DERIVED CATEGORIES AND LIE ALGEBRAS 0 1 JIE XIAO,FANXUANDGUANGLIANZHANG 0 2 Dedicated to Professor George Lusztig n a J 7 2 Abstract. Let Db(A) be the derived category of a finite dimensional basic algebra A with finite global dimension. We construct the Lie algebra arising ] from the 2-periodic version K2(P(A)) of Kb(P(A)) in term of constructible A functions onvarietiesattached toK2(P(A)). Q . h 1. Introduction t a 1.1. Inthe lastthirty yearsofthe twentiethcentury,thereweretwoparallelfields m in mathematics got extensively developed. One is the infinite dimensional Lie the- [ ory,inparticular,the Kac-MoodyLiealgebras. Oneisthe representationtheoryof 2 finite dimensional algebras, in particular, the representations of quivers. The close v relation between the two subjects was discovered in a very early stage. Gabriel in 4 [G] found that the quivers of finite representation type were given by the Dynkin 6 graphsinLietheory,andthedimensionvectorsprovidethebijectivecorrespondence 5 4 between the isomorphism classes of indecomposable representations of the quiver 0 and the positive root system of the semisimple Lie algebra. After the Gabriel the- 6 orem,a lot of progresson the connectionbetween the representationsof quiversor 0 hereditary algebras and Lie algebras had been made, for example, by Bernstein- / h Gelfand-Ponomarev[BGP]andDlab-Ringel[DR]. Thefinalandmostgeneralresult t is the Kac theorem [K1] which extends to consider the quiver and the symmetric a m Kac-Moodyalgebraof arbitrarytype. It states that the dimension vectors of inde- composablerepresentationsareexactlythepositiveroots,auniqueindecomposable : v correspondsto each realroot and infinitely many to eachimaginary root; the mul- i X tiplicity of the imaginary root, which is conjectured by Kac in [K2], is given by a geometric parameter in terms of representations of the quiver. A new progress on r a the Kac conjecture is by Crawley-Boevey and Van den Bergh in [CBV]. 1.2. Ringel in [R2] discovered his Hall algebra structure by giving an answer to the following fundamental question: how to recover the underlying Lie algebra structure directly from the category of representations of the quiver. Let Q be a quiver, A=F Q the path algebra of Q over F : the finite field with q q q elements. Set = isoclasses of representations of Q .Foranyα chooseV α P { } ∈P TheresearchwassupportedinpartbyNSFofChinaandbythe973ProjectoftheMinistry ofScienceandTechnology ofChina. 2000 Mathematics Subject Classification. Primary 18E30, 17B37, 16G10; Secondary 16G20, 14L30,17B67. Key words and phrases. Derived category, orbit space, constructible function, Lie algebra, Kac- Moody. 1 2 JIEXIAO,FANXUANDGUANGLIANZHANG to be a representative in the class α. Given three classes λ, α, β , let gλ be ∈ P αβ the order of the finite set {W (cid:1)Vλ|W ∼= Vβ,Vλ/W ∼= Vα}. By taking v = √q and the integral domain Q(v), the (twisted) Ringel-Hall algebra ∗(A) can be defined H to be a free Q(v)-module with basis u λ and multiplication is given by λ { | ∈P} u u =vhα,βi gλ u for all α,β . α∗ β αβ λ ∈P λX∈P One may consider the subalgebra of ∗(A) generated by u = u , for i I(= H i αi ∈ Q ) where α is the isoclass of simple A-module at vertex i. The subalgebra is 0 i ∈P called the composition algebra and it is denoted by ∗(A). On the other hand, the C indexsetI ofsimpleA-modulestogetherwiththesymmetricEulerform(I,( , )) − − ofAisaCartandatuminthesenseofLusztig[L4]. ForaCartandatum(I,( , )), − − the