b ) 1 + . . . + m n − 1 p ( b ) n − 1 in H H 0 ( A ( b )). • Because e K 0 ( F ) is a group isomorphism, there exists a matrix N = ( n ij ) ∈ GL n − 1 ( Z ) suc h that for all 1 ≤ i ≤ n − 1 w e ha v e e K 0 ( F )  [ P ( a ) i ] − [ A ( a )]  = n 1 ,i  [ P ( b ) 1 ] − [ A ( b )]  + . . . + n n − 1 ,i  [ P ( b ) n − 1 ] − [ A ( b )]  . It results from th e definition of the m i ’s that the matrix asso ciated to K 0 ( F ) with resp ect to the bases  [ P ( a ) 1 ] − [ A ( a )] , . . . , [ P ( a ) n − 1 ] − [ A ( a )] , [ A ( a )]  and  [ P ( b ) 1 ] − [ A ( b )] , . . . , [ P ( b ) n − 1 ] − [ A ( b )] , [ A ( b )]  is      N m 1 . . . m n − 1 0 1      . 2.6.2 Link b etw een the matrices of e K 0 ( F ) and H H 0 ( F ) W e still consider the comm utativ e d iagram (3). T hen we get for all 1 ≤ i ≤ n − 1 H H 0 ( F )  tr([ P ( a ) i ] − [ A ( a )])  = tr  e K 0 ( F )([ P ( a ) i ] − [ A ( a )])  , that is H H 0 ( F )( p ( a ) i ) = tr  n 1 ,i ([ P ( b ) 1 ] − [ A ( b )]) + . . . + n n − 1 ,i ([ P ( b ) n − 1 ] − [ A ( b )])  , so α 1 ,i p ( b ) 1 + . . . + α n − 1 ,i p ( b ) n − 1 = n 1 ,i p ( b ) 1 + . . . + n n − 1 ,i p ( b ) n − 1 . Sin ce the p ( b ) i ’s are linearly 7 indep en d an t, w e get α k ,i = n k ,i for all 1 ≤ k , i ≤ n − 1, that is, the matrices asso ciated to e K 0 ( F ) and H H 0 ( F ) in our c hosen bases are equal. 2.6.3 Computing H H 0 ( F ) Because the d iagram (3) is comm utativ e, we hav e f or all 1 ≤ i ≤ n − 1: H H 0 ( F )  tr([ P ( a ) i ])  = tr  K 0 ( F )([ P ( a ) i ])  . (4) The left part of this equation is equal to H H 0 ( F )(1 a + p ( a ) i ). Lemma 2.6.1. The fol lowing formulas hold r esp e ctively in H H 0 ( A ( a )) and H H 0 ( A ( b )) 1 a = − n − 1 X i =1 ( λ i − λ n ) p ( a ) i ; 1 b = − n − 1 X j =1 ( µ j − µ n ) p ( b ) j . (5) Pro of . W e giv e the pro of for a ( h ), the pro of for b ( h ) b eing completely similar. So w e omit the upp er indices ( a ) in the follo wing. Recall from the notations in tro du ced in 2.5 that p i ( h ) = ( σ − 1)( v i ( h )), with v 1 ( h ) , . . . , v n ( h ) the Lagrange in terp olation p olynomials associated to λ 1 , . . . , λ n . Reasoning in k n − 1 , we hav e h = P n i =1 λ i v i ( h ) and 1 = P n i =1 v i ( h ), so that h = P n − 1 i =1 ( λ i − λ n ) v i ( h ) + λ n . W e conclude by n oticing th at 1 = − ( σ − 1)( h ). ⊓ ⊔ No w w e hav e H H 0 ( F )(1 a + p ( a ) i ) = H H 0 ( F ) n − 1 P j =1 ( − λ j + λ n + δ ij ) p ( a ) j ! = n − 1 P j =1 n − 1 P k =1  ( − λ j + λ n + δ ij ) α k j p ( b ) k  . On the other han d , w e ha v e tr  K 0 ( F )([ P ( a ) i ])  = tr  [ P ⊗ A ( a ) P ( a ) i ]  = tr  n − 1 P k =1 n k ,i ([ P ( b ) k ] − [ A ( b )]) + [ A ( b ) P ]  = n − 1 P k =1 n k ,i p ( b ) k + 1 b + n − 1 P k =1 m k p ( b ) k = n − 1 P k =1 ( n k ,i + m k + ( − µ k + µ n )) p ( b ) k . So E q u ation (4) gives rise for all 1 ≤ k ≤ n − 1 to n − 1 X j =1 ( − λ j + λ n + δ ij ) α k j = n k i + m k + ( − µ k + µ n ) . Thanks to § 2.6.2 we can rewrite the p receding equ ation only in terms of the n i ’s, and finally summarize the results of th is section in the follo wing 8 Theorem 2.6.2. Set a = ( h − λ 1 ) . . . ( h − λ n ) , b = ( h − µ 1 ) . . . ( h − µ n ) ∈ k [ h ] two p olynomials of de gr e e n such that λ i − λ j 6∈ Z , µ i − µ j 6∈ Z f or al l i 6 = j . Define the fol lowing c olumn ve ctors : Λ = ( λ n − λ 1 , . . . , λ n − λ n − 1 ) t , Ω = ( µ n − µ 1 , . . . , µ n − µ n − 1 ) t ∈ k n − 1 . A ssume the algebr as A ( a ) and A ( b ) ar e Morita e quivalent. Then ther e exist a matrix N = ( n ij ) ∈ GL n − 1 ( Z ) and a c olumn ve ctor of inte gers M = ( m 1 , . . . , m n − 1 ) t ∈ Z n − 1 such that: N . Λ = Ω + M (6) Pro of . It resu lts fr om the preceding computations that for all 1 ≤ i, k ≤ n − 1 w e ha v e the follo wing equation n − 1 X j =1 ( − λ j + λ n + δ ij ) n k j = n k i + m k + ( − µ k + µ n ) . The term n k i app ears on ce on b oth sides of this equaliy , so cancels, and i d o es not app ear anymo re in the equation. Then the s tatement of the theorem is just a rephrasing of these facts in terms of matrices. ⊓ ⊔ Remark 2.6.3. • Since GL 1 ( Z ) = { 1 , − 1 } , condition (6) can b e consid ered as an extension in degree n of the condition obtained b y Ho dges in [8] (see Theorem 2.1.3). • Condition (6) is actually sa ying that the Z -la ttice generated in k by th e λ n − λ i ’s has to b e the same as the one generated by the µ n − µ j ’s. There is a c anonical wa y to asso ciate a noncomm utativ e torus to a la ttice (see [14, 10]), and w e will discuss this in Section 4. 3 Discussion on the case of degree 3 In this section and the follo wing one we assume that k = C . Consider t w o p oly- nomials a ( h ) = ( h − λ 1 )( h − λ 2 )( h − λ 3 ) and b ( h ) = ( h − µ 1 )( h − µ 2 )( h − µ 3 ) b oth satisfying the criterion (2). 3.1 Notations • Set P = A ( b ) P A ( a ) as in the p revious sectio n. W e already kno w that [ P ] = [ A ( b )] + m 1 ([ P ( b ) 1 ] − [ A ( b )]) + m 2 ([ P ( b ) 2 ] − [ A ( b )]) and tr ( b ) ( P ) = 1 b + m 1 p ( b ) 1 + m 2 p ( b ) 2 for some m 1 , m 2 ∈ Z , and that there exists a matrix N =  n 1 n 2 n 3 n 4  ∈ GL 2 ( Z ) suc h that e K 0 ( F )([ P ( a ) 1 ] − [ A ( a )]) = n 1 ([ P ( b ) 1 ] − [ A ( b )]) + n 3 ([ P ( b ) 2 ] − [ A ( b )]) e K 0 ( F )([ P ( a ) 2 ] − [ A ( a )]) = n 2 ([ P ( b ) 1 ] − [ A ( b )]) + n 4 ([ P ( b ) 2 ] − [ A ( b )]) . Theorem 2.6.2 tr anslates in the follo wing wa y in the presen t setting. 