Dedicated to Claus Michael Ringel on the occasion of his 60th birthday

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1 GENERIC EXTENSIONS AND CANONICAL BASES FOR CYCLIC QUIVERS BANGMING DENG, JIE DU AND JIE XIAO Abstract. We use the monomal bass theory developed n [4] to present an elementary algebrac constructon of the canoncal bases for both the Rngel Hall algebra of a cyclc quver and the +-part U + of the quantum affne sl n. Ths constructon reles on analyss of quver representatons and the ntroducton of a new ntegral PBW-lke bass for the Lusztg Z[v, v 1 ]-form of U +. Dedcated to Claus Mchael Rngel on the occason of hs 60th brthday 1. Introducton As a landmark n Le theory, G. Lusztg ntroduced n [18] the canoncal bass of the quantum envelopng algebra of a smple complex Le algebra and showed that ths bass has some remarkable propertes such as the postvty property for structure constants (cf. the work [15] for Hecke algebras), the compatblty wth varous natural fltratons, and the fact that ths bass s well adapted to all fnte dmensonal rreducble representatons. In ths case Lusztg actually gave two constructons of the canoncal bases, namely, the elementary algebrac constructon, nvolvng analyss of quver representatons, and the geometrc constructon, based on perverse sheaves on representaton varetes of a quver. Nevertheless, the key steps n the proof of the exstence of the canoncal bases are the use of Rngel s Hall algebra assocated to the representaton category of a quver [25, 27]. The geometrc constructon was soon extended n [19, 20] to an arbtrary Kac-Moody algebra (see, e.g., [23]). Though there are other elementary constructons ncludng Kashwara s crystal bass approach [14] for arbtrary Kac-Moody algebras and, n the affne case, the constructons gven n [1] and [2], the algebrac constructon for the general case nvolvng analyss of quver representatons remans unclear. In ths paper, we wll present such a constructon for cyclc quvers. The man ngredent n ths constructon s the strong monomal bass property establshed n [4]. Ths property s a systematc constructon of many monomal bases for the subalgebra, the composton algebra, generated by smple modules of the generc (twsted) Rngel Hall algebra of a cyclc quver. It s proved n [26] that the composton algebra s somorphc to the postve part U + of the quantum envelopng algebra U of the affne Le algebra ŝl n. Ths realzaton together wth the strong monomal bass property allows us to ntroduce ntegral monomal/pbw-lke bases for the Lusztg Z-form U + Z (Z = Z[v, v 1 ]) of U + and to see the trangular relatons of the bar nvoluton on these bass elements. In ths way, a new bass s constructed through a standard lnear algebra method. We then prove that ths bass agrees wth Lusztg s canoncal bass; cf., e.g., [16, 30]. We further extend the approach to produce a Date: January 13, Mathematcs Subject Classfcaton. 17B37, 16G20. Supported partally by ARC (Grant: DP ) and the NSF of Chna. The research was carred out whle Deng and Xao were vstng the Unversty of New South Wales. The hosptalty and support of UNSW are gratefully acknowledged. 1

2 2 BANGMING DENG, JIE DU AND JIE XIAO smlar constructon for the canoncal bass of the whole Rngel Hall algebra. Note that the PBW bases constructed n ths paper do not nvolve brad group actons. It would be nterestng to fnd a relaton between our PBW bases and those constructed n [2]. Note also that the constructon for the cyclc quver case s a key step towards the completon of a smlar constructon sutable for all affne Kac-Moody algebras wth symmetrc generalzed Cartan matrces; see [17]. The paper s organzed as follows. We start wth nlpotent representatons of a cyclc quver and ther assocated Rngel Hall algebra H n 2. We nvestgate n 3 the generc extenson monod M of through a mnmal set I e of generators consstng of smple and sncere semsmple representatons. Thus we obtan a monod epmorphsm from the free monod over I e to M. Wth, we construct n 4 a dstngushed word n every fbre of, and dscuss the Strong Monomal Bass Property for Rngel Hall algebras n 5. From 6 on wards, we use the twsted Rngel Hall algebra H Z and ts composton algebra C Z as a realzaton of U + Z ( 6) to ntroduce a new ntegral PBW bass from whch we construct a so-called IC bass for U + Z ( 7). In 8, we show that ths elementarly constructed IC bass concdes wth the (geometrcally constructed) canoncal bass for U + Z, and n the last secton, we further extend the constructon to the whole Rngel Hall algebra H Z. Some notaton. For a fnte dmensonal quver representaton (or a fnte dmensonal module over an algebra) M, let soc 1 M = soc M (resp. rad 1 M = radm) denote the socle (resp. radcal) of M. Let soc 0 M = 0, rad 0 M = M and, for > 1, soc M be the nverse mage of soc(m/soc 1 M) n M under the natural projecton M M/soc 1 M and rad M = rad(rad 1 M). We also set top M = M/radM. Let Ll(M) denote the Loewy length of M, that s, Ll(M) = mn{s rad s M = 0} = mn{t soc t M = M}. Then M admts two natural fltratons: the radcal fltraton and the socle fltraton M radm rad l 1 M rad l M = 0 M = soc l M soc l 1 M soc 1 M 0, where l = Ll(M). We have obvously the followng lemma. Lemma 1.1. For each 0 s l, soc s M s the unque maxmal submodule of M of Loewy length s, whle M/rad s M s the unque maxmal quotent module of M of Loewy length l s. In other words, any fltraton M = M 0 M 1 M l 1 M l = 0 satsfyng the property that M s (resp. M/M s ) s a maxmal submodule (resp. quotent module) of Loewy length s (resp. l s) concdes wth the socle (resp. radcal) fltraton of M. 2. Nlpotent representatons and Rngel Hall algebras Let = (n) be the cyclc quver n n 2 n 1 wth vertex set I := Z/nZ = {1, 2,..., n} and arrow set { n}, and let k be the path algebra of over a feld k. For a representaton M = (V, f ) of, let dmm = n =1 dmv and dm M = (dm V 1,...,dmV n ) N n denote the dmenson and