quantizedenvelopingalgebra defined by Drinfeld [Dr]andJimbo [J]is asso- q U ciated with it. The positive part + is generated by E , i I with subject to the Uq i ∈ quantum Serre relations. Thereisausualwaytodefinethegenericform ∗(Q)ofthecompositionalgebra C ∗(A) by considering the representations of Q over infinitely many finite fields. C Then ∗(Q) is a Q(v)-algebra where v becomes a transcendental element over Q. Put u(C∗n) = u∗in for i I and n N and let ∗(Q) be the integral form of i [n]i! ∈ ∈ C Z ∗(Q), which is generated by u(∗n), i I, n N over the integral domain = C i ∈ ∈ Z Z[v,v−1].Alsothequantumgroup + hastheintegralform +,whichisgenerated Uq UZ by E(n), i I, n N over . Then by Ringel [R1] and Green [Gr], the canonical i ∈ ∈ Z map ∗(Q) + by sending u(∗n) to E(n) for i I and n N leads to a C Z → UZ i i ∈ ∈ -algebra isomorphism, if the two algebras share a common Cartan datum. Z Let ind = isoclasses of indecomposable representations of Q . Then gλ can P { } αβ be regarded as a function on q for α,β,λ ind . In fact, Ringel in [R2] proved ∈ P that gλ is an integral polynomial on q when Q is of finite type. One can take αβ the integral value gλ (1) by letting that q tends to 1. Ringel [R2], for Q of finite αβ type, prove that the u ,α ind , spanned a Lie subalgebra of ∗(Q) with α q=1 ∈ P C | Lie bracket [u ,u ]= (gλ (1) gλ (1))u α β αβ − βα λ λ∈XindP forα,β ind . This realizedthe positive partn+ ofthe semisimple Lie algebrag. ∈ P Of course, Ringel’s approach also works for Q of arbitrary type. In general, there existsthegenericcompositionLiesubalgebra of ∗(Q) generatedbyu ,i I q=1 i L C | ∈ andind isnolongertoindexabasisof .Now iscanonicallyisomorphictothe P L L positivepartn+ ofthesymmetricKac-MoodyLiealgebrag.Forarealizationofthe whole g, not just its positive part, Peng and Xiao in [PX3] have constructed a Lie algebrafromatriangulatedcategorywiththe2-periodicshiftfunctorT,i.e,T2 =1. If,specially,considerthe2-periodicorbitcategoryofthederivedcategoryofafinite dimensional hereditary algebra, the Lie algebra obtained in [PX3] gives rise to the globalrealizationofsymmetrizableKac-Moodyalgebraofarbitrarytype. In[PX3] they consider the triangulated categories over finite fields. Replacing counting the order of the filtration set, they calculate the order of the orbit space of a triangle. By a hard work, they obtained a Lie ring g over Z/(q 1) for the prime (q−1) − powers q = F . Then they performed their work over finite field extensions of q | | DERIVED CATEGORIES AND LIE ALGEBRAS 3 arbitrarily large order and construct a generic Lie algebra which is similar to the generic composition Lie subalgebra done by Ringel in [R3]. A transcendental Lie algebra was finally obtained. 1.3. QuicklyaftertheworkofRingel[R1],peoplerealizedthatageometricsetting ofRingel-Hallalgebrais possible by using the convolutionmultiplication (see [Sch] and [L1]). Let Q be a quiver and α= a i N[I] a dimension vector. We fix i∈I i ∈ a I-graded space Cα =(Cai)i∈I. ThenP Eα = HomC(Cas(h),Cat(h)) M h:s(h)→t(h) is an affine space. Set G =Π GL(a ,C). α i∈I i Forany(x ) E andg =(g ) G , wedefine theactiong (x )=(g x g−1 ). h ∈ α i ∈ α · h t(h) h s(h) For any Q-representationM with dimM =α, let E be the G -orbit of M. M α α O ⊂ For an algebraic variety X overC, a subset A of X is said to be constructible if it is a finite union of locally closed subsets. A function f :X C is constructible → if it is a finite C-linear combinationof characteristicfunctions 1 for