DMV Lectures on Representations of quivers, preprojective algebras and deformations of quotient singularities. William Crawley-Boevey

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1 DMV Lectures on Representatons of quvers, preprojectve algebras and deformatons of quotent sngulartes Wllam Crawley-Boevey Department of Pure Mathematcs Unversty of Leeds Leeds, L2 9JT, UK Contents Lecture 1. Representatons of quvers An ntroducton to representatons of quvers and Kac s Theorem on the possble dmenson vectors of ndecomposable representatons. Dscusson about the Dynkn and extended Dynkn cases. Lecture 2. Preprojectve algebras The "deformed preprojectve algebras", ther modules, and the connecton wth representatons of the underlyng quver. The reflecton functors and ther use n provng part of Kac s Theorem. Lecture 3. Module varetes and skew group algebras Basc propertes of module varetes for algebras relatve to a semsmple subalgebra. kew group algebras. Realzng solated sngulartes usng module varetes. Lecture 4. Deformng skew group algebras ymplectc forms. Two famles of algebras parametrzed by the centre of the group algebra. McKay quvers. The connecton wth deformed preprojectve algebras. Lecture 5. The Klenan case nvestgaton of the deformatons for Klenan sngulartes. Homologcal propertes. Use of smple modules to characterze the commutatve deformatons. A conjecture about the rght deals. 1

2 ntroducton n May 1999 there was a German Mathematcal ocety (DMV) emnar on Quantzatons of Klenan sngulartes at the Mathematcal Research nsttute n Oberwolfach. The organzers were Ragnar Buchwetz, who spoke about deformaton theory, Peter lodowy, who spoke about Klenan sngulartes, and myself. These are a slghtly revsed and expanded verson of the notes that prepared for the meetng. assume throughout that K s an algebracally closed feld of characterstc zero and work wth algebras and varetes over K. The am s to study quotent sngulartes V/, where V s a smooth affne varety and s a fnte group actng on V. n partcular the Klenan sngulartes 2 K / wth L (K). For example one would lke to construct and 2 understand deformatons, quantzatons and desngularzatons of these sngulartes. The key dea s to try to realze V/ as a modul space of modules for the skew group algebra K[V]# formed by the acton of on the coordnate rng of V. Ths dea seems to be mplct n some of the recent work on the hgher dmensonal McKay correspondence, where t sometmes leads to a desngularzaton of V/. To get deformatons and quantzatons of V/, we look for deformatons of the algebra K[V]#. t s easy to wrte some down n case V s a vector space and preserves a symplectc form on V. To determne the propertes of our deformatons, we restrct to the Klenan sngularty case, when K[V]# s Morta equvalent to a "preprojectve algebra", and one can use representatons of quvers. The lectures therefore begn wth an ntroducton to these topcs. Much of ths materal comes from W.Crawley-Boevey and M.P.Holland, Noncommutatve deformatons of Klenan sngulartes, Duke Math. J. 92 (1998), Wllam Crawley-Boevey Aprl 2 2

3 Lecture 1. Representatons of quvers 1.1. QUVER. Let Q be a quver wth vertex set. Thus Q s a drected graph wth a fnte number of arrows and vertces. Each arrow a has ts tal at a vertex t(a) and ts head at a vertex h(a). We also wrte a: j to ndcate that = t(a) and j = h(a). A representaton of Q conssts of a vector space M for each vertex and a lnear map M M for each arrow a: j. A homomorphsm between two j representatons M N conssts of a lnear map M N for each vertex, such that for each arrow a: j the square M N M N j j commutes. Clearly n ths way one obtans a category of representatons, and the somorphsms turn out to be those homomorphsms n whch all lnear maps M N are nvertble. The dmenson vector of a representaton M s the vector whose components are gven by = dm M. The notaton = dm M s often used. Choosng bases for the vector spaces, any representaton of dmenson s gven by an element of Rep(Q, ) = Mat(,K). a: j j We wrte K for the representaton correspondng to a pont x Rep(Q, ). x Thus the vector space at a vertex s K and the lnear maps are gven by the matrces. Another common notaton s to use the correspondng captal letter X for the representaton. The group GL( ) = GL(,K) acts on Rep(Q, ) by conjugaton. The group elements n whch all matrces 3

4 are the same nonzero multple of the dentty matrx act trvally, so the quotent group G( ) = GL( )/K acts. Clearly the orbts (for ether group acton) correspond to the somorphsm classes of representatons of Q of dmenson. Moreover the stablzer of x n GL( ) s evdently the automorphsm group Aut(K ), so the stablzer of x n G( ) s Aut(K )/K. x x 1.2. PATH ALGEBRA. The path algebra KQ assocated to a quver Q s the assocatve algebra wth bass the paths n Q. Ths ncludes a trval path e for each vertex. For example the path algebra of the quver 1 a c b has bass e, e, e, e, a, b, c, ca, cb. The multplcaton n KQ s gven by composton of paths f they are compatble, or zero f not. n the example we have a.b =, c.b = cb, e.c = c, e.c =, etc. Note that 4 3 our conventon for the order of arrows s to compose them as f they were functons. Clearly the e are orthogonal dempotents, and the sum of them s an dentty element for KQ. The path algebra s fnte-dmensonal f and only f Q has no orented cycles. tudyng representatons of Q s essentally the same as studyng KQ-modules. (By default ths means left modules.) The connecton s as follows. - f M s a representaton of Q, so gven by vector spaces M for each vertex, and lnear maps, then M = M can be turned nto a KQ-module as follows. f s a vertex, then multplcaton by e acts as the projecton onto M. f a: j s an arrow, then multplcaton by a acts as the composton a M M M M. j - Conversely, f M s a KQ-module, there s a representaton M wth M = e M, and wth the lnear map M M correspondng to an arrow a: j j gven by left multplcaton by a. 4

5 Ths defnes an equvalence of categores, but perhaps t s not an somorphsm. Nevertheless, n future we shall blur the dfference between representatons and modules NDECOMPOABLE. Recall that a module M s sad to be ndecomposable f t cannot be wrtten as a drect sum of two proper submodules M = X Y. For fnte-dmensonal modules for an algebra, whch s our nterest n these notes, there are two key results: - Fttng s Lemma says that a module s ndecomposable f and only f every endomorphsm s of the form 1 + where K and s nlpotent. - Any fnte-dmensonal module can clearly be wrtten as a drect sum of ndecomposable submodules, but such a decomposton s not unque. However, the Krull-chmdt Theorem says that any two decompostons have the same number of ndecomposable summands, and the summands can be pared off so that correspondng summands are somorphc TANDARD REOLUTON. f s an algebra and V s an --bmodule then the tensor algebra of V over s T V = V (V V) (V V V)... wth the natural multplcaton. f A = T V, there s a canoncal exact sequence f A V A A A A m where f(a v a ) = av a - a va and m(a a ) = aa. The path algebra KQ s a specal cases of a tensor algebra, wth the a: j commutatve semsmple algebra = K and V = K, consdered as an --bmodule va svs = (s v s ). n fact any tensor algebra T V j a a: j wth,v fnte dmensonal and commutatve semsmple arses ths way. 5