3 GENERIC EXTENSIONS AND CANONICAL BASES 3 dmenson vector of M, respectvely, and let [M] denotes the soclass (=somorphsm class) of M. Further, for each a 1, we wrte am := M } {{ M }. a If a = 0, we let am = 0 by conventon. A representaton M = (V, f ) of over k (or a k -module) s called nlpotent f the composton f n f 2 f 1 : V 1 V 1 s nlpotent, or equvalently, one of the f 1 f n f 1 f : V V (2 n) s nlpotent. Let T k = T k (n) denote the category of fnte-dmensonal nlpotent representatons of over k, and let S = (S ) k, I (resp. S [l] k, I and l 1) be the rreducble (resp. ndecomposable) objects n T k. Here S [l] k s the (unque) ndecomposable object wth top (S ) k and length (.e., dmenson) l. Followng [16], a (cyclc) multsegment s a formal fnte sum π = π,l [; l), I, l 1 where π,l N. Let Π denote the set of all multsegments. Then each multsegment π =,l π,l[; l) Π defnes a representaton n T k M k (π) = π,l S [l] k. I, l 1 In ths way we obtan a bjecton between Π and the set of soclasses of representatons n T k. Note that ths bjecton s ndependent of the feld k. Thus, throughout, the subscrpts k are often dropped for notatonal smplcty. We shall also wrte End (M), Hom(M, N), etc. for End k (M), Hom k (M, N), etc. Remark 2.1. In [28, 4], nlpotent representatons of are parametrzed by n-tuples of parttons. In fact, f we dentfy a multsegment π =,l π,l[; l) Π as the n-tuple (π (1), π (2),..., π (n) ) of parttons, where, for each 1 n, π () s the partton dual to the partton (m π,m,...,2 π,2, 1 π,1 ), where m s the maxmal l for whch π,l 0 (. e., m = Ll(M(π))), then the two parametrzatons concde. A multsegment π =,l π,l[; l) n Π s called aperodc 1 (see [19, p.417]) f, for each l 1, there s some I such that π,l = 0. Otherwse, π s called perodc. By Π a we denote the set of aperodc multsegments. A representaton M n T s called aperodc (resp. perodc) f M = M(π) for some π Π a (resp. π Π\Π a ). For d N n, let Π d = {λ Π dm M(λ) = d} and Π a d = Πa Π d. Assocated to a cyclc quver, or more precsely, to T, Rngel ntroduced an assocatve algebra, the Rngel Hall algebra, whch can be defned at the two levels: the ntegral level and the generc level. Let k be a fnte feld of q k elements and, for L, M, N n T k, let FMN L be the number of submodules V of L such that V = N and L/V = M. More generally, gven modules M, N 1,..., N m n T k, we let FN M 1... N m be the number of the fltratons M = M 0 M 1 M m 1 M m = 0 1 The correspondng n-tuple of parttons s called separated n [26, 4.1].

4 4 BANGMING DENG, JIE DU AND JIE XIAO such that M t 1 /M t = Nt for all 1 t m. By [26] and [13], F M N 1... N m s a polynomal n q k. In other words, for π, µ 1,..., µ m n Π, there s a polynomal ϕ π µ 1... µ m (q) A := Z[q] such that for any fnte feld k of q k elements ϕ π µ 1... µ m (q k ) = F M k(π) M k (µ 1 )... M k (µ m). The (generc) Rngel-Hall algebra H = H A (n) of (n) s by defnton the free A-module wth bass {u π π Π} and multplcaton gven by u µ u ν = π Π ϕ π µν(q)u π. By specalzng q to the prme power q k, we obtan the ntegral Rngel Hall algebra assocated to T k. In practce, we sometmes wrte u π = u [M(π)] n order to make certan calculatons n terms of modules. Denote by C = C A (n) the subalgebra of H generated by u := u [S ], I. Ths s called the (generc) composton algebra of (n). It s easy to see that C s a proper subalgebra of H. Moreover, both H and C admt a natural N n -gradng by dmenson vectors: (2.1.1) H = d N nh d and C = d N nc d where H d s spanned by all u λ wth λ Π d and C d = C H d. 3. The generc extenson monod of a cyclc quver Let M (resp. M c ) be the set of all soclasses of representatons (resp. aperodc representatons) n T. Gven two objects M, N n T, there exsts a unque (up to somorphsm) extenson G of M by N wth mnmal dm End (G) ([3, 24, 4]). The extenson G s called the generc extenson 2 of M by N and s denoted by G =: M N. Thus, f we defne [M] [N] = [M N], then t s known from [4] that s assocatve and (M, ) s a monod wth dentty [0]. Every semsmple module n T has the form S a = n =1 a S for some a = (a ) N n. We shall see below that every module n T s a sequence of generc extensons by semsmple modules. For each multsegment π =,l π,l[; l) and each I, we defne π = π [ + 1;l 0 ) + [; l 0 + 1), where l 0 s maxmal such that π +1,l0 0. Then, by [4, Proposton 3.7], we have S M(π) = M( π). Further, for each I, we set π () = l 1 π,l[; l). Then π = π (1) + π (2) + + π (n). Fnally, for every a = (a ) N n, we defne a π = I }{{} a π (+1). 2 Geometrcally, when k s algebracally closed, each soclass [M] of dmenson vector d = dm = (d ) N n corresponds to a unque GL(d)-orbt O M n the representaton varety R(d) = n =1 Hom k(k d, k d +1 ) on whch GL(d) = n =1 GL d (k) acts by conjugaton. Thus, M N of dmenson vector d = d M + d N corresponds to the unque dense orbt O (of maxmal dmenson) n the extenson varety E(M, N) = {x R(d) x defnes an extenson of M by N}.

5 GENERIC EXTENSIONS AND CANONICAL BASES 5 Lemma 3.1. Let a N n and π Π. Then we have M(a π) = S a M(π). Dually, a smlar result holds for the generc extenson of a module by a semsmple one. Proof. For each 1 n, we set M (π) = M(π () ) = π,l S [l]. l 1 Then M(π) = n =1 M (π). Snce Ext 1 (S, M j (π)) = 0 for all j + 1, we have S a M(π) = I Applyng [4, Proposton 3.7] repeatedly gves (a S ) M +1 (π). (a S ) M +1 (π) = S S }{{} M(π (+1) ) = M(} {{ } π (+1) ). a a Hence, M(a π) = S a M(π). By Lemma 3.1, we obtan the followng (cf. [22]). Corollary 3.2. Let M T wth l = Ll(M). Then we have and M = (M/radM) (radm/rad 2 M) (rad l 2 M/rad l 1 M) (rad l 1 M) M = (M/soc l 1 M) (soc l 1 M/soc l 2 M) (soc 2 M/soc M) (soc M). In partcular, we have for each 0 < s l, (M/rad s M) (rad s M) = M = (M/soc s M) (soc s M). A sem-smple module S a = n =1 a S s called sncere f all a 1. Clearly, sncere sem-smple modules are n one-to-one correspondence wth sncere vectors a = (a ) N n. Let I e = I {all sncere vectors n N n }. We have already proved the followng result (cf. [22]). Proposton 3.3. The generc extenson monod M s generated by [S a ], a I e, and ths generatng set s mnmal. In [22], the structure of the monods M and M c n terms of generators and relatons s nvestgated. Let Σ (resp. Ω) denote the set of all words on the alphabet I e (resp. I). For each w = a 1 a 2 a m Σ, we set M(w) = S a1 S a2 S am. Then there s a unque π Π such that M(w) = M(π), and we set (w) = π. In ths way we obtan a surjectve map : Σ Π, w π = (w). Note that the map s ndependent of the feld k and that nduces a surjecton : Ω Π a (see [4, Theorem 4.1]).