constructible O subsets . O WedefineM (Q)tobethespaceofconstructibleG -invariantfunctionsE Gα α α → C, andlet M (Q)= M (Q). Let indE (Q) to be the constructible subset G α∈NI Gα α ofEαconsistingofallLpointsxwhichcorrespondtoindecomposableQ-representations, and let indM (Q) to be the space of constructible G -invariant functions over Gα α indE . We may regard as indM (Q) = f M (Q)suppf indE , and α Gα { ∈ Gα | ⊆ α} indM (Q) = indM (Q), where, by Kac theorem, R+ is the positive root G ⊕α∈R+ Gα system of the Kac-Moody Lie algebra corresponding to Q. The space M (Q) = G M (Q) cabbe endowedwith the associativealgebrastructure by the con- α∈NI Gα vLolution multiplication: 1 1 (y)=χ( y ) O1 ∗ O2 FO1O2 for any G -invariant constructible set and G -invariant constructible set α 1 β 2 with α,β NI, where y = x O M(x) M(y) and M(y)/M(x) O ∈ FO1O2 { ∈ O2 | ⊆ ∈ O1} and χ(X) denotes the Euler characteristic of the topological space X. As in [Rie] and [DXX], it can be proved that the space indM (Q) has a Lie algebra structure G under the usual Lie bracket [1 ,1 ]=1 1 1 1 . O1 O2 O1 ∗ O2 − O2 ∗ O1 Applying this setup to the case Q being a tame quiver, Frenkel-Malkin-Vybornov [FMV] gave an explicit realization of the positive parts of affine Lie algebras. 1.4. The great progress is made by Lusztig, who apply the Hall algebras in a geometric setting to study the quantum groups (see [L1] and [L2]) and the en- veloping algebras (see [L5]). The canonical bases of the quantum groups and the semicanonicalbasesoftheenvelopingalgebraswereoriginallyconstructedinterms of representations of quivers. However Lusztig [L2] has pointed out that a more suitable choice is the preprojective algebra, which is given by the double quiver of Q with the Gelfand-Ponomarev relations. Further progress in this direction is the studyofNakajima[N]onhisquivervarieties,whichleadstoageometricrealization of the representation theory of Kac-Moody algebras. 4 JIEXIAO,FANXUANDGUANGLIANZHANG Inspired by Ringel’s work on Hall algebras and Lusztig’s geometric approach to quantumgroups,the aimofthis paperis to givea globalandgeometricrealization of the Lie algebras arising from the derived categories,which is a generalization of our earlier work [PX3]. 1.5. If we consider the module category of A = CQ/J, we have the algebraic va- riety E (Q,R) for A-modules with a fixed dimension vector d and it is a G-variety d where G = G (Q) is a reductive group. According to the work of C.de Concini d and E.Strickland in [CS] and M.Saorin and B.Huisgen-Zimmermann in [SHZ], this geometry can be generalized to over the chain complexes of A-modules. Section 2 is devoted to do this. Let K ( b(A)) be the Grothendieck group of b(A) and 0 D D dim be the canonical map from the abelian group of dimension vector sequences to K ( b(A)). Given d K ( b(A)) and d dim−1(d), the set b(A,d) of 0 0 D ∈ D ∈ C all complexes of A-modules with the dimension vector sequences d and its subset b(A,d) of all projective complexes can be endowed with the affine variety struc- P tures. Let b(A,d)(resp. b(A,d))bethedirectlimitof b(A,d)(resp. b(A,d)) C P C P for d dim−1(d). Here, we associate to b( (A)) its quotient space b(A,d) ∈ K P QP which is the direct limit of the quotient spaces b(A,d)) of b(A,d) under the QP P action of some algebraic group Gd. Our main aims in Section 2 are to study the relation between b(A,d) and b(A,d), the action of derived equivalence on Q QP b(A,d) and b(A,d) and characterize the orbit space b(A,d) of