9 Prop osition 3.1.1. Set a = ( h − λ 1 )( h − λ 2 )( h − λ 3 ) , b = ( h − µ 1 )( h − µ 2 )( h − µ 3 ) ∈ k [ h ] two p olynomia ls of de gr e e 3 such that λ i − λ j 6∈ Z , µ i − µ j 6∈ Z for al l i 6 = j . Assume the algebr as A ( a ) and A ( b ) ar e M orita e qu ivalent. Then ther e exist a matrix  n 1 n 2 n 3 n 4  ∈ M 2 ( Z ) and inte gers m 1 , m 2 ∈ Z such that n 1 n 4 − n 2 n 3 = ± 1 ( − λ 1 + λ 3 ) n 1 + ( − λ 2 + λ 3 ) n 2 = m 1 + ( − µ 1 + µ 3 ) ( − λ 1 + λ 3 ) n 3 + ( − λ 2 + λ 3 ) n 4 = m 2 + ( − µ 2 + µ 3 ) (7) ⊓ ⊔ W e shall note that in th e “generic” case, knowing λ i ’s, µ j ’s and m k ’s satisfying (6), the matrix N is uniqu ely determined. More precisely , giv en λ 1 , λ 2 , λ 3 , µ 1 , µ 2 , µ 3 , m 1 , m 2 and 2 matrices N and N ′ satisfying (6), assu me th at ( λ 3 − λ 1 ) / ( λ 3 − λ 2 ) 6∈ Q . (8) Since the vect or (( λ 3 − λ 1 ) / ( λ 3 − λ 2 ) , 1) sh ou ld b e in the ke rnel of the matrix N − N ′ , this matrix has to b e null, that is N = N ′ . 3.2 Reduction of the matrix H H 0 ( F ) . W e present in this sectio n the matrices H H 0 ( F ) asso ciated to some elementa ry op erations on the ro ots of the p olynomial a ( h ). 3.2.1 Exc hanging λ 1 and λ 2 . W e consider the p olynomial b ( h ) = ( h − λ 2 )( h − λ 1 )( h − λ 3 ), that is w e set µ 1 = λ 2 , µ 2 = λ 1 and µ 3 = λ 3 . Obvio usly A ( a ) = A ( b ) = A , and the Morita equiv alence ma y b e give n by P = A ( b ) A ( b ) A ( a ) . T hen T r ( b ) ( P ) = 1, and m 1 = m 2 = 0. Also w e ha v e K 0 ( F ) = Id, and P ( a ) 1 = Ax + A ( h − λ 1 ) = Ax + A ( h − µ 2 ) = P ( b ) 2 , so that n 1 = 0 , n 3 = 1. Then equations (7) lead to n 2 = 1 , n 4 = 0, and we fin ally get  n 1 n 2 n 3 n 4  =  0 1 1 0  = N 1 . 3.2.2 Exc hanging λ 2 and λ 3 . W e consider the p olynomial b ( h ) = ( h − λ 1 )( h − λ 3 )( h − λ 2 ), that is w e set µ 1 = λ 1 , µ 2 = λ 3 and µ 3 = λ 2 . Once aga in A ( a ) = A ( b ) = A , and the Morita equiv alence ma y b e giv en by P = A ( b ) A ( b ) A ( a ) , so th at T r ( b ) ( P ) = 1, and m 1 = m 2 = 0. W e ha v e K 0 ( F ) = Id , and P ( a ) 1 = Ax + A ( h − λ 1 ) = P ( b ) 1 , so n 1 = 1 , n 3 = 0. Then equati ons (7) lead to n 2 = − 1 , n 4 = − 1, and w e finally g et  n 1 n 2 n 3 n 4  =  1 − 1 0 − 1  = N 2 . 10 3.2.3 λ 1 7→ λ 1 + 1 By [ 9, Theorem 2.3 and Lemma 2.4], P = P ( b ) 1 pro vides a M orita equiv alence b et we en A ( a ) and A ( b ), w ith µ 1 = λ 1 + 1 , µ 2 = λ 2 , µ 3 = λ 3 . By definition of P , w e get m 1 = 1 , m 2 = 0. Then the identit y matrix I 2 satisfies the equations (7). 3.2.4 λ i 7→ − λ i + 1 It is kno wn after [3 ] that A ( a ) is isomorph ic to A ( b ) for b ( h ) = a (1 − h ). So once again using P = A ( b ) A ( b ) A ( a ) in this co nt ext w e get m 1 = m 2 = 0. The matrix − I 2 satisfies the equations (7). Moreo ver the isomorphism is giv en by x 7→ y , y 7→ x, h 7→ 1 − h . 