6 6 BANGMING DENG, JIE DU AND JIE XIAO Besdes the monod structure, M has also a poset structure. For two representatons M, N T, we say that M degenerates to N (or N s a degeneraton of M), followng [3], and wrte M deg N, f dm Hom(X, M) dm Hom(X, N) for all X n T (see also [31]). Snce the order relaton s ndependent of the feld k, we may turn Π nto a poset wth the opposte partal order := ( deg ) op defned by settng 3 µ λ M(λ) deg M(µ). 4. Dstngushed words and dstngushed decompostons We recall from [26, 2.3] and [4, Secton 5] the defntons of a reduced fltraton and dstngushed words n Ω. We now generalze them to the words n Σ. For a I e, we set u a = u [Sa]. Let w = a 1 a 2 a m be a word n Σ and let ϕ λ w(q) be the Hall polynomal ϕ λ µ 1... µ m (q) wth M(µ r ) = S ar. Then w can be unquely expressed n the tght form w = b e 1 1 be 2 2 bet t, where e r = 1 f b r I e \I, and e r s the number of consecutve occurrences of b r f b r I. A fltraton M = M 0 M 1 M t 1 M t = 0 of a nlpotent representaton M s called a reduced fltraton of type w f M r 1 /M r = er S br for all 1 r t. By γw(q) λ we denote the Hall polynomal ϕ λ µ 1... µ t (q), where M(µ r ) = e r S br. Thus, for any fnte feld k of q k elements, γw(q λ k ) s the number of the reduced fltratons of M k (λ) of type w. A word w s called dstngushed f the Hall polynomal γw (w) (q) = 1. Note that w s dstngushed f and only f, for an algebracally closed feld k, M k ( (w)) has a unque reduced fltraton of type w. For each multsegment π =,l π,l[; l), we defne p(π) = max{l π,l 0 for all 1 n}. If no such an l exsts, we set p(π) = 0. Ths s exactly the case where π s aperodc. In partcular, a multsegment π s called strongly perodc f π,l = 0 for all I and l > p(π). Clearly, we have (4.0.1) p(a π) = p(π) + 1 whenever a N n s sncere. Let π Π wth p = p(π) and consder the submodule M = soc p M(π) of M(π). Then M := M(π)/M = π,l S [l p]. I Let π, π Π be such that M(π ) = M and M(π ) = M. Then, obvously, π s strongly perodc, π s aperodc, and both π and π are unquely determned by π. We call (π, π ) the assocated par of π. We have the followng. Lemma 4.1. Mantan the notaton ntroduced above. We have M(π) = M(π ) M(π ). Moreover, M s the unque submodule of M(π) somorphc to M(π ). Proof. The somorphsm follows from Corollary 3.2, whle the unqueness follows from Lemma 1.1 snce M = soc p M(π). We have the followng characterzaton of a strongly perodc multsegment. 3 Geometrcally, ths orderng concdes wth the Bruhat type orderng: µ λ f and only f OM(µ) O M(λ), the closure of O M(λ) ; see [4, 3]. l>p

7 GENERIC EXTENSIONS AND CANONICAL BASES 7 Lemma 4.2. Let π Π and M = M(π). Then π s strongly perodc wth p = p(π) f and only f p = Ll(M) and every subquotent S as = soc p s+1 M/soc p s M, 1 s p, n the socle fltraton of M s sncere. Moreover, puttng y π = a 1 a p, we have (y π ) = π, and any fltraton M = M 0 M 1 M p 1 M p = 0 satsfyng M s 1 /M s = Sas for all 1 s p s the socle fltraton of M. Proof. The suffcent part follows from (4.0.1). To see the necessary part, we apply nducton on p; the case for p = 1 s trval. Assume now p > 1 and let π =,l π,l[; l). Then a 1 = (π 1,p,..., π n,p ) s sncere. Puttng π 1 = π,l π,p [; p) +,l π,p [ + 1;p 1), we have π = a 1 π 1, and π 1 s strongly perodc wth p(π 1 ) = p 1. Hence S a1 M(π 1 ) = M(π) by Lemma 3.1. Clearly, M(π 1 ) s somorphc to a maxmal submodule M 1 of M(π) wth Loewy length p 1. Hence, M 1 = soc p 1 M(π) by Lemma 1.1, and the asserton follows from nducton. For an aperodc π Π a, we have the followng whch was not explctly stated n [4]. Proposton 4.3. For any π Π a, there exsts a dstngushed word w π = j e 1 1 jet t Ω 1 (π) where j r 1 j r, e r 1 for all r, that s, M(π) has a unque fltraton M(π) = M 0 M 1 M t 1 M t = 0 satsfyng M r 1 /M r = er S jr for all 1 r t. Proof. Let π =,l π,l[; l) be an aperodc multsegment. For each I, we set M = l 1 S [l]. Then there s a j I (not necessarly unque) such that Ll(M j ) > Ll(M j+1 ). We wrte M j = S j [l 1 ] S j [l 2 ] S j [l r ] wth l 1 l 2 l r 1. Choose e 1 such that l e > l e+1 and l e > Ll(M j+1 ), and defne µ = π ( [j; l 1 ) + + [j; l e ) ) + ( [j + 1;l 1 1) + + [j + 1;l e 1) ). By [4, Lemma 5.4], there s a unque submodule X of M(π) such that X = M(µ) and M(π)/X = es j. We may assume that µ s aperodc. For example, takng the maxmal ndex e wth the property l e > l e+1 and l e > Ll(M j+1 ) ensures that µ s aperodc. By nducton, there s a dstngushed word w 1 Ω 1 (µ). Then w := j e w 1 s a dstngushed word n Ω 1 (π), as desred. Note that by [4, Theorem 5.5], every dstngushed word n 1 (π) can be obtaned n the above way. Example 4.4. Let n = 3 and π = [1; 4) + [1; 3) + [2; 2) + [2; 1) + 2 [3; 1). Then π s aperodc. From the proof of Proposton 4.3, we can take j 1 = 1 or 2. Moreover, f j 1 = 1, then e 1 = 1 or 2, and f j 1 = 2, then e 1 = 1. If we fx j 1 = 1 and e 1 =2, then j 2 = 2 and e 2 = 1 or 3. Contnung ths process, we fnally get all the 7 dstngushed words n 1 (π): , , , , , ,

8 8 BANGMING DENG, JIE DU AND JIE XIAO In general, for any π Π wth p = p(π), let (π, π ) be the assocated par, where π s strongly perodc wth p(π ) = p and π s aperodc. By Lemma 4.2 and Proposton 4.3, there are dstngushed words (4.4.1) w π = j e 1 1 jet t Ω 1 (π ) and y π = a 1 a 2 a p Σ 1 (π ) assocated to π and π. Thus, we obtan a word (4.4.2) and a decomposton w π = w π y π = j e 1 1 jet t a 1 a p 1 (π), M(π) = e 1 S j1 e t S jt S a1 S ap. We shall call such a decomposton a dstngushed decomposton because of the followng. Proposton 4.5. For any π Π, the word w π defned n (4.4.2) s dstngushed. Proof. The exstence of a reduced fltraton of type w π obtaned by refnng M(π) M = soc p M(π) 0, follows from Lemmas 4.1 and 4.2, and Proposton 4.3. Suppose now that M(π) has another fltraton M(π) = N 0 N 1 N t 1 N t N t+p 1 N t+p = 0 satsfyng N s 1 /N s = es S js for 1 s t and N t+ 1 /N t+ = Sa for 1 p. Then we have Ll(N t ) p. Snce M s the maxmal submodule of M(π) of Loewy length p, we nfer N t M, and consequently, N t = M as dm N t = dm M. Now the unqueness of the fltratons gven n Lemma 4.2 and Proposton 4.3 forces that the fltraton above must be unque. Hence, w π s dstngushed. 5. The Strong Monomal Bass Property For m 1, let [[m]]! = [[1]][[2]] [[m]], where [[e]] = qe 1 q 1. For any w = a 1 a 2 a m Σ, let u w = u a1 u a2 u am. The proof of the followng s entrely smlar to that of [4, Proposton 9.1]. Lemma 5.1. For each w Σ wth the tght form b 1 e 1 b 2 e2 b t e t, we have (5.1.1) t u w = [[e r ]]! γw(q)u λ λ. r=1 λ (w) In other words, we have the relaton ϕ λ w(q) = t r=1 [[e r]]! γ λ w(q). Moreover, the coeffcents appearng n the sum are all non-zero. For any π Π, choose an arbtrary w π 1 (π). We shall call the set {w π π Π} a secton of Σ over Π. Smlarly, we may defne a secton of Ω over Π a. A secton s called dstngushed f all ts members are dstngushed words. By the nvertblty of the matrx arsng from (5.1.1) over the component H d, Lemma 5.1 mples mmedately the followng strong monomal bass property for the Rngel Hall algebra assocated to a cyclc quver; see [4, 8.1] and [5, 1.1] for the quantum group case.