b(A,d) Q QP QP P under the actionofthe directlimit Gd ofalgebraicgroupsGd. Thereforethe main point in Section 2 is that, we can regardthe Gd-invariant geometry in b(A,d) as P the moduli space in which the orbits index the isomorphism classes of objects in the derived category. In Section 3, we consider the inverse limit of the C-space of Ge-invariant con- structiblefunctionsover b(A,e)foranye dim−1(d). Anyelementintheinverse P ∈ limitcanbe viewedas aGd-invariantconstructiblefunction over b(A,d). We de- P finetheconvolutionbetweenGd-invariantconstructiblefunctions. Ourmainresult in Section 3 is that our convolution rule is well-defined. In order to prove it, we need to define the na¨ıve Euler characteristic of the orbit spaces induced by trian- glesinthetriangulatedcategoryasin[Jo1,Section4.3]. ThetheoremofRosenlicht [Ro] for the algebraic group action on varieties is crucial for us. Section 4 is just to transfer the results in Section 3 to the 2-periodic orbit categories of the derived categories. Section 5 is devoted to verifying the Jacobi identity. In [PX3], by counting the Hall numbers FL for the triangles of the form X L Y X[1], it has XY → → → been provedthat the Jacobiidentity can be deduced from the octahedralaxiom of the triangulated categories. However we need to prove that the correspondences among the various orbit spaces in the derived categories are actually given by the algebraicmorphismsofalgebraicvarieties. Wethinkthisgeometricmethodismore transparent to reflect the hidden symmetry in the derived category. Additionally, wegetthetwopropertieswhichisunknownin[PX3]. Firstlytheproperassumption in[PX3]isnotnecessary,infact,itiseasytogiveexamplessuchthatdimX =0for somenonzeroindecomposableX in b(A). Secondlyweshowthatthe Lie algebras D arisingfromthe 2-periodicorbitcategoriesofthe derivedcategoriesalwayspossess the symmetric invariant form in the sense of Kac [K3], which is essentially non- degenerated. Section6istoapplytheconstructiontothe2-periodicorbitcategories of the derived categories of representations of quivers, particularly, tame quivers. DERIVED CATEGORIES AND LIE ALGEBRAS 5 This gives rise to a global realization of the symmetric (generalized) Kac-Moody algebras of arbitrary type. In particular, an explicit realization of the affine Lie algebras. 1.6. FinallyweshouldmentionrecentadvancesbyTo¨en[T]andJoyce[Jo2]. To¨en defined an associative algebra, called the derived Hall algebra associated to a dg category over a finite field. A direct proof for To¨en’s theorem is given in [XX]. Joyceconsideredanew Ringel-Halltype algebraconsistingoffunctions overstacks associated to abelian categories. Their results can be viewed as improvements of the Ringel-Hall type algebra with respect to categorification and geometrization. However, it is unknown how to define an analogue of the derived Hall over the complex field (see [L6] [N]) or an analogue of the derived algebra for the 2-period versionof a derivedcategory(see [T] and [XX]). Hence, it is still an open question to define an associative multiplication which induces the Lie bracket in this paper and supplies the realization of the corresponding enveloping algebra. 