6 LEMMA (tandard resoluton). Any KQ-module X has a projectve resoluton KQe X KQe X X a: j j n partcular gl.dm KQ 1. PROOF. Here X = e X s the vector space at vertex n the correspondng representaton, and the tensor products are over K. Thus KQe X s somorphc as a KQ-module to a drect sum of copes of KQe, ndexed by a bass of X. Now snce e s dempotent, KQe s a projectve KQ-module, and hence so s the drect sum. Thus the terms are ndeed projectve modules. The sequence s obtaned by applyng - X to the canoncal exact sequence KQ for KQ = A = T V. The canoncal exact sequence s a sequence of A-A-bmodules, and t s clearly splt as a sequence of rght A-modules, so t remans exact under the tensor product. nce any left KQ-module has a projectve resoluton wth two terms, we deduce that KQ has left global dmenson at most 1. But the opposte algebra of KQ s also a path algebra, of the opposte algebra, so the same apples for rght global dmenson BLNEAR FORM. The Rngel form for Q s the blnear form on defned by <, > = -. a: j j The Tts form s the quadratc form q( ) = <, >. The correspondng symmetrc blnear form s (, ) = <, > + <, >. f X,Y are (f.d.) representatons of Q then there s Rngel s formula: 1 dm Hom(X,Y) - dm Ext (X,Y) = <dm X,dm Y>. 6

7 whch follows from applyng the functor Hom(-,Y) to the standard resoluton for X and usng the fact that Hom(KQe,Y) Y to compute dmensons ROOT. Let denote the coordnate vector at vertex. The matrx A = (, ) s a Generalzed Cartan Matrx (at least when Q has no j j loops), and so there s an assocated Kac-Moody Le algebra. Ths algebra has a root system assocated to t. We need the same combnatorcs. f s a loopfree vertex n Q (meanng that there s no arrow wth head and tal at ), then there s a reflecton s :, s ( ) = - (, ) The Weyl group s the subgroup W Aut( ) generated by the s. The fundamental regon s F = { :, has connected support, and (, ) for all } By defnton the real roots for Q are the orbts of coordnate vectors (for loopfree) under W. The magnary roots for Q are the orbts of (for F) under W. f s a root, then so s -. Ths s true by defnton for magnary roots. t holds for real roots snce s ( ) = - f s a loopfree vertex. t can be shown that every root has all components or. Ths can be deduced from Le Theory, but one could also prove t usng the methods of Lecture 2. Thus one can speak of postve and negatve roots. t s easy to check that q(s ( )) = q( ), so that the Weyl group preserves the Tts form. t follows that the real roots have q( )=1, and the magnary roots have q( ). n general, however, not all vectors wth these propertes are roots. (But see 1.9.) A nonzero element of s sad to be ndvsble f gcd( ) = 1. Clearly any real root s ndvsble, and f s a real root, only are roots. On the other hand every magnary root s a multple of an ndvsble root, and all other nonzero multples are also roots. 7

8 1.7. KAC THEOREM. () f there s an ndecomposable representaton of Q of dmenson, then s a root. () f s a postve real root there s a unque ndecomposable of dmenson (up to somorphsm). () f s a postve magnary root then there are nfntely many ndecomposables of dmenson (up to somorphsm). n Lecture 2 we shall prove () and (). n the rest of ths lecture we shall assume the truth of () and (), dscuss Dynkn and extended Dynkn quvers, and prove a very specal case of (). Thus, although we do not prove all of Kac s Theorem, we do prove everythng we need for Klenan sngulartes DYNKN AND EXTENDED DYNKN QUVER. The extended Dynkn quvers are ~ ~ ~ ~ ~ those whose undelyng graph s one of A, D, E, E, E (wth n+1 n n vertces). n each case we ve ndcated a specal vector by markng each vertex wth the component. ~ ~ A : D : n 1 1 n (n ) (n 4) ~ ~ ~ E : E : 2 E : ~ Thus A conssts of one vertex and one loop. An extendng vertex s one wth = 1. The Dynkn quvers A,D,E,E,E are obtaned by deletng an n n extendng vertex. We have the followng observatons. (1) Let Q be an arbtrary quver. By defnton the radcal of the Tts form q s Rad(q) = { : (, )= for all }. Wrtng n for the j number of edges j (loops count twce), we have Rad(q) (2 - n ) = n for all. j j j 8

9 (2) f s a radcal vector wth > for all, then by calculaton 2 j q( j ) = n - <j j 2 j for any. t follows that q s postve semdefnte (meanng that q( ) for all ). Assumng that Q s connected the only vectors on whch q vanshes are the elements of, so ths s also the radcal of q. (3) One can easly check that Rad(q) for Q extended Dynkn. Thus q s postve semdefnte and Rad(q) =. (4) t follows mmedately that q s postve defnte for Q Dynkn (meanng that q( ) > for all ). (5) A case-by-case analyss shows that any connected quver whch s not Dynkn or extended Dynkn must properly contan an extended Dynkn quver, and ths mples that q s ndefnte for such quvers (so takes both postve and negatve values) ROOT FOR DYNKN AND EXTENDED DYNKN QUVER. Let Q be a Dynkn or extended Dynkn quver. (1) We show that f and q( ) 1, then s ether postve or negatve. Wrte = - wth, havng dsjont support. For a + - contradcton suppose that and are both nonzero. Now q( ) = q( ) + q( ) - (, ) q( ) + q( ) + - but q( ) and q( ) are ntegers and q s postve semdefnte, so one + term must vansh, say q( ) =. Ths mples that Q s extended Dynkn and + s a nonzero multple of. But all components of are nonzero, so we - must have =. Contradcton. (2) The roots for Q are exactly the wth q( ) 1. Certanly any root has these propertes. On the other hand, f has these propertes then we apply a sequence of reflectons to mnmze. f s now a multple of a coordnate vector at a loopfree vertex, then snce q( ) 1 9

10 we see that the multple s 1, so s a real root. Otherwse, f s any loopfree vertex then the reflecton at cannot change the sgn of, for otherwse t leads to a vector wth both postve and negatve components. By mnmalty ths mples that (, ). Thus q( ), so s a multple of, so n the fundamental regon, and hence an magnary root. (3) Clearly the magnary roots for an extended Dynkn quver are exactly the multples of. (4) Clearly a Dynkn quver has only real roots. n fact t has only fntely many roots, for they form a dscrete subset of the compact set { :q( )=1}. Thus Kac s Theorem mples Gabrel s Theorem, that the quvers wth only fntely many ndecomposables are the Dynkn quvers LEMMA (Rngel). An ndecomposable f.d. KQ-module whch s not a brck has a submodule whch s a brck wth self-extensons. (By defnton a brck s a module X wth End(X) = K, and X has 1 self-extensons f Ext (X,X) ). PROOF. By nducton t suffces to prove that X has an ndecomposable proper submodule wth self-extensons. For a contradcton, suppose not. Let End(X) be a nonzero endomorphsm wth = m( ) of mnmal dmenson. By hypothess s ndecomposable, so has no self-extensons. 2 2 Now =, for m( ), and f they are equal then the composton X s an somorphsm, so s a drect summand of X. Thus Ker( ). Wrte Ker( ) as a drect sum of ndecomposables, say Ker( ) = K, and let : Ker( ) K be the projectons. For some j we must have (). j 1 uppose for a contradcton that Ext (K,K ) =. j j Mnmalty mples that s njectve (consderng the composton j X K X). Applyng Hom(-,K ) to the short exact sequence j j K K / j j 1