9 GENERIC EXTENSIONS AND CANONICAL BASES 9 Theorem 5.2. Let H Q(q) = H A Q(q) and, for w Σ, let u (w) = 1 t r=1 [e r ]!u w. (1) If {w π π Π} s a secton of Σ over Π. Then the set {u wπ π Π} forms a bass for H Q(q). In partcular, the Rngel Hall algebra H Q(q) s generated by u a, a I e. (2) If the secton {w π π Π} s dstngushed, then {u (wπ) π Π} forms an ntegral bass for H. 6. Twsted Rngel Hall algebras and quantum affne sl n Let Z = Z[v, v 1 ] be the Laurent polynomal rng over Z n ndetermnate v. For each m 1, let [m] = vm v m v v 1 and [m]! = [1][2] [m]. The twsted Rngel-Hall algebra H Z = H Z (n) of (n) s by defnton the free Z-module wth bass {u π = u [M(π)] π Π} and multplcaton defned by u µ u ν = v ε(µ,ν) (u µ u ν ) = v ε(µ,ν) π Π ϕ π µν(v 2 )u π. Here ε(µ, ν) = ε(dm M(µ),dm M(ν)) s the Euler form ε(, ) : Z n Z n Z assocated wth the cyclc quver and defned by n n ε(a,b) = a b a b +1, =1 for a = (a 1,..., a n ), b = (b 1,..., b n ) (notng n + 1 = 1 n I). It s well-known that for two representatons M, N T, there holds ε(dm M,dm N) = dm k Hom(M, N) dm k Ext 1 (M, N). The Z-subalgebra C Z of H Z generated by u (m) := um, I and m 1, s called the [m]! twsted composton algebra. Then C Z s also generated by u [ms ], I, m 1, snce u (m) = v m(m 1) u [ms ]. Clearly, both H Z and C Z nhert the gradng gven n (2.1.1). Let =1 H = H Z Z Q(v) and C = C Z Z Q(v). Let U = U v (ŝl n) be the quantum ŝl n (n 2) over Q(v), and let E, F, K ±1 ( I) be ts generators; see, e.g., [21]. Then U admts a trangular decomposton U = U U 0 U + where U + (resp. U, U 0 ) s the subalgebra generated by the E s (resp. F s, K ±1 s). We denote by U + Z the Lusztg form of U+, that s, U + Z s generated by all the dvded powers E (m) := Em. We have the followng mportant results. [m]! Theorem 6.1. (1) ([26]) There s a Z-algebra somorphsm C Z U + Z, u(m) E (m), I, m 1, and hence a Q(v)-algebra somorphsm U + = C. (2) ([29]) The algebra H s somorphc to U + Q(v) Q(v)[x 1, x 2,...] where Q(v)[x 1, x 2,...] s an nfnte polynomal algebra over Q(v) wth x r central of degree (r,..., r).

10 10 BANGMING DENG, JIE DU AND JIE XIAO In the sequel, we wll dentfy the two algebras U + Z and C Z. In partcular, we shall dentfy u (m) wth E (m), etc. Note that the Rngel Hall algebra notaton u λ wll be used to facltate calculatons nvolvng modules. The elementary constructon of the canoncal bases for U Z and H Z uses (ntegral) monomals whch we now defne. For each w = 1 m = j e 1 1 jet t Ω wth j r 1 j r for all r, let 4 (6.1.1) m w = u 1 u m = E 1 E m and m (w) = u (e 1) j 1 u (et) j t = E (e 1) E (et) j t. Then we have by Lemma 5.1 (6.1.2) m w = v δ 1(w) u w = v δ 1(w) λ (w) where δ 1 (w) = 1 r<s m ε(dm S r,dm S s ). If we put (6.1.3) δ 2 (w) = t r=1 e r (e r 1) 2 and then t r=1 [e r]! = v δ 2(w) t r=1 [[e r]]!, and t m (w) = ( [e r ]! ) 1 m w = ( (6.1.4) r=1 = v δ(w) λ (w) ϕ λ w(v 2 )u λ, j 1 δ(w) = δ 1 (w) + δ 2 (w), t [[e r ]]! ) 1 v δ 1(w)+δ 2 (w) u w r=1 γ λ w(v 2 )u λ. We now defne (ntegral) monomals n H Z. For each a = (a ) N n, we set a = a2 and a = a, and defne In partcular, for I and e 1, we have ũ a = v dm End(Sa) dm Sa u a = v a a u a H Z. ũ e = v e2 e u e = u (e), where u e = u [es ]. Note that f a = (a ) N n s nsncere, say a = 0, then ũ a = ũ a 1 ( 1) ũ a1 1 ũ ann ũ a+1 (+1) s a monomal n U + Z defned above. e In general, for a gven word w = a 1 a 2 a m Σ wth the tght form b 1 e2 e 1 b 2 b t t, we defne monomals n H Z (cf. (6.1.1)) and Agan by Lemma 5.1, we obtan (6.1.5) m w = u a1 u am and m (w) = ũ e1 b 1 ũ etb t. m w = v 1 r<s m ε(dm Sar,dm Sas) m (w) = v δ (w) 4 The element mw s denoted as E w n [5]. λ (w) λ (w) γ λ w(v 2 )u λ, ϕ λ w(v 2 )u λ