3.3 A subgroup of S L 2 ( Z ) The necessary condition app earing in Prop osition 3.1.1 is still w eak er than the sufficien t condition of Pr op osition 2.1. In the follo wing w e sh o w that the necessary condition (7) cannot b e sufficien t in d egree 3, at least n ot without the extra assumption that the p olynomials a and b b oth satisfy (2). Giv en t w o p olynomials a and b , a p ermutatio n of th e first t w o r o ots of b leads to m ultiplication on the right of H H 0 ( F ) by the matrix N 1 . Th an k s to this, w e may assume that H H 0 ( F ) ∈ S L 2 ( Z ), that is n 1 n 4 − n 2 n 3 = 1. Notation. Let G b e the subgroup consisting of matrices N ∈ S L 2 ( Z ), such that for al l triples ( λ 1 , λ 2 , λ 3 ) and ( µ 1 , µ 2 , µ 3 ) satisfying (7), the algebras A ( a ) and A ( b ) are Morita equiv alen t, with a = ( h − λ 1 )( h − λ 2 )( h − λ 3 ) and b = ( h − µ 1 )( h − µ 2 )( h − µ 3 ). It is clear from paragraph 3.2 that − I 2 and N 1 N 2 b elong to G . These t wo el- emen ts generate a sub group G 6 isomorphic to Z / 2 Z × Z / 3 Z . The 6 eleme nts of this subgroup are the identit y m atrix I 2 , its opp osite − I 2 =  − 1 0 0 − 1  , N 1 N 2 =  0 − 1 1 − 1  , − N 1 N 2 =  0 1 − 1 1  , ( N 1 N 2 ) 2 =  − 1 1 − 1 0  = N 2 N 1 and − N 2 N 1 =  1 − 1 1 0  . Prop osition 3.3.1. The matric es N 1 N 2 and − I 2 gener ate G ; that is: G = G 6 . Pro of . Let N =  n 1 n 2 n 3 n 4  b e an elemen t of S L 2 ( Z ). W e will sho w that if N is not one of the 6 matrices ab o v e, then there exist triples ( λ 1 , λ 2 , λ 3 ) and ( µ 1 , µ 2 , µ 3 ) satisfying (7), suc h that the algebra A ( a ) is simp le with fin ite global dimension and the algebra A ( b ) is not, with a = ( h − λ 1 )( h − λ 2 )( h − λ 3 ) and b = ( h − µ 1 )( h − µ 2 )( h − µ 3 ). S o N 6∈ G . • Assume | n 1 n 2 | > 1. S ince N ∈ S L 2 ( Z ), this implies n 1 6 = n 2 . Consider now the triple λ 1 = 1 / (2 n 1 ), λ 2 = 1 / (2 n 2 ) and λ 3 = 0. It r esults from the hyp othesis that 11 0 < | λ i − λ j | < 1 for all i 6 = j . S o the algebra A ( a ) is simple and of finite global dimension. But for a triple µ 1 , µ 2 , µ 3 satisfying (7) we get µ 3 − µ 1 = − m 1 − 1 ∈ Z , so the algebra A ( b ) is n ot simple, or not of fi nite global dimension if m 1 = − 1. • The case | n 3 n 4 | > 1 is dealt with similarly . So a m atrix in the group G has all its entries in the set { 0 , 1 , − 1 } . • n 1 = 0. Then necessarily n 2 n 3 = − 1. Assume fi rst th at n 2 = − n 3 = 1. If n 4 = 0 then denote x =  0 1 − 1 0  the corresp onding matrix. Considering the triples ( λ 1 = 3 / 4 + i, λ 2 = 1 / 4 − i, λ 3 = 0) and ( µ 1 = − 1 / 4 + i, µ 2 = 3 / 4 + i, µ 3 = 0) leads as b efore to a simple and a non simple algebras. If n 4 = − 1 then consider ( λ 1 = 3 / 4 , λ 2 = 1 / 4 , λ 3 = 0) and ( µ 1 = 1 / 4 , µ 2 = − 1 , µ 3 = 0). Note that a similar example w ill do as so on as n 3 n 4 = 1, or b y symmetry of the problem as so on as n 1 n 2 = 1, and that n on e of th e matrices in G 6 satisfies su c h an hypothesis. At last, taking n 4 = 1 giv es N = − N 1 N 2 , w hic h b elongs to G 6 . If n 2 = − n 3 = − 1 then m ultiplying by − I 2 leads to similar conclusions. So x 6∈ G . • n 1 = 1. W e consider three su b cases, dep end ing on the v alue of n 2 . 