2. Topological spaces attached to derived categories 2.1. Module varieties. GivenanassociativealgebraAoverthe complex fieldC, in this paper, we always assume that A is both finite dimensional and finite global dimensional. ByaresultofP.Gabriel([G])thealgebraAisgivenbyaquiverQwith relations R (up to Morita equivalence). Let Q = (Q ,Q ,s,t) be a quiver, where 0 1 Q and Q are the sets of vertices and arrows, respectively, and s,t : Q Q 0 1 1 0 → are maps such that any arrow α starts at s(α) and terminates at t(α). For any dimension vector d=(d ) , we consider the affine space over C i i∈Q0 Ed(Q)= HomC(Cds(α),Cdt(α)) αM∈Q1 Any element x = (x ) in E (Q) defines a representation (Cd,x) where Cd = α α∈Q1 d i∈Q0Cdi. A relation in Q is a linear combination ri=1λipi, where λi ∈ C and Lpi are paths of length at least two with s(pi) = s(Ppj) and t(pi) = t(pj) for all 1 i,j r. For any x=(x ) E and any path p=α α α in Q we set x≤= x ≤x x . Thenαxαsa∈tQis1fi∈es adrelation r λ p if1 2r···λmx = 0. If R p α1 α2··· αm i=1 i i i=1 i pi isasetofrelationsinQ,thenletEd(Q,R)be thePclosedsubvaPrietyofEd(Q)which consists of all elements satisfying all relations in R. Any element x = (x ) α α∈Q1 in E (Q,R) defines in a natural way a representation M(x) of A = CQ/J with d dimM(x) = d, where J is the admissible ideal generated by R. We consider the algebraic group G (Q)= GL(d ,C), d i ı∈YQ0 which acts on E (Q) by (x )g = (g x g−1 ) for g G and (x ) E . It d α t(α) α s(α) ∈ d α ∈ d naturally induces the action of G (Q) on E (Q,R). The induced orbit space is d d denoted by E (Q,R)/G (Q). There is a natural bijection between the set (A,d) d d M ofisomorphismclassesofC-representationsofAwithdimensionvectordandtheset oforbitsofG (Q)inE (Q,R).So wemayidentify (A,d)withE (Q,R)/G (Q). d d d d M 6 JIEXIAO,FANXUANDGUANGLIANZHANG 2.2. Categoriesofcomplexes. Firstweconsiderthecategoryofcomplexes (A). C Its objectsaresequencesM• =(M ,∂ ) offinite dimensionalA-modules andtheir n n homomorphisms (2.1) ... ∂n−1 M ∂n M ∂n+1 M ∂n+2 ... n n+1 n+2 −−−−→ −−−−→ −−−−→ −−−−→ suchthat∂ ∂ =0foralln. Amorphismφ• :M• M′•betweentwocomplexes n+1 n → is a sequence of homomorphismsφ• =(φ :M M′) suchthat the following n n → n n∈Z diagram is commutative. ... ∂n−1 M ∂n M ∂n+1 M ∂n+2 ... n n+1 n+2 −−−−→ −−−−→ −−−−→ −−−−→ (2.2) φn φn+1 φn+2    ... ∂n′−1 My′ ∂n′ My′ ∂n′+1 My′ ∂n′+2 ... −−−−→ n −−−−→ n+1 −−−−→ n+2 −−−−→ One says that such a morphism is homotopic to zero if there are homomorphisms σ : M M′ such that φ = σ ∂ +∂′ σ for all n Z. The factor n n → n−1 n n+1 n n−1 n ∈ category (A) of (A) modulo the ideal of morphisms homotopic to zero is called K C the homotopic category of A-modules. For each n the n-th homology of a complex is defined as H (M•) = Ker∂ /Im∂ . Obviously, a morphism φ• of complexes n n n−1 induces homomorphisms of homologies H (φ•) : H (M•) H (M′•) and if φ• n n n → is homotopic to zero, it induces zero homomorphisms of homologies. One call a morphism φ• in (A) or in (A) quasi-isomorphism if the induced morphisms C K H (φ•)areisomorphismsforall n.Now the derivedcategory (A) isdefinedto be n D the categoryof fractions (A)[ −1], where is the set of allquasi-isomorphisms, K N N whichisobtainedfrom (A)by inversingallmorphismsin . Onecallsacomplex K N right bounded (left bounded, bounded, respectively) if there is n such that M = 0 n 0 for n > n ( there is n such that M = 0 for n < n , or there are both, 0 1 n 1 respectively). The corresponding categories are denoted by −(A), −(A), −(A) C K D (by +(A), +(A), +(A), orby b(A), b(A), b(A), respectively). Inthis paper C K D C K D we mainly deal with the bounded situation. ThecategoryA-modoffinitedimensionalA-modulescanbenaturallyembedded into (A) (even in b(A)): a module M is identified with the complex M• such D D that M =M and M =0 for n=0. 