11 where δ (w) = GENERIC EXTENSIONS AND CANONICAL BASES 11 m ( e 2 r b r e r b r e r(e r 1) ) + 2 r=1 1 r<s m ε(dm S ar,dm S as ). Note that, f w Ω, then all b r = b r = 1, and so δ (w) = δ(w). Snce δ extends δ, we wll use the same letter δ for δ n the sequel. Here are the twsted verson and the (non-ntegral) quantum group verson of the Strong Monomal Bass Property (Theorem 5.2). The ntegral quantum group verson of ths property was not gven n [4] and wll be dscussed n next secton as a key step to the elementary constructon. Theorem 6.2. (1) For each π Π, choose a word w π 1 (π). Then the set {m wπ π Π} s a Q(v)-bass of H. Moreover, f all w π are chosen to be dstngushed, then the set {m (wπ) π Π} s a Z-bass of H Z. (2) ([4, 8.1]) Let {w π π Π a } be a secton of Ω over Π a. Then the set {m wπ π Π a } s a Q(v)-bass of U +. By [30, Proposton 7.5]), there s a Z-lnear rng nvoluton ι : H Z H Z satsfyng ι(v) = v 1 and ι(m (wπ) ) = m (wπ). Remark 6.3. The constructon of the rng nvoluton ι s not algebrac and elementary, though ts restrcton to U + Z can be seen easly through the Drnfeld-Jmbo presentaton. However, f we note that the rng homomorphsm condton s not requred n the (lnear algebra) constructon of IC bases, then we may use the bass {m (w) w D} for H Z descrbed n Theorem 6.2(1) to defne a sem-lnear nvoluton ι(d) on H Z, and then to construct an IC bass wth respect to ι(d) (see, e.g., [9]). By Theorem 9.2, we shall see that the resultng IC bases constructed from the sem-lnear maps ι(d) are the same. Ths n turn shows that the defnton of ι = ι(d) s ndependent of the selecton of dstngushed sectons (cf. Corollary 8.3). Hence, ths defnton for ι s also somehow natural. 7. Integral PBW and canoncal bases for quantum affne sl n In ths secton, we gve two applcatons of Theorem 6.2(2). Frst, t can be used to prove that the Z-form U + Z s Z-free wth many monomal bases determned by dstngushed words. Second, from every such a monomal bass, we may construct an ntegral PBW bass for U + Z from whch the canoncal bass can be constructed by a standard lnear algebra argument. Lemma 7.1. Let P be the subspace of H spanned by all u λ wth λ Π\Π a. Then as a vector space H = U + P Proof. If suffces to prove that, for each d N n, H d = U + d P d, where P d s the Q(v)-subspace of H d spanned by all u λ wth λ Π d \Π a d. Frst, we show U + d P d = 0. Take an x U + d P d and suppose x 0. Snce x U + d, we use the bass {m wπ π Π a d } for U+ d constructed n Theorem 6.2 to wrte x = a π m wπ π Π a d

12 12 BANGMING DENG, JIE DU AND JIE XIAO for some a π Q(v). Let now µ Π a d be maxmal such that a µ 0. Usng (6.1.2), we can rewrte x = λ Π d b λ u λ. By the maxmalty of µ, we have b µ = a µ v δ 1(w µ) ϕ µ w µ (v 2 ) 0. Ths contradcts the fact that x P d. Hence, U + d P d = 0. Now a dmenson comparson forces H d = U + d P d. For each π Π a, we now fx a dstngushed word w π Ω 1 (π) (see 4.3). Snce γ π w π (v 2 ) = 1, we may rewrte (6.1.4) as (7.1.1) m (wπ) = v δ(wπ) u π + v δ(wπ) λ<π γ λ w π (v 2 )u λ. Defnton 7.2. For each gven dstngushed secton D = {w π π Π a }, we defne nductvely the elements E π = E π (D), π Π a, as follows. For any d N n and π Π a d, f π s mnmal, put E π = m (wπ) U + d := U d U + Z. Assume n general that E λ U + d have been defned for all λ Πa d wth λ < π. Then we defne (7.2.1) E π = m (wπ) v δ(wπ) δ(wλ) γw λ π (v 2 )E λ U + d. λ Π a d, λ<π In other words, we have (7.2.2) m (wπ) = E π + λ Π a d, λ<π v δ(wπ) δ(wλ) γ λ w π (v 2 )E λ. Lemma 7.3. Let {w π π Π a } be a gven dstngushed secton. For each d N n and each π Π a d, we have E π = v δ(wπ) u π + ξλ π u λ λ Π d \Π a d, λ<π for some ξλ π Z. Proof. By (7.1.1), we have m (wπ) v δ(wπ) γw λ π (v 2 )u λ = λ Π a d, λ π λ Π d \Π a d, λ<π v δ(wπ) γ λ w π (v 2 )u λ P. On the other hand, replacng m (wπ) n the left hand sde by (7.2.1) yelds m (wπ) v δ(wπ) γw λ π (v 2 )u λ = v δ(wπ) δ(wλ) γw λ π (v 2 )(E λ v δ(wλ) u λ ) P. λ Π a d, λ π λ Π a d, λ π Now an nductve argument concludes E π v δ(wπ) u π P. Hence, E π = v δ(wπ) u π + ξλ π u λ for some ξλ π Z, λ Π d \Π a d, λ<π as requred. Example 7.4. Let n = 3 and d = (1, 2, 3). Then Π d conssts of 18 elements,.e., there are 18 soclasses of nlpotent representatons of (3) of dmenson vector d. Let π = [1; 3) + [2; 1) + 2 [3; 1) Π d,.e., M(π) = S 1 [3] S 2 2S 3. Then Π π d := {π Π d π π} = {π, λ, µ, ν, τ} such that M(λ) = S 1 [2] S 2 [2] 2S 3, M(µ) = S 1 [2] S 2 3S 3, M(ν) = S 1 S 2 [2] S 2 2S 3, M(τ) = S 1 2S 2 3S 3.

13 GENERIC EXTENSIONS AND CANONICAL BASES 13 Clearly, π, λ, µ are aperodc, ν, τ are perodc, and the Hasse dagram of (Π π d, ) has the form π λ µ ν τ Take dstngushed words w 1 = (π), w 2 = (λ), and w 3 = (µ). Then we get m (w 1) = v 2 u π + v 2 u λ + (v 4 + v 2 )u µ + v 2 u ν + (v 4 + v 2 )u τ m (w 2) = v 3 u λ + v 3 u µ + v 3 u ν + (v 5 + v 3 )u τ m (w 3) = v 6 u µ + v 6 u τ. Thus, wth respect to the chosen dstngushed words w 1, w 2, w 3, we obtan E π = v 2 u π v 4 u τ E λ = v 3 u λ + v 3 u ν + v 5 u τ E µ = v 6 u µ + v 6 u τ and m (w 1) = E π + v 1 E λ + (v 2 + v 4 )E µ m (w 2) = E λ + v 3 E µ m (w 3) = E µ Theorem 7.5. For each gven dstngushed secton D = {w π π Π a } of Ω over Π a, each of the followng sets forms a Z-bass for U + Z. (1) {m (wπ) π Π a }; (2) {E λ λ Π a }, where E λ = E λ (D). In partcular, U + Z s a free Z-module. Proof. By Theorem 6.2(2), m (wπ), π Π a, are Z-lnearly ndependent. It suffces to prove that, for any dmenson vector d N n, the Z-module U + d = U+ Z U+ d s spanned by {m (wπ) π Π a d }, or equvalently, spanned by {E π π Π a d } by (7.2.2). Let x U + d and wrte x a π u π (mod P), π Π a d where a π Z. Then we get by Lemma 7.3 that x v δ(wπ) a π E π = v δ(wπ) a π (v δ(wπ) u π E π ) U + d P. π Π a d π Π a d Snce U + d P = 0 by Lemma 7.1, we have x π Π a v δ(wπ) a π E π = 0, as requred. d Wth the bass {E π } π Π a, we may follow the standard lnear algebra method to defne (unquely) an IC bass {C π } π as follows (see, e.g., [10]). The nvoluton ι : H Z H Z defned at the end of the last secton restrcts to an nvoluton ι : U + Z U+ Z takng E(m) E (m) and v v 1. For each polynomal f Z, we wll denote ι(f) by f. For each fxed dmenson vector d N n, by restrctng to Π a d, (7.2.2) gves a transton matrx (f λ,π ) λ,π Π a d. Ths matrx has an nverse (g λ,π ) λ,π Π a d satsfyng g λ,λ = 1 and g λ,π = 0