11 gves a long exact sequence 1 f Ext (K,K ) Ext (,K ) Ext (K /,K )... j j j j j 2 and the Ext term vanshes, so f s onto. Now consder the pushout of the short exact sequence Ker( ) X along, say j Ker( ) X g h K Y j f t splts, then h has a retracton, and ts composton wth g s a retracton for the ncluson of K n X. But ths mples that K s a j j 1 drect summand of X, whch s nonsense. Thus we must have Ext (,K ). j 1 t follows that Ext (K,K ) j j. Contradcton LEMMA. For Q extended Dynkn, the general element of Rep(Q, ) s a brck, and there are only fntely many other orbts. PROOF. There are only fntely many orbts of decomposable modules snce there are only fntely many roots whch are less than real roots. Now usng the fact that q( )=,, and they are all 2 2 dm Rep(Q, ) = =, dm G( ) = - 1, a: j j so there must be nfntely many orbts. Thus the general element of Rep(Q, ) must be ndecomposable. Now Rngel s Lemma says that each ndecomposable s ether a brck, or t has a proper submodule M whch s a brck wth self-extensons. But then 11

12 1 q(dm M) = dm End(M) - dm Ext (M,M), so dm M s a multple of. Ths s mpossble FURTHER READNG. The best reference for Kac s Theorem s hs last paper on the topc, V.G.Kac, Root systems, representatons of quvers and nvarant theory, n: nvarant theory, Proc. Montecatn 1982, ed. F. Gherardell, Lec. Notes n Math. 996, prnger, Berln, 1983, Another useful reference s H.Kraft and Ch.Redtmann, Geometry of representatons of quvers, n: Representatons of algebras, Proc. Durham 1985, ed. P. Webb, London Math. oc. Lec. Note eres 116, Cambrdge Unv. Press, 1986, The defntve reference for extended Dynkn quvers s ecton 3.6 of C.M.Rngel, Tame algebras and ntegral quadratc forms, Lec. Notes n Math. 199, prnger, Berln, Lecture 2. Preprojectve algebras Let Q be a quver wth vertex set PREPROJECTVE ALGEBRA. The double of Q s the quver obtaned by adjonng an arrow a :j for each arrow a: j n Q. The preprojectve algebra s the assocatve algebra (Q) = KQ / ( [a,a ]). a Q More generally, the deformed preprojectve algebra of weght K s (Q) = KQ / ( [a,a ] - e ) a Q 2.2. REMARK. (1) The preprojectve algebra frst appeared wth the relaton (aa +a a) =. t s easy to see that ths gves an somorphc a Q algebra provded the quver s bpartte, meanng that the vertces can be dvded nto two sets and no arrow has both head and tal n the same set. 12

13 (2) f Q has no orented cycles then KQ s a fnte-dmensonal algebra. For such algebras there are Auslander-Reten operators DTr and TrD, and t can be shown that n (Q) (TrD) (KQ). n= Ths means that KQ-modules. (Q) s the sum of all ndecomposable preprojectve (3) (Q) doesn t depend on the orentaton of Q. Just reverse the role of a and a, and change the sgn of one of them. (4) f r s the defnng relaton for (Q), then r = e re, and Thus e re = a a - a a - e. h(a)= t(a)= (Q)-modules correspond to representatons of Q n whch the lnear maps satsfy the relatons a a - a a = d h(a)= t(a)= for all. Wth ths dentfcaton we can speak of the dmenson vector of a (Q)-module. (5) f there s a (Q)-module of dmenson then = must be equal to zero. To see ths, take the traces of all the relatons, and sum. On the left hand sde every term tr(aa ) s cancelled by a term -tr(a a). On the rght hand sde the traces add up to MOMENT MAP. The relatons for the deformed preprojectve algebra arse from a moment map. Let V be a vector space wth a symplectc form, a skew symmetrc blnear form V V K whch s non-degenerate n the sense that (u,v)= for all v mples u=. Let an algebrac group G act on V preservng. Dfferentaton gves an acton of the Le algebra V V. nce G preserves, t follows that 13

14 ( v,v ) = - (v, v ) for all and v,v V. By defnton the moment map n ths stuaton s 1 the map :V defned by (v)( ) = (v, v) for v V and. t has the 2 requred property of moment maps n symplectc geometry: ts dervatve d :V at v V satsfes v 1 d (v )( ) = ( (v, v)+ (v, v )) = (v, v ). v 2 To apply ths to quvers we equp Rep(Q, ) wth the symplectc form comng from ts dentfcaton wth the cotangent bundle T Rep(Q, ). Explctly (x,y) = tr(x y ) - tr(x y ) a Q a a a a for x,y Rep(Q, ). The group G( ) = GL( )/K preserves. acts by conjugaton and (gx) a: j -1 = g x g a j ts Le algebra s dentfed wth End( )/K where End( ) = and the acton s gven by Mat(,K), ( x) = x - x. a: j a a j Let End( ) = { End( ) : tr( )=}. The trace parng gves an somorphsm End( ) (End( )/K), ( tr( )). The moment map s thus : Rep(Q, ) End( ) gven by x ( x x - x x ) h(a)= a a t(a)= a a Now G( ) acts by conjugaton on End( ), and the nvarant elements are those n whch each component s a multple of the dentty matrx. We -1 dentfy these wth elements of { K : =}. Then ( ) s dentfed wth the space of (Q)-modules of dmenson. 14

15 2.4. LEMMA. f x Rep(Q, ) and X s the correspondng KQ-module, then there s an exact sequence 1 op f t Ext (X,X) Rep(Q, ) End( ) End(X) where t( )( ) = tr( ) comes from the trace parng, and j a Q a a dfferent ways of extendng the acton of KQ on X to an acton of f(y) = [x,y ]. Thus the fbre of f over K conssts of the (Q). PROOF. Apply Hom (-,X) to the standard resoluton of X, dualze, and use KQ trace parngs to dentfy terms THEOREM. f X s a KQ-module then the acton of KQ on X can be extended to an acton of (Q) f and only f dm Y = for any KQ-module summand Y of X. PROOF. uppose that the acton extends. Let X be gven by x Rep(Q, ). Then n the lemma we have m(f), so t( ) =, so tr( ) = for any End (X). f Y s a KQ-module summand of X, apply ths wth the KQ projecton onto Y to see that dm Y =. For the converse, t suffces to prove that an ndecomposable X wth dm X = lfts. By Fttng s Lemma End(X) conssts of multples of the dentty plus a nlpotent endomorphsm, so t s easy to see that t( ) = REMARK. Assumng Kac s Theorem, t follows that the possble dmenson vectors of (Q)-modules are exactly the sums of roots wth =. Note that although we don t prove all of Kac s Theorem, we prove enough to justfy ths clam for Q Dynkn or extended Dynkn, for when wrtng a vector as a sum of roots wth = you can take all these roots to be ether real roots or, and what we prove s suffcent. n fact one can prove that s the dmenson vector of a smple (Q)-module f and only f s a postve root, =, and 1-q( ) > (1-q( )) + (1-q( ))

16 whenever = a sum of postve roots wth = =...=. ee PROPOTON. For an extended Dynkn quver Q, all fbres of are 2 rreducble of dmenson 1 +. Thus s flat. PROOF. f End( ), let be the composton -1 ( ) Rep(Q, ) Rep(Q, ) f x Rep(Q, ) then (x) f ( ) n the sequence of Lemma 2.4, so t s 1 ether empty, or a coset of Ext (X,X). Thus t s ether empty or rreducble of dmenson dm End(X) + q( formula. ) = dm End(X) by Rngel s The brcks form a dense open set B Rep(Q, ). They have nonempty fbres Thus (B) s rreducble of dmenson dm B + 1 = + 1. Besdes the brcks, there are only fntely many other orbts of G( ) on Rep(Q, ). The stablzer of x s dentfed wth Aut(X)/K, so the orbt of x has dmenson dm G( ) - dm Aut(X)/K and ts nverse mage under (f non-empty) has dmenson 2 dm G( ) - dm Aut(X)/K + dm End(X) =. Now any rreducble component of a fbre of has dmenson at least 2 dm Rep(Q, ) - dm End( ) = t follows that each fbre s rreducble of dmenson + 1. Ths mples flatness snce s a map between smooth rreducble varetes REFLECTON FUNCTOR. f s a loopfree vertex, we have a reflecton 16

17 r : K K, r ( ) = - (, ). j j j dual to s. The dualty means that r = s for all,. We say that the reflecton s admssble for f. n ths case there s a Morta equvalence (Q)-modules r (Q)-modules whch acts as s on dmenson vectors. We call ths a reflecton functor. (Do not confuse ths wth a reflecton functor n the sense of Bernsten, Gelfand and Ponomarev - they are for KQ-modules, and are not equvalences.) EXAMPLE. f Q s the quver then a (Q)-module X s gven by vector spaces and lnear maps X 1 X X X X 2 satsfyng the deformed preprojectve relatons. For vertex =3, the lnear maps combne to gve maps X X X X X and (nsertng mnus sgns sutably) the relatons ensure that = d. 3 Now f ths mples that s the ncluson of a drect summand and 3 X X X = m( ) Ker( ) The functor sends X to the (Q)-module 17

18 X 1 Ker( ) X X 4 5 X 2 n whch the lnear maps to and from Ker( ) come from the two decompostons of X X X CONEQUENCE. Let the Weyl group W act on K va w = (w ) for all,. We clam that f W then (Q) and (Q) are Morta equvalent, that s, there s an equvalence (Q)-modules (Q)-modules Namely, wrte = r... r. n 1 Dong ths wth n as small as possble, the reflectons at each stage are admssble (for f = then r = ). The reflecton functors then gve the equvalence LEMMA. f there s a smple module for (Q) of dmenson, and s a vertex, then = or (, ) or. PROOF. uppose otherwse. nce (, )> there s no loop at. f X s the smple module, snce =, the lnear maps combne to gve maps X X X j j wth composton zero. (The drect sum s over all arrows ncdent at, and the correspondng term s the space X at the other end of the arrow.) j Now s njectve, for Ker( ) s a submodule of X, and f X = Ker( ) then X lves at, and smplcty mples that =. Dually s surjectve. But then dm X 2 dm X, j j 18

19 so (, ), a contradcton LEMMA. The dmenson vector of any smple (Q)-module s a root. PROOF. uppose that there s a smple of dmenson. Applyng a sequence of admssble reflectons, we follows the effect on and : r r r j... s s s j Because of the reflecton functors there s a smple (Q)-module of dmenson. Thus s postve. We choose the sequence to make as small as possble. Ths mples that (, ) for any vertex wth. The prevous lemma now mples that s ether a coordnate vector at a loopfree vertex or has (, ) for all vertces. Of course has connected support because of the exstence of a smple (Q)-module of dmenson. Thus n the latter case, s n the fundamental regon. t follows that s a root PROPOTON. f there s an ndecomposable for KQ of dmenson then s a root. PROOF. Wrte = k wth ndvsble. Choose K wth =, but for any whch s not a multple of. Ths s possble snce K has characterstc zero. The ndecomposable KQ-module extends to an ndecomposable (Q)-module, and any composton factor of ths must have dmenson m for some m. Thus m s a root. Now apply admssble reflectons to and m as n the proof of Lemma We pass to and a vector whch s easly seen to be of the form m for some ndvsble. The reflecton functors can also be appled to the ndecomposable (Q)-module of dmenson = k to gve an ndecomposable 19

20 (Q)-module of dmenson k. Now ether m fundamental regon. s a coordnate vector at a loopfree vertex, or n the n the frst case m=1, but also because there s an ndecomposable (Q)-module of dmenson k, we must have k=1. Thus s a root. n the second case and k are also n the fundamental regon, so that s agan a root PROPOTON. f s a postve real root then (up to somorphsm) there s a unque ndecomposable KQ-module of dmenson. PROOF. We use the fact that every root s postve or negatve. Wrte = s... s ( ) j n 1 wth j a loopfree vertex and n as small as possble. Then all ntermedate k terms = s... s ( ) are postve roots. Defne K by j k 1 (=j) =, 1 (else) k k k and let = r... r ( ). Now ( ) = = where k 1 k+1 k+1 = s...s ( ). 1 k k+1 Now, for snce s a real root t s postve or negatve, so the condton = mples that =, but then j = s... s ( ) = j k+1 k 1 k contradctng the mnmalty of n. Thus the reflecton at k+1 k admssble for. Thus there are reflecton functors s 2