1. n 2 = 0. Then necessarily n 4 = 1. So if n 3 = 0 then N = Id; if n 3 = 1 then we are in the case n 3 n 4 = 1 which can b e d ealt w ith as b efore; if n 3 = − 1 then one can easily c hec k N =  1 0 − 1 1  = x − 1 N 2 N 1 . So N 6∈ G 6 , otherwise w e wo uld hav e x ∈ G . 2. n 2 = 1. Th en n 1 n 2 = 1, and we already n oticed th at non e of the matrices satisfying suc h an h yp othesis is in G . 3. n 2 = − 1. Then n 4 + n 3 = 1, i.e. ( n 3 , n 4 ) ∈ { (1 , 0) , (0 , 1) } . The first case corresp onds to − N 2 N 1 , whic h b elongs to G 6 . On e c hec ks easily that the second case corresp onds to the matrix N = x ( N 2 N 1 ) − 1 , wh ic h cannot b elong to G , otherwise w e wo uld hav e x ∈ G . • n 1 = − 1. This case is strictly similar to the p receding one, up to multi plication b y the matrix − I 2 whic h b elongs to G 6 . ⊓ ⊔ 4 Links with quan tum tori As f or the p revious section, we assu me h ere that k = C . 4.1 Quan tum tori W e recall the follo wing Definition 4.1.1. L et n ≥ 1 b e an inte g e r and Q = ( q ij ) ∈ M n ( C ∗ ) b e a multi- plic atively antisymm etric matrix (i.e. q ij q j i = q ii = 1 ∀ i, j ). The quantum torus (or M acConnel l-Pettit algebr a [ 11 ] ) p ar ametrize d by Q is the C -algebr a gener ate d by X 1 , . . . , X n , with r elations X i X j = q ij X j X i , and their inverses X − 1 1 , . . . , X − 1 n . It is denote d T Q = C Q [ X ± 1 1 , . . . , X ± 1 n ] . 12 These algebras p la y a crucial role in quan tum algebra (see for example [5]), and ha v e b een extensively studied. Note that when n = 2 the matrix Q is un iquely determined b y the entry q = q 12 . In this case we ma y denote the asso ciated quan tum torus by T q or C q [ X ± 1 1 , X ± 2 2 ]. W e will fo cus in the sequel on th e follo wing prop erty . Prop osition 4.1.2 ([11 ], P r op osition 1.3) . L et T Q = C Q [ X ± 1 1 , . . . , X ± 1 n ] b e a quantum torus. The fol lowing c ond itions ar e e qui valent: 1. the c entr e of T Q is r e duc e d to C ; 2. T Q is a simple ring; 3. if ( m 1 , . . . , m n ) ∈ Z n satisfies n Y k =1 q m k k j = 1 , ∀ 1 ≤ j ≤ n (9) then m i = 0 for al l 1 ≤ i ≤ n . ⊓ ⊔ If n = 2 th en this condition is equiv ale nt to sa ying that q is n ot a ro ot of unity . Since we are d ealing with Morita equiv alence, we may mention also th e follo wing consequence of [11 , Theorem 1.4], [12, Th´ eor ` eme 4.2] and [13, Lemma 3.1.1]. Theorem 4.1.3. L et Q = ( q ij ) , Q ′ = ( q ′ ij ) ∈ M n ( C ∗ ) b e multiplic atively antisym- metric matric es. Assu me th at th e quantum tori T Q and T Q ′ p ar ametrize d by Q and Q ′ ar e simple. Then the fol lowing ar e e quivalent 1. T Q and T Q ′ ar e isomorphic; 2. ther e exists M = ( m ij ) ∈ GL n ( Z ) such that for al l i, j one has q ′ ij = Y t,k q m ki m tj k t ; 3. T Q and T Q ′ ar e bir ationna l ly e quivalent (i.e. have isomorph ic sk e w-fields of fr actions); 4. T Q and T Q ′ ar e Morita e q u ivalent. ⊓ ⊔ If n = 2 th en condition 2. is easily seen to b e equiv alent to q ′ = q or q − 1 . No w we will explain ho w this is related to GW As. Th e next subsection is dev oted to the case n = 2. 13 4.2 Rank 2 case Our motiv ation here is the survey pap er [10] by Y uri Manin. Ev en if the author is there interested in differentia l non commutat iv e geo metry and considers smo oth and rapidly decreasing f u nctions, w e will ke ep an algebraic p oint of view and only consider noncomm utativ e Laurent p olynomials. Consider a lattice of r ank t w o Z ⊕ θ Z ⊂ C , with θ ∈ C \ Q . T o this datum one asso ciates the quan tum torus T q , with q = q ( θ ) = e 2 iπ θ . F rom the pr eceding subsection w e see that T q ( θ ) is simple if and only if θ 6∈ Q , and that T q ( θ ) and T q ( θ ′ ) are isomorphic if and only if θ ′ = θ + m or θ ′ = − θ + m with m ∈ Z . This leads to the follo wing. Prop osition 4.2.1. Consider two GW As define d by p olynomials of de gr e e two a ( h ) = ( h − λ 1 )( h − λ 2 ) and b ( h ) = ( h − µ 1 )( h − µ 2 ) . Fix θ = λ 1 − λ 2 , θ ′ = µ 1 − µ 2 , q = e 2 iπ θ , q ′ = e 2 iπ θ ′ , and denot e by T q and T q ′ the asso ciate d quant um to ri. Then, if T q (r esp. T q ′ ) is si mple then A ( a ) (r esp. A ( b ) ) is simple and has finite glob al dimension. Assuming now that this c ondition holds for b oth q and q ′ in the fol lowing state- ments, then: • A ( a ) ≃ A ( b ) if and only if θ = ± θ ′ . • A ( a ) and A ( b ) ar e Morita e quivalent if and only if T q ≃ T q ′ . Pro of . The first assertion and th e fi rst item are straigh tforw ard from previous remarks. F or the last p oin t, jus t note that q ′ = q ± 1 if and only if θ ′ = ± θ + m , with m ∈ Z . ⊓ ⊔ Remark 4.2.2. Th e previous Prop ositio n pro vides an alternativ e approac h to Ho dges’ result concerning Morita equiv alence for GW A when n = 2. The follo wing sub section is devo ted to obtain some generalisations in any degree. 4.3 Rank n Notations. F or a p olynomial a ( h ) = Q n i =1 ( h − λ i ) we will d enote Θ( a ) = ( θ ij ) the matrix in M n ( Z ) d efined by θ ij = λ i − λ j . This m atrix is not uniquely d etermined, since it actually d ep ends on an indexing of the r o ots of a . In the sequel w e will alw a ys assu me that the p olynomial a is given w ith an in dexing of its ro ots (counted with th eir multipliciti es if n ecessary), and state our results up to a reindexin g of these ro ots (see for example next Prop osition). No w w e set q ij = e 2 iπ θ ij and Q ( a ) = ( q ij ) ∈ M n ( C ∗ ). The matrix Q ( a ) is multiplic ativ ely an tisymmetric, an d T Q ( a ) will d enote the quan tum torus asso ciated to these d ata. W e fi r st note the follo wing fact. Prop osition 4.3.1. With the notations ab ove, two g ener alize d Weyl algebr as A ( a ) and A ( b ) ar e isomorp hic if and only if Θ( b ) = ± S − 1 Θ( a ) S for a p ermutation matrix S . 14 Pro of . Denote by ( λ 1 , . . . , λ n ) and ( µ 1 , . . . µ n ) th e ro ots of a ( h ) and b ( h ), coun ted with their multi plicities. It r esults f rom T heorem 2.1.1 that A ( a ) and A ( b ) are isomorphic if and only if there exists a p erm utation σ , a scalar β and a sign ǫ su c h that µ i = ǫλ σi + β for all i . F rom this one deduces easily the “only if ” direction. F or the recipro cal, assum e that µ i − µ j = ǫ ( λ σi − λ σ j ). T hanks to Th eorem 2.1.1 one can assume that up to isomorphism ǫ = 1 and σ = Id, and that λ 1 = µ 1 = 0, and then a ( h ) = b ( h ). ⊓ ⊔ Corollary 4.3.2. If the algebr as A ( a ) and A ( b ) ar e isomorphic then the asso ciate d quantum tori T Q ( a ) and T Q ( b ) ar e isomorphic. Pro of . P erm uting the generators with resp ect to the m atrix S , o ne only has to p ro v e that the matrix Q ( a ) an d its transp ose defin e the same quantum torus. According to the notatio ns of Theorem 4.1.3 the isomorphism is defin ed thanks to the matrix M = ( m ij ) where m 11 = 0, m 1 j = 1 if j ≥ 2; m 22 = 0, m 2 j = 1 if j 6 = 2; m ij = − δ ij if i ≥ 2. One u s es the f act that λ ij λ j k = λ ik to verify that condition 2 of T heorem 4.1.3 is satisfied. W e lea v e the d etails to th e reader. ⊓ ⊔ Remark 4.3.3. This result strongly relies on the p articular form of the parametriza- tion matrices we hav e here, and the fact that λ ij λ j k = λ ik . F or instance, tak- ing λ, µ, ρ ∈ C ∗ algebraical ly indep endan t, the matrix 1 λ µ λ − 1 1 ρ µ − 1 ρ − 1 1 ! and its transp ose parametrize tw o quan tum tori wh ic h are not isomorphic, since the corre- sp ond ing matrix G = ( g ij ) ∈ GL 3 ( Z ) in Th eorem 4.1.3 sh ould satisfy g 2 11 g 2 22 g 2 33 = − 1. W e in tro du ce no w the follo wing condition on A ( a ). Definition 4.3.4. With the notations ab ove, a gener alize d Weyl algebr a A ( a ) wil l b e c al le d q -simp le if the asso ciate d quantum torus T Q ( a ) is simple. Prop osition 4 .3.5. Assume that the GW A A ( a ) is q - simple. Then it is simple and has finite glob al dimension. Pro of . By Prop osition 1.1 we only hav e to show that if λ i − λ j ∈ Z then the matrix Θ( a ) cannot satisfy condition (9 ). But this is clear b y us in g the v ector of Z n with 1 in the i th p lace, − 1 in th e j th p lace and 0 everywhere else. ⊓ ⊔ Remark 4.3.6. In the case n = 2, f or a p olynomial a = ( h − λ 1 )( h − λ 2 ), b eing q -simple is equiv alent to λ 1 − λ 2 6∈ Q . This