0 n 6 A complex P• =(P ,∂ ) is called projective if all P are projective A-modules. n n n Since the category A-mod has enough projective objects, one can replace, when consideringrightboundedhomotopicandderivedcategory,arbitrarycomplexesby projectiveones. Wedenoteby −(A)andby b(A)thefullsubcategoriesof −(A) P P C and b(A) which consist of right bounded and bounded projective complexes, re- C spectively. Actually, we have −(A) −( −(A)) −(A)/ , where is the D ≃ K P ≃ P I I ideal of morphisms homotopic to zero (see[GM]). Moreover, every finite dimen- sional A-module M has a projective cover, i.e., an epimorphism p : P(M) M M → suchthatP(M)isprojectiveandKerp radP(M),theradicalofP(M). There- M ⊆ fore, we can only consider minimal or radical projective complexes P• =(P ,∂ ) n n with the property: P is projective and Im∂ radP for all n. Let rad −(A) n n n+1 ⊆ P be the full subcategory of −(A) which consist of minimal projective complexes. P Since every projective complex in −(A) is quasi-isomorphic to a minimal projec- P tive complexes,we have −(A) rad −(A)/ , where is the ideal ofmorphisms D ≃ P I I homotopic to zero. One immediately checks that a morphism φ• between minimal DERIVED CATEGORIES AND LIE ALGEBRAS 7 projective complexes induces an isomorphism in −(A) if and only if φ• itself is D an isomorphism in rad −(A). If we further assume that the global dimension of P A is finite, then we have b(A) rad b(A)/ , since any bounded complex has a D ≃ P I bounded projective resolution. 2.3. Complexvarieties. ThegeometrizationofA-moduleswithdimensionvector d can be carried over, in the same spirit, to the complexes. Let the algebra A = CQ/J be of finite global dimension and the admissible ideal J is given by a set R of relations in Q. For a dimension vector d we understand as d : Q N. We 0 → set the Q -gradedC-space Cd = Cd(j). For a sequence of dimension vectors 0 j∈Q0 d=( ,d ,d ,d , )withonLlyfinitemanynon-zeroentries,wedefine b(A,d) ··· −1 0 1 ··· C to be the closed subset of (see [SHZ]) Ed (Q,R) HomC(Vdi,Vdi+1) i × iY∈Z Yi∈Z which consists of elements (xi,∂i)i, where xi Ed (Q,R) and M(xi)=(Cdi,xi) is ∈ i the corresponding A-module and ∂i HomC(Cdi,Cdi+1) is a A-module homomor- ∈ phismfromM(x )toM(x )withtheproperty∂ ∂ =0.Infact,(M(x ),∂ ) ,or i i+1 i+1 i i i i simplydenotedby(x ,∂ ) ,isacomplexofA-modulesanddiscalleditsdimension i i i vector sequence. The group Gd := i∈ZGdi(Q) acts on Cb(A,d) via the conjugation action Q (g ) (x ,∂ ) =((x )gi,g ∂ g−1) i i i i i i i+1 i i i wheretheaction(xi)gi wasdefinedasinSection2.1. Thereforetheorbitsunderthe actioncorrespondbijectivelytotheisomorphismclassesofcomplexesofA-modules. We fix a set P ,P , ,P to be a complete set of indecomposable projective 1 2 l ··· A-modules(uptoisomorphism). Let b(A)be the fullsubcategoryof b(A) which P C consistsofprojectivecomplexesP• =(Pi,∂ )suchthateachPi hasthedecomposi- i tionPi = l eiP . We denote by e(Pi) the vector (ei,ei, ,ei). The sequence ∼ j=1 j j 1 2 ··· l ( ,e(P−L1),e(P0),e(P1), ), denoted by e(P•), is called the projective dimen- ··· ··· sion sequence of P•. Put d(e)=(d ), where d =dimPi,. In the similar way as in i i [JSZ],forafixedprojectivedimensionsequencee=( ,e , ),wedefine b(A,e) ··· i ··· P to be the locally closed subset of b(A,d(e)) consisting of (xi,∂i)i with (Cdi,xi) C isomorphictoPiforanyi∈Z.TheactionofthealgebraicgroupGd(e) := i∈ZGdi on b(A,d(e)) induces an action on b(A,e). Q C P 2.4. In this subsection