21 (Q)-modules 1 2 (Q)-modules (Q)-modules... Clearly there s a unque (Q)-module of dmenson, and t s smple. j n Thus, lettng =, there s a unque (Q)-module M of dmenson and t s smple. Now M s ndecomposable as a KQ-module, for f t has an ndecomposable summand of dmenson, then n = = = where = s...s. 1 n But s a root by Proposton 2.12, hence so s, and the condton = mples that =. Thus = s...s ( ) =, so n fact j j n 1 =. Fnally the ndecomposable KQ-module of dmenson s unque snce any such module can be extended to a (Q)-module, but there s a unque (Q)-module of dmenson FURTHER READNG. The deformed preprojectve algebra and the reflecton functors were ntroduced n W.Crawley-Boevey, and M.P.Holland, Noncommutatve deformatons of Klenan sngulartes, Duke Math. J. 92 (1998), The constructon of the preprojectve algebra usng TrD s n D.Baer, W.Gegle and H.Lenzng, The preprojectve algebra of a tame heredtary Artn algebra, Commun. Algebra 15 (1987), However, to see that these two descrptons of the preprojectve algebra are the same, see C.M.Rngel, The preprojectve algebra of a quver, n: Algebras and modules, (Geranger, 1996), , CM Conf. Proc., 24, Amer. Math. oc., Provdence, R, 1998, or W.Crawley-Boevey, Preprojectve algebras, dfferental operators and a Conze embeddng for deformatons of Klenan sngulartes, Comment. Math. Helv. 74 (1999), Ths latter paper contans much more about deformed preprojectve algebras. The paper W.Crawley-Boevey, Geometry of the moment map for representatons of quvers, to appear n Composto Math., proves the characterzaton of the dmensons of smple modules for deformed preprojectve algebras. n an appendx t also contans the elementary deducton of much of Kac s Theorem gven here. 21

22 Lecture 3. Module varetes and skew group algebras We are nterested n fnte dmensonal left modules for a fntely generated K-algebra A (assocatve, wth 1) MODULE VARETE. There s an affne varety n Mod(A,n) = {A-module structures on K } = {K-algebra maps A Mat(n,K)} m = {(,..., ) Mat(n,K) : r(,..., )= for all r R} 1 m 1 m on chosng generators of A, and hence wrtng A = K<x,...,x >/R, where 1 m K<x,...,x > s the free assocatve algebra on generators x,...,x and R 1 m 1 m s an deal. We need a varaton on ths, relatve to a semsmple subalgebra. Let A be a f.g. K-algebra and A a f.d. semsmple subalgebra (or more generally let A be a homomorphsm). f M s a f.d. -module, let Mod (A,M) = {A-module structures on M extendng ts -module structure} t s an affne varety. (Choosng a bass of M, t can be dentfed wth a fbre of the map Mod(A,n) Mod(,n).) The group G(M) = Aut (M)/K acts on Mod (A,M). t orbts are n 1-1 correspondence wth somorphsm classes of A-modules X wth X M. We wrte O(X) for the orbt correspondng to X. The stablzer of a pont x O(X) can be dentfed wth Aut (X)/K. A pecal cases: (1) uppose A = T V s the tensor algebra on an --bmodule. Then by the unversal property of tensor algebras, Mod (T V,M) Hom (V M,M). (2) uppose that A s a path algebra KQ of a quver Q wth vertex set. Let = K... K spanned by the trval paths e. f defne 22

23 K = K. Ths s an -module, wth multplcaton by e the -th summand. Then we can dentfy Mod (KQ,K ) = Rep(Q, ), G(K ) = G( ) actng as projecton onto (3) f A s a quotent of a path algebra, let be the subalgebra generated by the trval paths as before. Then Mod (A,K ) s a closed subvarety of Rep(Q, ). n partcular, -1 Mod ( (Q),K ) = ( ). 3.2 TRACE FUNCTON. f a A and X s a f.d. A-module then defne tr(a,x) = trace of the map X X, x ax. Ths defnes a functon tr(a,-) : Mod (A,M) K. t s a G(M)-nvarant, so G(M) tr(a,-) K[Mod (A,M)]. Gven a module X, we wrte gr X for the semsmple module whch s the drect sum of the composton factors of X (wth the same multplctes). (1) tr(a,x) = tr(a,gr X). The trace of a matrx wth block form s the sum of the traces of the dagonal blocks, so f X X X s 1 2 an exact sequence of A-modules, then tr(a,x) = tr(a,x )+tr(a,x ) = tr(a,x X ). The asserton follows by nducton (2) f tr(a,x) = tr(a,y) for all a, then gr X gr Y. Ths s character theory! To prove t we may assume that X,Y are semsmple. Replacng A by A/ann (X Y) we may assume that A s semsmple. Now f A A Mat(n,K)... Mat(n,K), 1 r a s the -th dentty element, and s the j-th smple module, then j tr(a, ) = n. The clam follows. j j j 23

24 3.3. PROPOTON. The closure of any orbt O(X) contans a unque closed orbt, O(gr X). Thus the closed orbts n Mod (A,M) are exactly those of semsmple A-modules. PROOF. We show frst that f Y X then O(Y X/Y) O(X). By defnton X = M wth an A-module structure. Let C be an -module complement to Y n M. The acton of any a A on X s an element of End (Y C), so can be wrtten K as a 2 2 matrx a a a 22 wth a End (Y), a Hom (C,Y), a End (C). For t K, let g G(M) 11 K 12 K 22 K t correspond to the automorphsm of M whch s multplcton by t on Y and the dentty on C. The acton of a A on g X s gven by the matrx t a ta a 22 Thus the closure of the orbt of X must contan the element gven by matrces a 11 a 22 That s, Y X/Y. Now by nducton O(gr X) O(X). n partcular, f O(X) s closed then X must be semsmple. On the other hand, the closure of any orbt s a unon of orbts, so always contans a closed orbt, eg one of mnmal dmenson. Fnally note that f O(Y) s a closed orbt n O(X) then by contnuty X,Y have the same trace functons, so gr Y gr X QUOTENT. f a reductve group G acts on an affne varety V then the quotent V / G (the set of orbts) s not usually well-behaved topologcally. However the set V // G of closed orbts s. t s natually 24

25 G an affne varety wth coordnate rng K[V]. Ths apples to G(M) actng on Mod (A,M). nce M s semsmple, M wth the non-somorphc smples, and Aut (M) GL (K). Thus G(M) s reductve. Now by Proposton 3.3, somorphsm classes of semsmple Mod (A,M) // G(M) = A-modules X wth X M and there s a natural map Mod (A,M) Mod (A,M) // G(M), X gr X. More general quotents can be constructed as part of "Geometrc nvarant Theory". Let be an addtve functon {semsmple -modules}. One says that X s -semstable f (X) = and (Y) for all A-submodules Y X. -stable f (X) = and (Y) > for all A-submodules Y X. A -semstable module X naturally has assocated to t a module gr X whch s a drect sum of -stables, and one says X,X are -equvalent f gr X gr X. There s a GT quotent Mod (A,M) // (G(M), ) whose ponts correspond to { -semstables X wth X M} / -equvalence, and there s a proper map Mod (A,M) // (G(M), ) Mod (A,M) // G(M) MORTA EQUVALENCE. f e A s an dempotent then eae s an algebra wth dentty e, and there s a functor A-modules eae-modules, X ex. To apply ths to module varetes we assume that e. Then the functor nduces a morphsm : Mod (A,M) // G(M) Mod (eae,em) // G(eM). ee 25