shows that q -simplicit y is strictly stronger th an s implicit y and fin ite global d imension. No w we r estate condition 2. of Theorem 4.1.3 in terms of matrices Θ( a ) and Θ( b ) asso ciated to the ro ots of the p olynomials a and b . Prop osition 4.3.7. L et a ( h ) = Q n i =1 ( h − λ i ) and b ( h ) = Q n i =1 ( h − µ j ) b e tw o p olynomials su c h that the GW As A ( a ) and A ( b ) ar e q - simple. Then the quantum tori T Q ( a ) and T Q ( b ) ar e isomorphic if and only if ther e exi st two matric es M ∈ GL n ( Z ) and N ∈ M n ( Z ) such that M t Θ( a ) M = Θ( b ) + N . 15 ⊓ ⊔ It would b e interesting to relate this to Condition (6). W e end this d iscu ssion with some results in this direction concerning the case n = 3. 4.4 Case n = 3 The cond itions ab o v e conce rnin g the matrices can b e restated, using cofac tor matrices. More precisely , let c M ij b e the matrix obtained from M by d eleting line i and column j . Recall that if w e denote by cof ( M ) the matrix suc h that cof ( M ) ij = ( − 1) i + j det ( c M ij ) then M · cof ( M ) t = det ( M ) · I d , so det ( cof ( M )) = 1 (since n = 3 and det ( M ) = ± 1). W e also ha v e det ( cof ( M )) = det ( cof ( M )) t , and det ( cof ( M )) = det ( cof ′ ( M )), where ( cof ′ ( M )) ij = det ( c M ij ). W e rephr ase in this case th e conditions of the previous Prop osition in terms of cofactor matrices. Prop osition 4.4.1. Under the hyp otheses of the ab ove pr op osition, for n = 3 , the c ondition c onc erning matric es holds i f and only if cof ′ ( M ) t ·   λ 23 λ 13 λ 12   =   µ 23 µ 13 µ 12   +   γ 23 γ 13 γ 12   T aking in to account that λ 12 = λ 13 − λ 23 , and similarly for the µ ’s, w e are able to establish a r elation b et we en Morita equiv ale nces and isomorph isms of quantum tori for n = 3. Theorem 4.4.2. Fix n = 3 . If two g e ner alize d Weyl algebr as A ( a ) and A ( b ) of de gr e e n ar e Morita e quivalent, then their asso ciate d quantum tori ar e isomorphic. Pro of . Giv en t w o Morita equiv alent algebras A ( a ) and A ( b ), let N =  n 1 n 2 n 3 n 4  ∈ GL 2 ( Z ) b e a matrix as in Section 3. W e will construct a matrix b N ∈ GL 3 ( Z ) suc h that b N .   λ 23 λ 13 λ 12   =   µ 23 µ 13 µ 12   +   γ 23 γ 13 γ 12   . In fact, it is su fficien t to tak e b N =   n 4 n 3 0 n 2 n 1 0 c d 1   , where c = n 1 − n 3 + 1 and d = n 2 − n 4 − 1. It is then straightforw ard to find a matrix M ∈ GL 3 ( Z ) su c h that b N = cof ′ ( M ) t . ⊓ ⊔ 16 References [1] V.V. Ba vu la, Gener alize d Weyl algebr as and their r epr e sentations , S t. P eters- burg Math. J . 4 (1993), no. 1, 71–92. [2] V.V. Ba vula, Description of bilater al ide als in a class of nonc ommutative rings. I , Ukr ainian Math. J. 45 (1993 ), no. 2, 223–234. [3] V.V. Ba vula and D.A. 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