14 14 BANGMING DENG, JIE DU AND JIE XIAO unless λ π. Thus we have (7.5.1) E π = m (wπ) + λ<π g λ,π m (w λ). Applyng ι, we get ι(e π ) = m (wπ) + λ<π ḡ λ,π m (w λ) = E π + λ<π r λ,π E λ. By [18, 7.10] (see [10] for more detals), the system p λ,π = r λ,µ p µ,π for λ π, λ, π Π a d λ µ π has a unque soluton satsfyng p λ,λ = 1, p λ,π v 1 Z[v 1 ] for λ < π. Moreover, the elements C π = p λ,π E λ, π Π a d, λ π, λ Π a d form a Z-bass of U + d. We shall prove n next secton that {C π π Π a } s n fact the canoncal bass of U + constructed n [20]. 8. A comparson of canoncal bases for quantum affne sl n We frst recall the geometrc constructon of Lusztg s canoncal bass for the (generc twsted) Rngel Hall algebra H Z. For each π Π, we denote by O π the orbt correspondng to the module M(π) (see footnote 2). Let χ π be the characterstc functon of O π and put O π = v dm Oπ χ π. Thus, the Rngel Hall algebra HZ L defned geometrcally by Lusztg (see [16, 3.2]) has the (twsted) multplcaton O λ O µ = v α(λ,µ,π) ϕ π λµ (v 2 ) O π, π where α(λ, µ, π) = dm O λ + dm O µ dm O π + m(λ, µ) wth m(λ, µ) = n =1 λ µ + n =1 λ µ +1. If we defne for each π Π (8.0.2) then we have the followng. ũ π = v dm End(M(π)) dm M(π) u π, Lemma 8.1. For λ, µ, π Π, let ψλµ π (v) Z satsfy ũ λ ũ µ = π ψ π λµ (v)ũ π. Then ψλµ π (v) = v α(λ,µ,π) ϕ π λµ (v2 ). Thus, we have a rng somorphsm L : H Z HZ L sendng v to v 1 and ũ λ to O λ. Proof. By [16, 3.3(7)], we have α(λ, µ, π) = ( dm EndM(λ) + dm EndM(µ) dm EndM(π) ) + ε(λ, µ). Now the equalty follows from the defnton.

15 GENERIC EXTENSIONS AND CANONICAL BASES 15 We further recall the geometrc constructon of the canoncal bass for HZ L at v = 0. Let HO λ (IC Oπ ) be the stalk at a pont of O π of the -th ntersecton cohomology sheaf of the closure O λ of O λ, and let b L π = v +dm Oπ dm O λ dm HO λ (IC Oπ ) O λ., λ π Then the set {b L π π Π} s the canoncal bass (at v = 0) of HZ L ntroduced n [16, 30]. Denote by b π the correspondng bass for H Z, that s, b π s sent to b L π under the map L. Then the set {b π π Π} s the canoncal bass (at v = ) for H Z and the elements b π wth π Π are characterzed as the unque elements of L such that (8.1.1) ι(b π ) = b π, b π λ π Z[v 1 ]ũ λ and b π ũ π (mod v 1 L), where ι s an nvoluton on H Z satsfyng ι(v) = v 1 and ι(ũ a ) = ũ a for all a N n, and L s the Z[v 1 ]-submodule of H Z spanned by ũ π, π Π. In other words, for any λ π n Π, the Laurent polynomals P λ,π := v dm Oπ+dm O λ dmh O λ (IC Oπ ) satsfy P π,π = 1, P λ,π v 1 Z[v 1 ] for λ < π, and b π = λ π P λ,πũ λ. Note that t s shown n [20] that the subset {b π π Π a } over Π a s a bass for U Z and s called the canoncal bass of U + Z. We now use the unqueness to prove that the bass {C π } π Π a concdes wth the bass {b π π Π a }. We need a lemma. Lemma 8.2. Let π Π be aperodc. Then, for each dstngushed word w Ω 1 (π), we have δ(w) = δ 1 (w) + δ 2 (w) = dm End (M(π)) dmm(π). Proof. Let w = 1 2 m 1 (π) be dstngushed wth the tght form j e 1 1 je 2 2 jet t. We wrte j = j 1 and e = e 1, and let w = j e 1 j e 2 2 jet t. From the defnton of a dstngushed word, we have that w s agan dstngushed. We further set µ = (w ). Thus, S j M(µ) = M(π). We use nducton on the length m of w. If m = 1, t s clear. Let now m > 1. By nducton hypothess, we have for w δ(w ) = δ 1 (w ) + δ 2 (w ) = dm End(M(µ)) dmm(µ). On the other hand, we have clearly (see (6.1.2)) m δ 1 (w) = ε(dm S 1,dm S s ) + δ 1 (w ) = δ 1 (w ) + ε(dm S 1,dm M(µ)) and (see (6.1.3)) Thus, we obtan (8.2.1) s=2 δ 2 (w) = e(e 1) 2 + δ 2 (w ) (e 1)(e 2) 2 = δ 2 (w ) + e 1. δ(w) = δ(w ) + ε(dm S 1,dm M(µ)) + e 1. Let µ =,l µ,l[; l) and take l 0 maxmal such that µ j+1,l0 0. Then j µ = π mples ν := π [j; l 0 + 1) = µ [j + 1;l 0 ).