26 n case AeA = A the the functor s an equvalence, so that A and eae are Morta equvalent. t follows that the morphsm s a bjecton. n fact we have: THEOREM. f AeA = A then s an somorphsm of varetes. We prove ths n the next subsecton. Here we verfy t the specal case when = Mat(n,K)... Mat(n,K), and e = e, where e s the 1 r elementary matrx 1.. n the -th factor. Then e = and ee K... K. t s easy to see that A s somorphc to the algebra of r r matrces (C ) where each C s an j j n n matrx of elements of e Ae, and that eae s somorphc to the j j algebra of r r matrces (D ) wth each D e Ae. Thus has an nverse j j j r comng from the map whch sends an eae-module structure on em e M to =1 n r the A-module structure on (e M) gven by the acton of the block =1 matrces. G(M) 3.6. THEOREM. The rng of nvarants K[Mod (A,M)] s generated by the trace functons tr(a,-). PROOF. Presumably there s a drect proof of ths. Defntely t needs characterstc zero n ths generalty. t s proved by Le Bruyn and Proces for path algebras. f G acts on X and Y X s G-stable closed G G subset, then the restrcton map K[X] K[Y] s surjectve by the Reynolds operator. Thus t follows for quotents of path algebras, or equvalently for algebras A n whch the semsmple subalgebra s somorphc to K... K. Now the general case can be reduced to ths by the specal case of Morta equvalence proved above. We now prove that s an somorphsm n the general case when AeA=A. nce s a bjecton, t s suffcent to prove that t s a closed embeddng, or equvalently that the natural map 26

27 G(eM) G(M) K[Mod (eae,em)] K[Mod (A,M)] ee s surjectve. Now the space on the rght hand sze s generated by trace functons tr(a,-) wth a A. nce AeA = A we can wrte a = a ea tr(a,m) = tr(a ea,m) = tr(ea a e,m) = tr(b,em), k k k k where b = ea a e eae, so tr(a,-) s n the mage of the map. k k k. Then k 3.7. KEW GROUP ALGEBRA. f A s an algebra and s a fnte group actng as automorphsms of A, then the skew group algebra A# conssts of the g g formal sums a g wth the multplcaton satsfyng g (a g)(a g ) = a( a ) gg. Let K be the group algebra of. Observe that an A# -module conssts of an g A-module X whch s also a K -module, and such that g(ax) = ( a)(gx). g f X s an A-module then (A# ) X s somorphc as an A-module to X, A g g where X denotes the module X wth the acton of A twsted by g, and the acton of permutes the factors. Henceforth we wrte e for the dempotent 1 e = g K. g t has the property that ek = K e = Ke and e(a# )e = A e A LEMMA. gl.dm A# = gl.dm A. PROOF. Usng a projectve resoluton of X t s easy to see that Ext (A# X,Y) Ext (X, Y) A# A A A when X s an A-module and Y s an A# -module. 27

28 Now f Z s an A-module, then t s a drect summand of Y = (A# ) Z. t follows that gl.dm A gl.dm A#. A Y where A On the other hand, f X s an A# -module then t s somorphc to a summand of A# X, snce the multplcaton map A# X X has a secton A A 1-1 x g g x g Ths mples that gl.dm A# gl.dm A MPLE MODULE. Let be a fnte group actng on an affne varety V, so also on ts coordnate rng K[V] va -1 (gf)(v) = f(g v) for g, v V and f K[V]. The quotent V/ s an affne varety wth coordnate rng K[V] (snce all orbts are closed). We are nterested n smple modules for K[V]#. Now K[V] s contaned n the centre of K[V]#, wth equalty f the acton of on V s fathful. Thus any smple module for K[V]# s annhlated by a unque maxmal deal for K[V], so t s a module for (K[V]/K[V] )#. Thus t s a module for O K[V] / K[V] # (K )# O where O s the orbt n V correspondng to, K s the space of functons from O to K, and acts by permutng the factors. Ths s a semsmple algebra. pecal cases are: (1) acts freely on O. That s O = = N, say. n ths case O (K )# Mat(N,K). Thus there s a unque smple module for K[V]# whch s annhlated by. Note that K and that e. K O (2) O conssts of one fxed pont. Then (K )# K, so each smple K -module nduces a smple module for K[V]# annhlated by. 28

29 3.1. MORTA EQUVALENCE. There s a functor K[V]# -modules K[V] -modules, X ex and hence (snce ek K), a morphsm : Mod (K[V]#,K ) // G(K ) Mod (K[V],K) // G(K) V/. K K LEMMA. f acts freely on V then (K[V]# )e(k[v]# ) = K[V]#. Thus the functor s a Morta equvalence, and s an somorphsm. PROOF. f K[V]# / (K[V]# )e(k[v]# ) then there s a non-zero K[V]# -module M wth em=. t follows that there s a smple module wth ths property. But ths s not the case OLATED NGULARTE. Let act on a smooth rreducble varety V. Let V be a fxed pont, and assume that acts freely on V\{}. Then V/ s an solated sngularty. THEOREM. n ths case s also an somorphsm. PROOF. For each orbt of on V\{} there s a smple K[V]# -module structure on K. n addton, each smple K -module gves a smple K[V]# -module correspondng to the pont V. t follows that for each pont of V/ there s a unque semsmple K[V]# -module structure on K. Thus s a bjecton. For each v V we consder K as a K[V]# -module, wth fg = f(gv)g for f K[V], g. Ths nduces a map V Mod (K[V]#,K ) // G(K ) K whch s constant on -orbts so factors through V/. The resultng map V/ Mod (K[V]#,K ) // G(K ) K s an nverse for. 29

30 3.12. REMARK. Provded acts fathfully on V, so that the general orbt of on V s free, The same argument shows that even f V/ s not an solated sngularty, the quotent Mod (K[V]#,K ) // G(K ) has an rreducble K component somorphc to V/. Now usng other Geometrc nvarant Theory quotents Mod (K[V]#,K ) // (G(K ), ) K one can hope to obtan a desngularzaton of V/. Ths was used by Cassens and lodowy to construct the mnmal desngularzaton for Klenan sngulartes, and t s essentally the " -Hlbert scheme" Hlb (V) consdered by to and Nakamura, Nakajma, and others LEMMA. n the solated sngulartes case, K[V]# /(K[V]# )e(k[v]# ) s fnte-dmensonal. PROOF. t s a f.g. K[V] -module, whose only composton factor s K[V] /, where s the maxmal deal correspondng to the sngular pont FURTHER READNG. Module varetes have been extensvely studed for fnte dmensonal algebras. ee for example P.Gabrel, Fnte representaton type s open, n: Representatons of algebras, Proc Ottawa 1974, eds V. Dlab and P. Gabrel, LN 488; C.Geß, Geometrc methods n representaton theory of fnte dmensonal algebras, n: Canadan Math. oc. Conf. Proc., 19, 1996; K.Bongartz, ome geometrc aspects of representaton theory, n: Canadan Math. oc. Conf. Proc., 23, Geometrc nvarant Theory quotents are dscussed n A.D.Kng, Modul of representatons of fnte dmensonal algebras, Quart. J. Math. Oxford 45 (1994), The specal case = also covers the affne quotents. Note that Kng assumes that A s fnte-dmensonal n hs ecton 4, but ths s only necessary for Proposton 4.3. The fact that the nvarants for representatons of quvers are generated by traces of orented cycles s n L.Le Bruyn, and C.Proces, emsmple representatons of quvers, Trans. Amer. Math. oc. 317 (199), kew group algebras are a classcal topc. ee for example J.C.McConnell and J.C.Robson, Noncommutatve noetheran rngs. For the constructon of desngularzatons of Klenan sngulartes n ths way see H.Cassens and P.lodowy, On Klenan sngulartes and quvers, ngulartes (Oberwolfach, 1996), Brkhauser, Basel, 1998,

31 The hgher dmensonal McKay correspondence comes from M.Red, McKay correspondence, math.ag/ The -Hlbert scheme s n Y.to and.nakamura, Hlbert schemes and smple sngulartes, n: Algebrac Geometry (Proc. Warwck, 1996), eds K. Hulek et al. (Cambrdge Unv. Press 1999), Also sgnfcant s Y.to and H.Nakajma, McKay correspondence and Hlbert schemes n dmenson three, math.ag/ Lecture 4. Deformng skew group algebras 4.1. YMPLECTC FORM. Recall that a symplectc form on a vector space V s a blnear form :V V K whch s skew symmetrc and non-degenerate n the sense that (u,v)= for all v mples u=. One can thnk of as a skew symmetrc element of V V. One can choose symplectc coordnates p,q :V K such that = p q -q p. n partcular dm V must be even. Observe that nduces an somorphsm V V, v (v,-) so V also gets a symplectc form GROUP PREERVNG A YMPLECTC FORM. We gve some examples of group actons whch preserve a symplectc form. 2 (1) (Klenan case) Let V = K wth (x,y) = x y - x y. Then a subgroup GL (K) preserves f and only f L (K). 2 2 (2) f acts on U, then t also acts on the cotangent bundle T U = U U preservng the symplectc form defned by (f u,f u ) = f (u)-f(u ). (3) An rreducble representaton V of a fnte group wth Frobenus-chur ndcator -1 preserves a skew symmetrc blnear form on V. t must be a symplectc form snce V s rreducble DEFORMNG KEW GROUP ALGEBRA. uppose that a fnte group acts lnearly on a vector space V preservng a symplectc form. Observe that 31

32 K[V]# = (T(V ) / ( - :, V ))# (T(V )# )/( - :, V ), K[V] = e K[V]# e. 1 Here T(V ) s the tensor algebra of V (over K) and e = g. g For Z(K ), defne = T(V )# / ( - - (, ) :, V ), O = e e. c c Observe that f c K then and O O, usng the automorphsm of T(V )# whch multples each element of V by c AOCATED GRADED ALGEBRA. uppose that an algebra A s generated over a subalgebra A by fntely many elements x. There s a "standard" fltraton = A A A where A = span of elements a x a...x a wth a A and k n. n 1 k 1 k Whenever A s a fltered rng there s an assocated graded algebra gr A = A /A. = -1 For the standard fltraton, gr A s generated over A by elements x. t s well known that f gr A has one of the followng propertes, then so does A: doman prme noetheran fnte global dmenson 4.5. LEMMA. s fltered, wth assocated graded rng K[V]#. Thus t s prme, noetheran of fnte global dmenson. O s fltered, wth assocated graded rng K[V]. Thus t s a noetheran doman. 32

33 PROOF. s generated over K by a bass of V. Ths gves the fltraton. Now gr s generated over K by. These elements satsfy =. Thus there s a surjecton K[V]# gr. j j s t an somorphsm? You can use the relaton n to reorder monomals, modulo lower degree. Thus has bass the elements n n 1 m... g. The rest 1 m follows. (When s O commutatve, and how does ts global dmenson depend on? can only answer these questons n the Klenan case, when these propertes are related to preprojectve algebras.) 4.6. FNTE GENERATON. The followng lemma shows that e s a f.g. O -module and O s a f.g. K-algebra. LEMMA (Montgomery and mall). f A s an algebra, e A s dempotent and AeA s a f.g. left deal n A (eg A s noetheran), then ea s a f.g. eae-module. f n addton A s a f.g. K-algebra then eae s a f.g. K-algebra. PROOF. Let AeA = Ax and x = v ew wth v,w A. Then j j j j Aew = AeA, so eaew = eaea = ea, so the elements ew generate j j j j j ea as an eae-module. Now suppose that t,...,t generate A and let ea = eaex. 1 n Wrte et = ey ex and ex t = ez ex wth y,z A. j j k j jk j jk We clam that eae s generated by the elements ex e, ey e and ez e. j jk For, every element of eae s a lnear combnaton of terms et t...t e, j j j 1 2 l and, for example, et t e = ey ex t e = ey e ez ex e l 1 l2 l 4.7. LEMMA. There s a surjectve homomorphsm T(V )# / ( - ). PROOF. Recall that = T(V )# / ( - - (, )). 33

34 f V has symplectc coordnates p,q, then = p q - q p. Now (p,q ) = 1, so - = (p q - q p - (p,q )). 2 where =. Thus there s a natural map T(V )# /( - ). But dm V McKAY QUVER. Let be a fnte group actng lnearly on a vector space V. Let N ( ) be the smple K -modules, wth N the trval module. The McKay quver for and V has vertex set, and by defnton the number of arrows j s the multplcty of N n V N. j HENCEFORTH UPPOE that preserves a symplectc form. dentfy Z(K ) wth an element of K va = trace of on N. The rest of ths lecture s devoted to provng that T(V )# / ( - ) s Morta equvalent to (Q) for some quver Q wth vertex set and Q =. ome more notaton. Let dm N =, so that K Mat(,K). Let the elements E K (, 1 p,q ) correspond to the elementary matrces. pq 1 Let f = E and let f = f. Clearly f = e = g. Also 11 g f K f = K f K... K. Observe that N K f, whch has bass E. Dually, f K has bass E. p1 1q 1 = E E = E f E K f K.,p p1 1p,p p1 1p Thus fk f s Morta equvalent to K LEMMA. f B s a K -K -bmodule, then there s an somorphsm f T (B) f T (fbf) whch s the dentty on fk f, and sends an element K fk f b c fb Bf to the element be E c fbf fbf. K p p1 1p PROOF. Follows from the Morta equvalence. 34

35 ( 4.1. LEMMA. f acts lnearly on a vector space W preservng a symplectc form, then the restrcton of to V = w v W W then = ew ev W W. k k k k s a symplectc form, and f PROOF. s non-degenerate snce f w W then there s v W wth (w,v), but then (w,ev)= (w,v) and ev W. ay, (W ), so = (x,-), = (x,-) for some x,x W. Then, ) = (x,x ) by defnton of = (x,x ) = ( (x,-), (x,-)) by defnton of = (x,w ) (x,v ) snce = w v k k k k k = (x,ew ) (x,ev ) snce x,x W k k k = (x,ew ) (x,ev ) k k k = (ew ) (ev ). k k k PROPOTON. Let B = V K consdered as a K -K -bmodule va g(v g )g = gv gg g. Then fbf has a symplectc form wth j j = E ( E ) E ( E ) fbf fbf jpqk 1p k q1 1q k p1 where = V V. Moreover respects the decomposton k k fbf = f Bf n the sense that (b,b )= f b f Bf and b f Bf j j j j wth j or j. -1 PROOF. Let W = fk V K f, as a K -module va g(a b) = ag g gb. Consder the map m:w fbf, a v b a(v b). Observe that m(e(a b)) = m(a b). By dmensons t follows that m nduces a vector space somorphsm (fk V K f) fbf. Now fk V K f has a symplectc form gven by (a b,a b ) = (, ) for a f K, b K f, a f K, b K f, where, K are gven by ab = f j j 35

36 and a b = f. The acton of preserves. Thus gves a symplectc form on fxed ponts, so on fbf. Call t. Clearly ths respects the decomposton. Now as an element of the tensor square of fk V K f s j j = (E E ) (E E ) jpqk 1p k q1 1q k p1 Thus, usng that m(e(a b)) = a( b), we get j j = E ( E ) E ( E ) jpqk 1p k q1 1q k p THEOREM. f T(V)# / ( - ) f T (fbf) / ( - f ) fk f PROOF. Frst observe that T(V )# T B K - As vector spaces the LH s K V K V V K... - The RH s K V K (V K ) (V K )... K Thus T(V)# / ( - ) T (B) / ( - ) where, f = V V, then K k k = ( 1) ( 1) B B. k k K Now f T(V)# / ( - ) f f T (B) f / where s the deal K f T (B) ( - ) T (B) f = f T (B) K fk ( - ) K fk T (B) f K K K K Ths s generated as an deal n f T (B) f by fk ( - ) K f. K Now f g then g = (g g) ( 1) = (g 1) (g g) = (g 1) (g 1)g = g k k k k k k snce = g g because s -nvarant. Thus k k ( - )f (=j and p=q) j j E ( - ) E = ( - ) E E = 1p q1 1p q1 (else) Now f = / f snce acts on N as multplcaton by /. 36

37 Now we consder the somorphsm f T (B) f T (fbf). K fk f We have f = 1/ E E snce one can carry the E across. p 1p p1 1p The somorphsm thus sends f to j j 1/ E ( E ) E ( 1) E = 1/ f. jpqk 1p k q1 1q k p1 Thus the relaton ( - )f s sent to 1/ ( f - f ) COROLLARY. The McKay quver s the double of a quver Q, and there s an somorphsm f T(V)# / ( - ) f (Q) sendng f to e. Thus T(V )# / ( - ) s Morta equvalent to (Q). PROOF. fbf has a symplectc form whch respects the decomposton fbf = f Bf. j j t follows that you can choose a bass {a,a } wth each a belongng to some f Bf and a to f Bf n such a way that = a a - a a. j j a Let Q be the quver wth arrows the a. nce dm f Bf = dm Hom (K f,v K f ) = dm Hom (V N,N ), j K j K j t follows that the double of Q s the McKay quver NOTE. For a dscusson about assocated graded rngs see J.C.McConnell and J.C.Robson, Noncommutatve noetheran rngs, 1.6. Theorem 4.12 and Corollary 4.13 are new. 37

38 Lecture 5. The Klenan case ETUP. Let be a fnte subgroup of L (K) actng on V = K. t 2 preserves the symplectc form (x,y) = x y - x y. t acts freely on V\{}. An element Z(K ) gves rse to rngs and O. The Gelfand-Krllov dmenson of a f.g. K-algebra s denoted GK A. More generally GK dmenson s defned for f.g. A-modules. The GK dmenson of gr A s equal to the GK dmenson of A. Moreover, for commutatve rngs, the GK dmenson s equal to the usual Krull dmenson. t follows easly that and O have GK dmenson 2. The McKay quver s equal to Q for some Q. Moreover f f (Q) (so that and (Q) are Morta equvalent) snce when consderng the generators - - (, ) of the deal defnng, because V s 2-dmensonal there s only one element here, and t s essentally -. Thus also O = e e e (Q)e LEMMA (McKay!). Q s an extended Dynkn quver. The element defned by = dm N s the radcal generator for Q. PROOF. nce dm V = 2, t follows that (, ) = for all. Now = 1 snce N s the trval module LEMMA. / e s Morta equvalent to (Q ) where Q s the Dynkn part of Q and s the restrcton of. These algebras are f.d., and are zero f and only f for all Dynkn roots (e roots wth =). PROOF. / e s Morta equvalent to (Q)/ (Q)e (Q) (Q ). 38

39 Now / e s fnte dmensonal, for t s fltered, and the assocated graded algebra s a quotent of / e, whch s f.d. by Lemma Fnally a f.d. algebra s zero f and only f t has no f.d. modules, so 2.6 apples ORDER AND REFLEXVE MODULE. A prme Golde rng A has a smple artnan quotent rng Q, and then A s an order n Q, meanng that every -1-1 q Q can be wrtten as as and as t b wth a,b,s,t A, s,t unts n Q. An order s sad to be maxmal f A B and xby A for some unts x,y Q, mply that A=B. A commutatve ntegral doman s a maxmal order f and only f t s ntegrally closed. f A s an order n Q, smple artnan, and e A s a non-zero dempotent, then you can dentfy End (ea) = {q Q : eaq ea}, for eae End (ea) End (eq) End (eq) Q eae eae eqe 1 s 1-1. t comes from the fact that ea Q eq. A 2 s equalty snce by general theory eae s an order n eqe. 3 the homothety s an somorphsm snce Q s smple artnan. f n addton A s a maxmal order then A End (ea): eae We have A {q Q : eaq ea} Q. Now QeQ = Q, so AeA contans a unt s of Q. Then eaq ea sq A. Thus maxmalty mples A = {q Q : eaq ea} Also ea s a reflexve eae-module. (Recall that M s a reflexve R-module f M Hom (Hom (M,R),R). Namely, there are somorphsms R R Ae Hom (ea,ea)e Hom (ea,eae) eae eae ea e Hom (Ae,Ae) Hom (Ae,eAe). eae eae f A s a maxmal order then so s A#. Thus K[V]# s a maxmal order. Also O K[V] s ntegrally closed, so t too s a maxmal order. 39