we consider the topological structures which are endowed with b(A) and b(A). C P Let K ( b(A)), or simply by K , be the Grothendieck group of the derived 0 0 D category b(A),anddim: b(A) K ( b(A))thecanonicalsurjection. Itinduces 0 D D → D acanonicalsurjectionfromthe abeliangroupofdimensionvectorsequencesto K , 0 we still denote it by dim. We have known that the set b(A,d) of all complex of C fixed dimension vector sequence d in b(A) is an affine variety. For any d ,d C 1 2 ∈ dim−1(d), wewrited d ,ifthereexistsacomplexM•(d ,d )in b(A,d d ) 1 ≤ 2 1 2 C 2− 1 which is a direct sum of shifted copies of complexes of the form // // 1 // // // 0 S S 0 ··· ··· 8 JIEXIAO,FANXUANDGUANGLIANZHANG where S is a simple A-module. This defines a partial order on dim−1(d). Fix the set M•(d ,d ) d ,d dim−1(d),M•(d ,d ) M•(d ,d )=M•(d ,d ) . { 1 2 | 1 2 ∈ 1 2 ⊕ 2 3 1 3 } We have a morphism of varieties : Td1d2 :Cb(A,d1)→Cb(A,d2) mapping a complex X• to X• M•(d ,d ). Then we obtain a direct system ⊕ 1 2 ( b(A,d),Tdd′) d,d′ dim−1(d) and define { C | ∈ } b(A,d)= lim b(A,d) C C d∈di−m→−1(d) for d K0. We have a canonical morphism Td : b(A,d) b(A,d) for any ∈ C → C d dim−1(d). A subset U is open in b(A,d) if and only if T−1(U) is open in d ∈ C b(A,d) for any d dim−1(d). C ∈ Moreover,wealsodefinethequotientspace b(A,d)= b(A,d)/ ,wherex y Q C ∼ ∼ if and only if the correspondingcomplexes M(x)• and M(y)• are quasi-isomorphic to each other, i.e., they are isomorphic in b(A). The topology of b(A,d) is D Q quotient topology, i.e., let π : b(A,d) b(A,d) be the canonical surjection, U C → Q is an open (closed) set of b(A,d) if and only if π−1(U) is an open (closed) set of Q b(A,d). LetS beasubsetof b(A,d). Itsorbitspaceisdefinedas (S)= y y C C O { | ∼ x for some x S . ∈ } AcomplexX• =(X ,∂ ) iscalledcontractibleiftheinducedhomologicalgroups i i i H (X•) = ker∂ /Im∂ = 0 for all i Z. It is easy to see that any contractible i i+1 i ∈ projective complex is isomorphic to a direct sum of shifted copies of complexes of the form 0 f 0 ... 0 P P 0 ... −−−−→ −−−−→ −−−−→ −−−−→ −−−−→ with P is a projective A-module and f is an automorphism. We call an element x b(A,e) contractible if the corresponding projective complex is contractible. ∈ P Let e and e be projective dimension sequence such that d(e ) and d(e ) are 1 2 1 2 in dim−1(d). We write e e if there exists a contractible projective complex 1 ≤ 2 P•(e ,e ) b(A,e e ). We fix the set 1 2 ∈P 2− 1 P•(e ,e ) P•(e ,e ) P•(e ,e )=P•(e ,e ),d(e ),d(e ),d(e ) dim−1(d) { 1 2 | 1 2 ⊕ 2 3 1 3 1 2 3 ∈ } of contractible projective complexes. Then we have a canonical morphism of vari- eties te1e2 :Pb(A,e1)→Pb(A,e2) mapping X• to X• P•(e ,e ). Hence, we can define ⊕ 1 2 b(A,d)= lim b(A,e) P P d(e)∈d−→im−1(d) for any d K0. By definition, there are canonical morphisms te : b(A,e) ∈ P → b(A,d). We have the quotient space P b(A,d)= b(A,d)/ , QP P ∼ wherex yin b(A,d)ifandonlyifthecorrespondingprojectivecomplexesP(x)• ∼ P and P(y)• are quasi-isomorphic, i.e., they are isomorphic in b(A). The topology D DERIVED CATEGORIES AND LIE ALGEBRAS 9 for b(A,d) is the quotient topology from b(A,d). For any x b(A,d), then QP P ∈P the orbit is (x)= y b(A,d)y x . O { ∈P | ∼ } For e1 ≤e2, there exist a natural morphism of varieties fe1e2 :Gd(e1) →Gd(e2) g 0 mapping g =(g ) to i . Then we define i (cid:18) 0 1 (cid:19) Gd = lim Gd(e). d(e)∈d−→im−1(d) By definition, there are canonical morphisms fe : Ge Gd. The action of the → algebraic group Gd(e) on b(A,e) naturally induces an action of Gd on b(A,d). P P Letge Ge andxe′ b(A,e′).Thenthereexistse′′ suchthate e′′ande′ e′′. ∈ ∈P ≤ ≤ We define fe(ge).te′(xe′)=fe′′(fee′′(ge).te′e′′(xe′)). It is well-defined. We denote by b(A,e) and b(A,d) the orbit spaces of 1 QP Q P b(A,e) and b(A,d) under the actions of Gd(e) and Gd, respectively. We have P P the follow result. Proposition 2.1. With the above notations, we have b(A,d)= b(A,d)= lim b(A,e). 1 QP Q P QP d(e)∈−d→im−1(d) The proposition is the direct corollary of the following three lemmas (see [BD] or [JSZ]). Lemma 2.2. In b(A), any projective complex can be uniquely decomposed (up to P isomorphism) intothedirectsumofaminimalprojectivecomplexandacontractible projective complex. Lemma 2.3. Let f• : P• Q• be a morphism between two minimal projective → complex in b(A). Then f• is a quasi-isomorphism if and only if f• is an isomor- P phism. Lemma 2.4. Let f• : P• Q• be a morphism in b(A) and e(P•) = e(Q•). → P Then f• is a quasi-isomorphism if and only if f• is an isomorphism. 2.5. The aim of this subsection is to build the connection between b(A,d) and Q b(A,d). QP Lemma 2.5. Suppose the following diagram is a pullback of A-module, and g is 1 surjective, X f1 Y −−−−→ f2 g1   Zy g2 Wy −−−−→ then we have the following properties hold: (1) Kerf =Kerg ; 1 ∼ 2 (2) Y = W ; Imf1 ∼ Img2 (3) there exists the exact sequence: 0 X Y Z W 0. −→ −→ ⊕ −→ −→ 10 JIEXIAO,FANXUANDGUANGLIANZHANG Proof. The first and third statement just follow the definition of pullback, refer to [ARS]. For the second statement, we use the first statement again to get an isomorphism:Kerf =Kerg . By this, the conclusion follows. (cid:3) 2∼ 1 Foranydimensionvectorsequenced=(d ) ,weconstructforanycomplexM•in i i∈Z b(A,d)aprojectivecomplexF• suchthatF• is quasi-isomorphicto M•. Assume C dim A=n and gl.dim.A=m. C Let M• be a complex with the following form: (2.3) 0 M ∂2 ... ∂r−1 M ∂r M 0 1 r−1 r −−−−→ −−−−→ −−−−→ −−−−→ −−−−→ which dim M =d for any i Z. Here, if d =(dj) , then d = dj. C i i ∈ i i j∈Q0 i j∈Q0 i SinceMr hasdimensiondr,wehavethesurjectivemap: πr :Adr P Mr. Along −→ the differential ∂ and the above π , we form the pullback X : r r r−1 ... Mr−2 Xr−1 ∂ˆr Adr 0 −−−−→ −−−−→ −−−−→ −−−−→ (2.4) id πˆr πr    ... My My ∂r My 0 r−2 r−1 r −−−−→ −−−−→ −−−−→ −−−−→ Depending on Lemma 2.6, we have the following exact sequence: 0 X Adr M M 0 r−1 r−1 r −→ −→ ⊕ −→ −→ The dimension of X , denoted by l , is d +d (n 1). Similarly, we have r−1 r−1 r−1 r − the surjective map: π˜r : Alr−1 Xr−1, and πˆr is also surjective by the lemma. −→ So we can also form the pullback X as showed in the following diagram: r−2 (2.5) ... //Mr−3 //Xr−2 ∂ˆr−1 //Alr−1 //Alr //0 πˆr−1 π˜r (cid:15)(cid:15) (cid:15)(cid:15) ... //Mr−3 // Mr−2 // Xr−1 ∂ˆr //Adr //0 πˆr πr (cid:15)(cid:15) (cid:15)(cid:15) ... //Mr−3 ∂r−2 // Mr−2 ∂r−1 // Mr−1 ∂r //Mr //0 Inductively, we get a complex of ‘almost’ free A-module F• as follows: (2.6) 0 P ... Al1 ∂ˆ2π˜2 ... Alr 0 −−−−→ −−−−→ −−−−→ −−−−→ −−−−→ −−−−→ where l +l = nl +d for i = 2 m, ,r, in particular, l = d and P is i i−1 i i−1 r r − ··· a projective module of dimension nl l . Every term of this complex is a 2−m 2−m − free A-module except the first term. By the construction, there exists a projective dimension sequence e only depending on the choice of dimension vector sequence d such that F• b(A,e). Moreover, This complex is quasi-isomorphism to M•. ∈ P First, H (F•)= Alr = Alr = Mr r Im∂ˆrπ˜r Im∂ˆr ∼ Im∂r