16 16 BANGMING DENG, JIE DU AND JIE XIAO In other words, M(π) = M(ν) S j [l 0 + 1] and M(µ) = M(ν) S j+1 [l 0 ]. Thus, we have dm Hom(M(ν), S j [l 0 + 1]) dm Hom(M(ν), S j+1 [l 0 ]) = ν j,l = (π j,l0 +1 1) + π j,l. l>l 0 l>l 0 +1 Snce w s dstngushed, M(π) has a unque submodule somorphc to M( (j e 2 2 jet t )). Equvalently, M j (π) = M(π (j) ) has a unque submodule N wth M j (π)/n = es j. By [4, Lemma 5.4], the unqueness mples e = l>l 0 π j,l. Hence, dm Hom(M(ν), S j [l 0 + 1]) dm Hom(M(ν), S j+1 [l 0 ]) = e 1. From the maxmalty of l 0, we compute dm Hom(S j [l 0 + 1], M(ν)) dm Hom(S j+1 [l 0 ], M(ν)) = s t, where s denotes the multplcty of S j n soc M(ν) and t = l 1 ν j+1,l. Ths s because each map from S j [l 0 + 1] nto socm(ν) s zero when restrcted to S j+1 [l 0 ]; whle each surjectve map from S j+1 [l 0 ] onto each summand S j+1 [l] of M(ν) can not be lfted to S j [l 0 + 1]. Further, we have { dm End (Sj+1 [l dm End (S j [l 0 + 1]) = 0 ]) + 1 f n l 0 dm End (S j+1 [l 0 ]) otherwse snce soc S j [l 0 + 1] = S j f and only f n l 0. Altogether, we obtan { dm End(M(µ)) + s + e t f n l0 dm End (M(π)) = dm End(M(µ)) + s + e t 1 otherwse. We also have ε(dm S j,dm M(µ)) = dm Hom(S j, M(µ)) dm Ext 1 (S j, M(µ)) { s t f n l0 = s t 1 otherwse. Fnally, puttng everythng nto (8.2.1), we obtan that as requred. δ(w) = δ 1 (w) + δ 2 (w) = dm End (M(π)) dmm(π), For each π Π a, we pck a dstngushed word w π Ω 1 (π) to form a dstngushed secton D = {w π π Π a }, and let {E π π Π a } be the bass of U + Z defned wth respect to D n 7.2. Then by Lemmas 7.3 and 8.2 we have for each π Π a d (8.2.2) E π = ũ π + m (wπ) = E π + λ Π d \Π a d, λ<π η π λũλ λ Π a d, λ<π f λ,π E λ (η π λ Z), (f λ,π = v δ(wπ) δ(wλ) γ λ w π (v 2 ) Z). Corollary 8.3. The bass {E π π Π a } s ndependent of the selecton of dstngushed sectons. Proof. Suppose D and D are two dstngushed sectons. Then E π (D) E π (D ) s a lnear combnaton of u λ, λ Π\Π a,.e., E π (D) E π (D ) U + P. Hence t s zero by Lemma 7.1. Lemma 8.4. For π Π a d and λ Π\Πa wth λ < π, we have η π λ v 1 Z[v 1 ], that s, E π L and E π ũ π (mod v 1 L).

17 GENERIC EXTENSIONS AND CANONICAL BASES 17 Proof. Frst, let π Π a d be mnmal, then E π b π = (ηλ π P λ,π)ũ λ U + d P d. λ Π d \Π a d, λ<π By Lemma 7.1, E π b π must be zero, that s, E π = b π and ηλ π = P λ,π v 1 Z[v 1 ] for all λ < π wth λ Π a d. Let now π Πa d and assume that the result s true for all µ Πa d wth µ < π, that s, for such a µ, we have η ν µ v 1 Z[v 1 ] for all ν Π d \Π a d wth ν < µ. Consder the element b π P λ,π E λ = (ũ π E π ) + P µ,π (ũ µ E µ ) + P σ,π ũ σ λ Π a d, λ π µ Π a d, µ<π σ Π d \Π a d, σ<π = η π µ λũλ νũ ν + P σ,π ũ σ λ Π d \Π a d, λ<π σ Π d \Π a d, σ<π P µ,πη µ Π a d, µ<π ν Πa \Π a d, ν<µ whch s clearly n U + d P d. Agan, by Lemma 7.1, we must have b π λ Π a d, λ<π P λ,πe λ = 0, that s, η λũλ π µ = νũ ν + P σ,π ũ σ. λ Π d \Π a d, λ<π σ Π d \Π a d, σ<π P µ,πη µ Π a d, µ<π ν Π d \Π a d, ν<µ Ths mples by nducton η π λ v 1 Z[v 1 ] for all λ < π, λ Π a d. Wth what we have done above, the followng comparson now follows easly. Theorem 8.5. For each π Π a, we have C π = b π. Proof. From the constructon, we have ι(c π ) = C π and C π = E π + p λ,π E λ, λ<π,λ Π a where p λ,π v 1 Z[v 1 ]. By Lemma 8.4, we see that C π Z[v 1 ]ũ λ and C π E π ũ π (mod v 1 L). λ π Thus, {C π π Π a } also satsfes the three propertes n (8.1.1). Hence, C π = b π for each π Π a. Remark 8.6. (1) The bass {E π π Π a } plays a role as a PBW bass. It would be nterestng to know f the PBW type bass (for affne type A) constructed n [2, 3.9, 3.39], nvolvng brad group actons, s the same as the bass E π presented here. It would be also nterestng to know the meanng of the coeffcents ηλ π gven n (8.2.2). (2) Ths elementary constructon s an mportant component n a more general elementary constructon [17] of canoncal bases for quantum groups assocated to all symmetrc affne Kac-Moody Le algebras. It s expected that one can extend ths elementary constructon to the symmetrzable affne case usng the theory developed n [7, 8], or the new approach developed n [6].

18 18 BANGMING DENG, JIE DU AND JIE XIAO 9. An algebrac constructon of the canoncal bass for H Z In ths secton, we shall use dstngushed words of the form n (4.4.2) to present an algebrac constructon of the canoncal bass for the whole Rngel Hall algebra H Z. Let π Π and (π, π ) be ts assocated par. Choose dstngushed par as n (4.4.1): w π = j e 1 1 jet t Ω 1 (π ) (j r 1 j r, r) and y π = a 1 a p Σ 1 (π ) and form w π = w π y π. By (6.1.4) and (6.1.5), we have m (w π ) = ũ e1 j 1 ũ etj t = v δ(w π ) µ π γ µ w π (v2 )u µ, and m (y π ) = ũ a1 ũ ap = v δ(y π ) ν π γ ν y π (v2 )u ν, where (9.0.1) Fnally, we get (9.0.2) p δ(y π ) := ( a s a s ) + ε(dm S as,dm S at ). s=1 1 s<t p m (wπ) = m (w π ) m (y π ) = v δ(w π )+δ(y π )+ε(dm M(π ),dm M(π )) γw λ π (v 2 )u λ. λ π A key step n such a constructon s to prove that the coeffcent of ũ π (see (8.0.2)) n (9.0.2) s 1. Proposton 9.1. Let π = I, l 1 π,l[; l) Π wth p = p(π) and (π, π ) be the assocated par. For each dstngushed word w π Ω 1 (π ), we have dm End (M(π)) dm M(π) = δ(w π ) + δ(y π ) + ε(dm M(π ),dm M(π )). Proof. We prove the proposton by nducton on p. If p = 0,.e., π s aperodc, ths s the case treated n Lemma 8.2. Suppose now p 1 and wrte M = M(π). Let y π = a 1 a 2 a p. Then soc M = S ap. Let µ Π be such that M(µ) = M/soc M. Then µ = π and y µ = a 1 a 2 a p 1. In partcular, w π y µ s a dstngushed word n 1 (µ). Snce p(µ) = p 1, we have by nducton that dm End (M(µ)) dmm(µ) = δ(w π ) + δ(y µ ) + ε(dm M(π ),dm M(µ )). It s clear that and dmm = dmm(µ) + dm socm = dmm(µ) + a p, δ(y π ) = δ(y µ ) + a p a p + ε(a a p 1, a p ), ε(dm M(π ),dm M(π )) = ε(dm M(π ),dm M(µ )) + ε(dm M(π ),dm soc M). On the other hand, snce each ndecomposable summand of M s unseral, we have dm End (M) = dm End (M/soc M) + dm Hom(M, soc M) = dm End (M(µ)) + dm Hom(topM, soc M). Note that a p = dm End (socm), a p = dm soc M and ε(a a p 1, a p ) + ε(dm M(π ),dm soc M) = ε(dm M(µ),dm soc M).

19 Hence, t remans to show that GENERIC EXTENSIONS AND CANONICAL BASES 19 dm Hom(top M, soc M) = ε(dm M(µ),dm soc M) + dm End (socm). Let now l = Ll(M) and for each 1 r l, set soc l r+1 M/soc l r M = S dr for some d r N n. In partcular, socm = S dl,. e., d l = a p. Now, for a = (a ), b = (b ) N n, we defne n τa = (a n, a 1,..., a n 1 ) and a b = a b. Then, we have by defnton that ε(a, b) = a b τa b = (a τa) b. Further, we have top M = S c wth c = dm top M = ( l π 1,l,..., l π n,l ) = =1 l (π 1,l,..., π n,l ). l =1 Hence, c = d 1 + (d 2 τd 1 ) + + (d l τd l 1 ). Fnally, we obtan l 1 dm Hom(topM, soc M) = c d l = (d r τd r ) d l + d l d l = ε(d d l 1,d l ) + d l = ε(dm M/soc M,dm soc M) + dm End(socM), as desred. r=1 We have now had all the ngredents for the elementary constructon of a canoncal bass. Frst, the Rngel Hall algebra H Z admts the nvoluton ι; see (8.1.1). Second, we use the bass {ũ π π Π} as a PBW bass. To see the trangular relaton when applyng ι to ũ π, we use a monomal bass of the form {m (wπ) π Π} constructed n (9.0.2) whose members are fxed by ι. Thus, for each π Π wth the assocated par (π, π ), we fx a dstngushed word w π Ω 1 (π ). By Proposton 4.5, the word w π = w π y π s also dstngushed. By Theorem 9.1, (9.0.2) becomes (9.1.1) m (wπ) = ũ π + λ<π θ λ,π ũ λ, where θ λ,π = v dm End(M(π)) dm End(M(λ)) γ λ w π (v 2 ). Solvng (9.1.1) gves ũ π = m (wπ) + λ<π ζ λ,π m (w λ). Now, applyng the standard constructon at the end of 7 yelds a new bass {c π π Π} of H Z satsfyng c π = λ π σ λ,π ũ λ, where σ π,π = 1 and σ λ,π v 1 Z[v 1 ] for λ < π. Snce the bass {b π π Π} satsfyng the same property (see (8.1.1)), the unqueness of the canoncal bass mples the followng theorem (cf. Theorem 8.5). Theorem 9.2. For each π Π, we have c π = b π. In partcular, we have, for each π Π a, c π = C π.

20 20 BANGMING DENG, JIE DU AND JIE XIAO Acknowledgement. The authors would lke to thank the referee for the nce proof of Proposton 9.1 and the suggeston of usng the multsegment parametrzaton of nlpotent representatons, whch smplfes the deas n the proofs nvolvng the calculaton of generc extensons throughout the paper. References [1] J. Beck, V. Char and A. Pressley, An algebrac characterzaton of the affne canoncal bass, Duke Math. J. 99 (1999), [2] J. Beck and H. Nakajma, Crystal bases and two sded cells of quantum affne algebras, Duke Math. J. 123 (2004), [3] K. Bongartz, On degeneratons and extensons of fnte dmensonal modules, Adv. Math. 121 (1996), [4] B. Deng and J. Du, Monomal bases for quantum affne sl n, Adv. Math. 191 (2005), [5] B. Deng and J. Du, On bases of quantzed envelopng algebras, Pacfc J. Math. 220 (2005), [6] B. Deng and J. Du, Frobenus morphsms and representatons of algebras, Trans. Amer. Math. Soc. 358 (2006), [7] V. Dlab and C.M. Rngel, On algebras of fnte representaton type, J. Algebra 33 (1975), [8] V. Dlab and C.M. Rngel, Indecomposable representatons of graphs and algebras, Memors Amer. Math. Soc. 6 no. 173, [9] J. Du, A matrx approach to IC bases, Representatons of algebras (Ottawa, ON, 1992), , CMS Conf. Proc., 14, Amer. Math. Soc., Provdence, RI, [10] J. Du, IC bases and quantum lnear groups, Proc. Sympos. Pure Math. 56 (1994), [11] J. Du and B. Parshall, Monomal bases for q-schur algebras, Trans. Amer. Math. Soc. 355 (2003), [12] I. Grojnowsk and G. Lusztg, A comparson of bases of quantzed envelopng algebras, Contemp. Math. 153 (1993), [13] J. Y. Guo, The Hall polynomals of a cyclc seral algebra, Comm. Algebra 23 (1995), [14] M. Kashwara, On cystal bases of the q-analogue of unversal envelopng algebras, Duke Math. J. 63 (1991), [15] D. Kazhdan and G. Lusztg, Representatons of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), [16] B. Leclerc, J.-Y. Thbon and E. Vasserot, Zelevnsky s nvoluton at roots of unty, J. rene angew. Math. 513 (1999), [17] Z. Ln, J. Xao and G. Zhang, Representatons of tame quvers and affne canoncal bases, to appear. [18] G. Lusztg, Canoncal bases arsng from quantzed envelopng algebras, J. Amer. Math. Soc. 3 (1990), [19] G. Lusztg, Quvers, perverse sheaves, and the quantzed envelopng algebras, J. Amer. Math. Soc. 4 (1991), [20] G. Lusztg, Affne quvers and canoncal bases, Inst. Hautes Études Sc. Publ. Math. 76 (1992), [21] G. Lusztg, Introducton to Quantum Groups, Brkhäuser, Boston, [22] A. Mah,, Generc extenson monods for cyclc quvers, PhD Thess, UNSW, [23] R.V. Moody and A. Panzola, Le algebras wth trangular decompostons, John Wley and Sons, New York (1995). [24] M. Reneke, Generc extensons and multplcatve bases of quantum groups at q = 0, Represent. Theory 5 (2001), [25] C. M. Rngel, Hall algebras and quantum groups, Invent. math. 101 (1990), [26] C. M. Rngel, The composton algebra of a cyclc quver, Proc. London Math. Soc. 66 (1993), [27] C. M. Rngel, Hall algebras revsted, Israel Mathematcal Conference Proceedngs, Vol. 7 (1993), [28] C. M. Rngel, The Hall algebra approach to quantum groups, Aportacones Matemátcas Comuncacones 15 (1995), [29] O. Schffmann, The Hall algebra of a cyclc quver and canoncal bases of Fock spaces, Internat. Math. Res. Notces (2000), [30] M. Varagnolo and E. Vasserot, On the decomposton matrces of the quantzed Schur algebra, Duke Math. J. 100 (1999),

21 GENERIC EXTENSIONS AND CANONICAL BASES 21 [31] G. Zwara, Degeneratons for modules over representaton-fnte bseral algebras, J. Algebra 198 (1997), School of Mathematcal Scences, Bejng Normal Unversty, Bejng , Chna. E-mal address: School of Mathematcs and Statstcs, Unversty of New South Wales, Sydney, NSW 2052, Australa. E-mal address: Home Page: jed Department of Mathematcal Scences, Tsnghua Unversty, Bejng , Chna E-mal address: