aa r X i v : . [ m a t h . R T ] J un Poisson automorphisms and quivermoduli
Markus ReinekeFachbereich C - MathematikBergische Universit¨at WuppertalD - 42097 Wuppertal, Germanye-mail: [email protected]
Abstract
A factorization formula for certain automorphisms of a Poisson alge-bra associated with a quiver is proved, which involves framed versionsof moduli spaces of quiver representations. This factorization formula isrelated to wall-crossing formulas for Donaldson-Thomas type invariantsof M. Kontsevich and Y. Soibelman [4].
In [4], a framework for the definition of Donaldson-Thomas type invariants forCalabi-Yau categories endowed with a stability structure is developed. One ofthe key features of this setup is a wall-crossing formula for these invariants,describing their behaviour under a change of stability structure in terms of afactorization formula for automorphisms of certain Poisson algebras defined us-ing the Euler form of the category.In this paper, we study such factorization formulas using quiver representations,their moduli spaces, and Hall algebras (for different such approaches, see [1, 3]).With a quiver without oriented cycles, we associate a Poisson structure on aformal power series ring and study factorizations of a certain automorphisminto infinite ordered products associated with stabilities for the quiver (see sec-tion 2 for precise definitions). Our main result, Theorem 2.1, describes thesefactorizations in terms of generating series for Euler characteristics of framedversions of moduli spaces of representations of the quiver, called smooth modelsin [2]. This result yields a weak form of an integrality conjecture for Donaldson-Thomas type invariants of [4].We approach this theorem by producing the desired factorizations in the Hallalgebra of the quiver, in terms of the Harder-Narasimhan recursion of [6]; sec-tion 4 therefore adapts some of the material of [6] to the present setup. Using1n evaluation map, simple identities in the Hall algebra often lead to inter-esting identities in a skew formal power series ring associated with the quiver(see [7] for some instances of this principle). In the present setup, the latterring is viewed as a quantization of the Poisson algebra; we develop a simplealgebraic setup for constructing automorphisms of the Poisson algebra from aclass of formal series in section 3. The key feature is that certain series in theskew formal power series ring induce non-trivial automorphisms of the Poissonalgebra via conjugation. That this property applies to the generating seriesinduced from the Harder-Narasimhan recursion follows from a formula for theBetti numbers of smooth models of [2]; this is explained in section 5. We endwith two classes of examples in section 6: considering generalized Kroneckerquivers, we can prove a weak form of the integrality conjecture of [4]. We alsorelate the factorization formula Theorem 2.1 to a factorization formula of [3] interms of Gromov-Witten theory. Finally, we consider Dynkin quivers, for whichthe relevant moduli spaces are trivial.
Acknowledgments:
I would like to thank J. Alev, T. Bridgeland, B. Keller,S. Mozgovoy, R. Pandharipande, B. Siebert, Y. Soibelman and V. Toledano-Laredo for interesting discussions concerning the material in this paper. I amindebted to Y. Soibelman for providing me with a preliminary version of [4],and for pointing out a gap in an earlier version of this paper.
Let Q be a finite quiver, given by a finite set of vertices I and a finite set ofarrows, written as α : i → j for i, j ∈ I . We assume throughout the paper that Q has no oriented cycles, thus we can order the vertices as I = { i , . . . , i r } in such away that k > l provided there exists an arrow i k → i l . Set Λ = Z I with naturalbasis i ∈ I , and consider the sublattice Λ + = N I . Let Θ ∈ Λ ∗ be a functional(called a stability), and define the slope of a non-zero d = P i ∈ I d i i ∈ Λ + as µ ( d ) = Θ( d )dim d , where dim d = P i ∈ I d i . For µ ∈ Q , define Λ + µ as the subsemigroup of Λ + of all0 = d ∈ Λ + such that µ ( d ) = µ , together with 0 ∈ Λ.Define a bilinear form h , i , the Euler form, on Λ via h d, e i = X i ∈ I d i e i − X α : i → j d i e j for d, e ∈ Λ . Denote by { , } the skew-symmetrization of h , i , thus { d, e } = h d, e i − h e, d i .Define b ij = { i, j } for i, j ∈ I .We consider the formal power series ring B = Q [[Λ + ]] = Q [[ x i : i ∈ I ]] with2opological basis x d = Q i ∈ I x d i i for d ∈ Λ + . The algebra B becomes a Poissonalgebra via the Poisson bracket { x i , x j } = b ij x i x j for i, j ∈ I. Define automorphisms T d of B by T d ( x j ) = x j · (1 + x d ) { d,j } for all d ∈ Λ + and j ∈ I . A direct calculation shows that these are indeedPoisson automorphisms of B (see Lemma 3.5 for a conceptual proof).We study a factorization property in the group Aut( B ) of Poisson automor-phisms of B involving a descending product Q ← µ ∈ Q indexed by rational num-bers. In the following sections, we will see that we actually work in a subgroup ofAut( B ) where such products are well-defined a priori (see the remark followingDefinition 4.2). Theorem 2.1
In the group
Aut( B ) , we have a factorization T i ◦ . . . ◦ T i r = ← Y µ ∈ Q T µ , where T µ ( x j ) = x j · Y i ∈ I Q iµ ( x ) b ij for formal series Q iµ ( x ) ∈ Z [[Λ + µ ]] . These series Q iµ ( x ) are given by the gener-ating function Q iµ ( x ) = X d ∈ Λ + µ χ ( M Θ d,i ( Q )) x d , where χ ( M Θ d,i ( Q )) denotes the Euler characteristic in singular cohomology of aframed moduli space of semistable representations of Q of dimension vector d (see section 5.1 for the precise definition). In this section, we quantize the Poisson algebra B to a skew formal power seriesring A . We define a class of invertible elements of A which induce well-definednon-trival automorphisms of B via conjugation. Definition 3.1
For a commutative ring S and an invertible element q ∈ S ∗ ,define a skew formal power series ring S q [[Λ + ]] as follows: as an S -module, S q [[Λ + ]] has a topological basis consisting of elements t d for d ∈ Λ + ; multipli-cation is defined by t d · t e = q −h e,d i t d + e .
3e apply this definition to the base ring K = Q ( q ), the field of rational functionsin q , and to R = Z [ q, q − ]. This yields A = Q ( q ) q [[Λ + ]], with a natural R -sublattice A = R q [[Λ + ]]. Lemma 3.2
Specialization at q = 1 identifies B with A / ( q − A as a Poissonalgebra. Proof:
That specialization at q = 1 induces an isomorphism of Q -algebrafollows from the definitions. Moreover, we have { t d , t e } = t d t e − t e t d q − q −h e,d i − q −h d,e i q − t d + e = { d, e } t d + e mod ( q − . (cid:3) For a series P ( t ) = P d a d ( q ) t d ∈ A , we denote by P ( x ) its specialization at q = 1 in B , that is, P ( x ) = X d a d (1) x d . Any R -algebra automorphism of A induces a Poisson automorphism of B viaspecialization; we define a class of automorphisms given by conjugation withinvertible elements of A mapping A to itself.For a functional η ∈ Λ ∗ , define a twisted form P ( q η t ) ∈ A of P ( t ) by P ( q η t ) = X d ∈ Λ + q η ( d ) a d t d . We view any n ∈ Λ as a functional n · ∈ Λ ∗ by n · d = P i ∈ I n i d i . A series P ( t ) ∈ A with P (0) = 1, i.e. with constant term equal to 1, is invertible in A . Proposition 3.3
Let P ( t ) ∈ A be a series with constant term equal to , anddefine Q η ( t ) = P ( q η t ) P ( t ) − for all η ∈ Λ ∗ .1. The following conditions on P ( t ) are equivalent:(a) Q η ∈ A for all η ∈ Λ ∗ ,(b) Q i · ∈ A for all i ∈ I .2. Conjugation by P ( t ) maps t d to t d · Q { ,d } ( t ) for all d ∈ Λ + .3. If the conditions of part 1 are fulfilled, conjugation by P ( t ) induces thefollowing Poisson automorphism of B : x d x d · Q { ,d } ( x ) .
4. We have Q η ( x ) = Q i ∈ I Q i · ( x ) η ( i ) for all η ∈ Λ ∗ .5. The set of all P ( t ) ∈ A with P (0) = 1 fulfilling the conditions of part 1forms a subgroup S of A ∗ . roof: We have Q η + ν ( t ) = P ( q η + ν t ) P ( t ) − = P ( q η + ν t ) P ( q ν t ) − P ( q ν t ) P ( t ) − = Q η ( q ν t ) Q ν ( t ) . We have Q ( t ) = 1 by the definitions, so 1 = Q η +( − η ) ( t ) = Q η ( q − η t ) Q − η ( t ) andthus Q − η ( t ) = Q η ( q − η t ) − . We conclude that every Q η ( t ) can be expressed as a product of twisted formsof the Q i · ( t ) for i ∈ I , proving part 1 and part 4.We have t e t d = q { e,d } t d t e by definition of A ; it follows that P ( t ) t d = X e a e ( q ) t e t d = t d X e q { e,d } a e ( q ) t e = t d P ( q { ,d } t ) , thus conjugation by P ( t ) maps t d to t d P ( q { ,d } t ) P ( t ) − , proving part 2. Nowpart 3 follows.To prove that S is a subgroup, suppose we are given P ( t ) , P ( t ) ∈ S , with whichwe associate elements Q η ( t ) , Q η ( t ) ∈ A as above. We write Q η ( t ) = P e b e ( q ) t e .Then P ( q η t ) P ( q η t ) P ( t ) − P ( t ) − = P ( q η t ) Q η ( t ) P ( t ) − = P ( q η t ) X e b e ( q ) t e P ( t ) − = X e b e ( q ) t e P ( q η + { ,e } t ) P ( t ) − = X e b e ( q ) t e Q η + { ,e } ( t )by the identities above. Now every summand in the last sum has coefficients in R , thus the sum belongs to A . For P ∈ S , write P ( q η t ) − P ( t ) = P d c d ( q ) t d .We conjugate this equation by P ( t ) and use part 2: Q η ( t ) − = P ( t ) P ( q η t ) − = P ( t ) P ( q η t ) − P ( t ) P ( t ) − = X d c d ( q ) t d Q { ,d } ( t ) | {z } ∈A . The leftmost term belonging to A , we see inductively that all c d ( q ) belong to R . This proves part 5. (cid:3) Corollary 3.4
With notation of the previous proposition, the map sending P ( t ) ∈ S to the automorphism x d x d · Q { ,d } ( x ) defines a group homomorphism Φ :
S →
Aut( B ) .
5s a first example of series in S , we choose a dimension vector d ∈ Λ + such that h d, d i = 1 (a real root for the root system associated with Q ), and consider theseries P d ( t ) = ∞ X n =0 q − n (1 − q − ) · . . . · (1 − q − n ) t nd ∈ A. The following can be proved using standard identities involving the q -binomialcoefficients h MN i = ( q M − · ... · ( q M − N +1 − q N − · ... · ( q − . We will give a more conceptual proofin Remark 5.2. Lemma 3.5
For any η ∈ Λ , we have Q ηd ( t ) = P d ( q η t ) P d ( t ) − = ∞ X n =0 (cid:20) η ( d ) n (cid:21) t nd . Using Corollary 3.4, we get in the notation of section 2:Φ( P d ( t )) = T d ∈ Aut( B ) . For later reference, we note the following property:
Lemma 3.6
Suppose P ( t ) ∈ S belongs to Q ( q ) q [[Λ + µ ]] for some µ ∈ Q , and let Q η ( t ) be as in Proposition 3.3. Then Q η ( t ) and Q ν ( t ) coincide if η − ν is arational multiple of the functional Θ − µ · dim . Proof:
The subsemigroup Λ + µ has the defining condition µ ( d ) = µ , that is,(Θ − µ · dim)( d ) = 0. Thus, the condition in the statement of the lemmais equivalent to η ( d ) = ν ( d ) for all d ∈ Λ + µ . In the definition of Q η ( t ), thefunctional η only enters through its values on Λ + µ by the choice of P ( t ); thelemma follows. (cid:3) With a quiver Q and a stability Θ as before, we associate a system of rationalfunctions defined recursively, and relate it to the cohomology of quiver modulivia Hall algebras; we adapt material of [6] to the present setup. Generatingseries for this system of rational functions will yield the automorphisms T µ ofTheorem 2.1 via the map Φ of Corollary 3.4. Definition 4.1
Define the following rational functions and their generating se-ries: . for d ∈ Λ + , define e d ( q ) ∈ Q ( q ) by e d ( q ) = q −h d,d i Y i ∈ I d i Y j =1 (1 − q − j ) − .
2. Define p d ( q ) ∈ Q ( q ) for d ∈ Λ + recursively as follows: if Θ is constant on supp( d ) = { i ∈ I : d i = 0 } , then p d ( q ) = e d ( q ) . Otherwise, define p d ( q ) = e d ( q ) − X d ∗ q − P k
The following explicit formula for p d ( q ) (a resolution of the definingrecursion) is proved in [6]: p d ( q ) = X d ∗ ( − s − q − P k ≤ l h d l ,d k i s Y k =1 Y i ∈ I d i Y j =1 (1 − q − j ) − ∈ Q ( q ) , where the sum ranges over all tuples d ∗ = ( d , . . . , d s ) of non-zero dimensionvectors such that d = d + . . . + d s and µ ( d + . . . + d k ) > µ ( d ) for all k < s . Definition 4.2
Given elements c µ ∈ Q ( q ) q [[Λ + µ ]] for µ ∈ Q such that c µ (0) =0 , we define ← Y µ ∈ Q (1 + c µ ) = X µ >...>µ s c µ · . . . · c µ s . Remark:
The sum on the right hand side is indeed well-defined, since calcu-lation of each t d -coefficient reduces to a finite sum. Applying this definition toseries R µ ∈ S ∩ Q ( q ) q [[Λ + µ ]], we see that decreasing products Q ← µ ∈ Q in the imageof Φ : S →
Aut( B ) are well-defined via ← Y µ ∈ Q Φ( R µ ) = Φ( ← Y µ ∈ Q R µ ) . Lemma 4.3
We have P i ( t ) · . . . · P i r ( t ) = P ( t ) = Q ← µ ∈ Q P µ ( t ) in A . Proof:
We first prove the second identity. By the definition of the functions p d ( q ), we have ← Y µ ∈ Q P µ ( t ) = X µ >...>µ s ( X d ∈ Λ + µ \ p d ( q ) t d ) · . . . · ( X d ∈ Λ + µs \ p d ( q ) t d )7 X ( d ,...,ds ) µ ( d >...>µ ( ds ) p d ( q ) t d · . . . · p d s ( q ) t d s = X ( d ,...,ds ) µ ( d >...>µ ( ds ) q − P k M, N .We denote by S i the one-dimensional simple representation supported at thevertex i ∈ I , and by P i its projective cover. Then every projective representa-tion of Q is isomorphic to P ( n ) = L i ∈ I P n i i for some n ∈ Λ + .For the following basic notions and facts on (semi-)stability of quiver represen-tations, see e.g. [7]. For a non-zero representation M , we define its slope as the8lope of its dimension vector, i.e. µ ( M ) = µ (dim M ). We call M semistable if µ ( U ) ≤ µ ( M ) for all non-zero proper subrepresentations U of M , and we call M stable if µ ( U ) < µ ( M ) for all such U . Moreover, we call M polystable if itis isomorphic to a direct sum of stable representations of the same slope.The semistable representations of a fixed slope µ ∈ Q form a full abelian subcat-egory mod µk Q , whose simple (resp. semisimple) objects are given by the stable(resp. polystable) representations of slope µ .For every representation M , there exists a unique Harder-Narasimhan filtration,by which we mean a filtration0 = M ⊂ M ⊂ . . . ⊂ M s = M such that all subfactors M i /M i − are semistable, and µ ( M /M ) > . . . > µ ( M s /M s − ) . Let k be a finite field. For d ∈ Λ + , fix k -vector spaces M i of dimension d i for i ∈ I , and let R d = M α : i → j Hom k ( M i , M j )be the space of all k -representations of Q on the vector spaces M i , on which thegroup G d = Y i ∈ I GL( M i )acts via base change ( g i ) i · ( M α ) α = ( g j M α g − i ) α : i → j , such that the G d -orbits in R d correspond naturally to the isomorphism classesof representations of Q . Let Q G d ( R d ) be the space of (arbitrary) G d -invariant Q -valued functions on R d , and define H k (( Q )) = Y d ∈ Λ + Q G d ( R d ) . This becomes a Λ + -graded algebra, the (completed) Hall algebra of Q , via theconvolution type product( f g )( M ) = X U ⊂ M f ( U ) g ( M/U )for functions f ∈ Q G d ( R d ), g ∈ Q G e ( R e ) and representations M ∈ R d + e . Notethat the sum over all subrepresentations U is finite and that the value of f on U (resp. of g on M/U ) is well-defined by the definitions.9et Q | k | [[Λ + ]] be defined as in Definition 3.1, where the cardinality | k | of k isviewed as an element of Q . The map Z : H (( Q )) → Q | k | [[Λ + ]]given by Z f = 1 | G d | X M ∈ R d f ( M ) t d for f ∈ Q G d ( R d ) is a Q -algebra morphism by [6].Define 1 d = 1 R d as the characteristic function of R d , and define 1 sstd = 1 R sstd asthe characteristic function of the locus R sstd of semistable representations in R d .We form generating functions of these elements by e := X d ∈ Λ + d , e µ = X d ∈ Λ + µ sstd for µ ∈ Q . Lemma 4.4 For every d ∈ Λ + , we have the following identity in H k (( Q )) : d = X d ∗ sstd · . . . · sstd s , the sum running over all decompositions d = d + . . . + d s into non-zero dimen-sion vectors such that µ ( d ) > . . . > µ ( d s ) . Consequently, we have e = Q ← µ ∈ Q e µ in H k (( Q )) . Proof: The existence and uniqueness of the Harder-Narasimhan filtration (seesection 4.2) can be rephrased as follows using the definition of the Hall algebra:for every k -representation M of Q , there exists a unique tuple ( d , . . . , d s ) ofdimension vectors such that µ ( d ) > . . . > µ ( d s ) and(1 sstd · . . . · sstd s )( M ) = 1(namely, the d i are the dimension vectors of the subfactors in the Harder-Narasimhan filtration). The first identity follows. Similar to the proof of Lemma4.3 above, this allows us to compute ← Y µ ∈ Q e µ = X µ >...>µ s ( X d ∈ Λ + µ \ sstd ) · . . . · ( X d ∈ Λ + µs \ sstd )= X ( d ,...,ds ) µ ( d >...>µ ( ds ) sstd · . . . · sstd s = X d ∈ Λ + d = e d . (cid:3) Since the rational functions e d ( q ) and p d ( q ) have no poles at q = | k | by Definition4.1, we can specialize the generating functions P ( t ) and P µ ( t ) to Q | k | [[Λ + ]].10 roposition 4.5 The series P ( t ) specializes to R e , and the series P µ ( t ) spe-cialize to R e µ for all µ ∈ Q . Proof: By the definitions of R d , G d and the Euler form h , i on Q , we have Z d = | R d || G d | = | k | P α : i → j d i d j Q i ∈ I | GL d i ( k ) | = | k | −h d,d i Y i ∈ I d i Y j =1 (1 − | k | − j ) − = e d ( | k | ) , proving the first statement. The second now follows since the elements R sstd satisfy the same recursion as the elements p d ( q ) by Lemma 4.4. (cid:3) Remark: By the definition of the function 1 sstd , we thus have p d ( | k | ) = | R sstd || G d | . In case d is coprime for Θ, this is used in [6] to prove that( q − · p d ( q ) = X i dim H i ( M std ( Q ) , Q ) q i/ , where M std ( Q ) denotes the moduli space of stable representations of Q of di-mension vector d (see section 5.1 for the definitions). The result holds since inthe coprime case, we have a smooth projective moduli space whose numbers ofrational points over finite fields k behave polynomially in | k | . Using the construction of (framed versions of) moduli spaces of representationsof quivers, we prove that the series P µ belong to S . This fact, together with theHarder-Narasimhan recursion of the previous section, proves Theorem 2.1. In this section, we work over the complex numbers. For every dimension vector d ∈ Λ + , there exists a smooth complex variety M std ( Q ) whose points parametrizeisomorphim classes of stable complex representations of Q of dimension vector d . It embeds as an open subset into a (typically singular) projective variety M sstd ( Q ) whose points parametrize isomorphism classes of polystable represen-tations of Q of dimension vector d . If d is coprime for Θ, by which we meanthat µ ( e ) = µ ( d ) for all 0 ≤ e < d , we have M std ( Q ) = M sstd ( Q ), consequently asmooth projective complex variety.For n ∈ Λ + , fix additional vector spaces V i of dimension n i for i ∈ I . Theorem 5.1 ([2]) There exists a projective variety M d,n ( Q ) parametrizingequivalence classes of pairs ( M, f ) consisting of a semistable representation M of on the vector spaces M i , together with a tuple of maps f = ( f i : V i → M i ) i ∈ I such that µ ( U ) < µ ( M ) whenever U ⊂ M is a proper subrepresentation of M containing the image of f (that is, the subrepresentation generated by the f i ( V i ) );such pairs are considered up to isomorphisms of the representations intertwiningthe additional maps, that is, ( M, f ) is equivalent to ( M ′ , f ′ ) if there exists anisomorphism ϕ : M → M ′ such that f ′ = ϕf . This is a framed version of moduli of (semistable) quiver representations, calledsmooth models in [2]. There it is also shown that the framing datum f induces amorphism from the projective representation P ( n ) to M such that µ ( U ) < µ ( M )for any proper subrepresentation U of M containing its image.There exists a canonical projective morphism π : M Θ d,n ( Q ) → M sstd ( Q ), whichby [2] is ´etale locally trivial for a suitable (Luna type) stratification of M sstd ( Q )with known fibres (they are isomorphic to certain nilpotent parts of smoothmodels for the trivial stability and quivers with oriented cycles).As a special case, we consider the quiver Q consisting of a single vertex andno arrows, with trivial stability. From the definitions, it is easy to see that M d,n ( Q ) ≃ Gr nd ( C ), the Grassmannian of k -planes in n -space. Let again k be a finite field. The semistable representations of Q of fixed slope µ ∈ Q form a full abelian subcategory mod µk ( Q ) of mod µk ( Q ). Thus, for asubrepresentation U ⊂ M of a representation M ∈ mod µk ( Q ), we can form theintersection h U i µ = \ V ∈ mod µk ( Q ) U ⊂ V ⊂ M V ∈ mod µk ( Q ) . This is the minimal subrepresentation of M containing U which is semistableof slope µ . For n ∈ Λ + , we denote by Hom Q ( P ( n ) , M ) the set of all morphisms f ∈ Hom Q ( P ( n ) , M ) such that h Im( f ) i µ = M . As mentioned above, the data( M, f ) of Theorem 5.1 can be viewed as pairs consisting of a representation M ∈ mod µk ( Q ), together with a map f ∈ Hom Q ( P ( n ) , M ).For d ∈ Λ + µ , define 1 sstd,n as the function taking value | Hom Q ( P ( n ) , M ) | = q n · d on M ∈ R sstd , and value 0 outside R sstd . Define f d,n ∈ H k (( Q )) as the functiontaking value | Hom Q ( P ( n ) , M ) | on M ∈ R sstd , and value 0 outside R sstd . We formgenerating functions e µ,n = P d ∈ Λ + µ sstd,n and h µ,n = P d ∈ Λ + µ f d,n .The following lemma is a special case of [2, Lemma 5.1]: Lemma 5.2 We have e µ,n = h µ,n · e µ in H k (( Q )) . Proof: For a representation M ∈ mod µk ( Q ), we have by the definitions( h µ,n · e µ )( M ) = X U | Hom ( P ( n ) , U ) | , U of M such that U ∈ mod µk ( Q ).But the set of all pairs ( U, f ) consisting of a subrepresentation U ⊂ M such that U ∈ mod µk ( Q ) and a morphism f ∈ Hom ( P ( n ) , U ) is naturally in bijection toHom( P ( n ) , M ), by associating to f : P ( n ) → M the pair ( h Im( f ) i µ , f : P ( n ) →h Im( f ) i µ ). Thus,( h µ,n · e µ )( M ) = | Hom( P ( n ) , M ) | = e µ,n ( M ) , proving the lemma. (cid:3) Proposition 5.3 The series P µ ( t ) belongs to S . The automorphism Φ( P µ ( t )) is given by x d x d · Y i ∈ I Q iµ ( x ) { i,d } , where Q iµ ( x ) = X d ∈ Λ + d χ ( M Θ d,i ( Q )) x d . Proof: By Proposition 4.5, we know that the series P µ ( t ) specializes to R e µ under specialization of q to k . By definition of e µ,n , the series P µ ( q n · t ) thusspecializes to R e µ,n for n ∈ Λ + . By the previous lemma, the function Q n · µ ( t ) = P µ ( q n · t ) P µ ( t ) − thus specializes to R e µ,n . By definition, the function f d,n inte-grates to Z f d,n = |{ ( M, f ) : M ∈ R sstd , f ∈ Hom ( P ( n ) , M }|| G d | . As in the proof of [2, Theorem 5.2], we can conclude, using the definition ofthe smooth model M Θ d,n ( Q ) and the comparison between numbers of rationalpoints and Betti numbers as in the final remark of section 4.3, that the Poincarepolynomial P i dim H i ( M Θ d,n ( Q ) , Q ) q i/ specializes to R f d,n at q = | k | . Conse-quently, we have Z e µ,n = X d ∈ Λ + µ X i dim H i ( M Θ d,n ( Q ) , Q ) q i/ t d . Thus, the coefficients b d ( q ) of Q n · µ ( t ) are rational functions taking integer valuesat all prime powers q , and thus are polynomials in q . By part 1 of Proposition3.3, we conclude that P µ ( t ) ∈ S . Using parts 3 and 4 of the same proposition, wederive the claimed formula for Φ( P µ ( t )) since the Poincare polynomial specializesto the Euler characteristic at q = 1 in absence of odd cohomology. (cid:3) Remark: Applying this result to the quiver Q , we get Lemma 3.5 in the case d = i . The assumption h d, d i = 1 there allows to generalize to such d using thedefinition of P d ( q ).Theorem 2.1 now follows immediately from Lemma 3.5, Lemma 4.3 and Propo-sition 5.3. 13 Examples Let Q = K m be the m -arrow Kronecker quiver with set of vertices I = { i, j } and m arrows from j to i . We have { ai + bj, ci + dj } = m · ( ad − bc ) and inparticular b ij = m . We choose the stability Θ = j ∗ . Writing a dimension vector d ∈ Λ + as d = ai + bj for a, b ∈ N , the automorphism T a,b = T d of the Poissonalgebra B m = Q [[ x i , x j ]] with Poisson bracket { x i , x j } = mx i x j is then givenby T a,b : (cid:26) x i x i (1 + x ai x bj ) − mb x j x j (1 + x ai x bj ) ma (cid:27) . We prove the following weak version of [4, Conjecture 1]: Theorem 6.1 In Aut( B m ) , there exists a factorization T i ◦ T j = ← Y b/a ∈ Q T d ( a,b ) a,b , where d ( a, b ) ∈ a,b ) Z for all a, b ∈ N . Proof: By Theorem 2.1, we have a factorization T i ◦ T j = Q ← µ ∈ Q T µ , where T µ is given by T µ : (cid:26) x i x i · Q jµ ( x ) − m x j x j · Q iµ ( x ) m (cid:27) for Q iµ ( x ), Q jµ ( x ) being the corresponding generating series for Euler character-istics of smooth models. Writing µ = b/ ( a + b ) for coprime a, b ∈ N , we haveΛ + µ = N · ( a, b ), and thus Q iµ ( x ) , Q jµ ( x ) ∈ Z [[ x ai x bj ]]. We choose integers c, d ∈ Z such that ac + bd = 1 and define F µ ( x ) = Q iµ ( x ) c Q jµ ( x ) d ∈ Z [[ x ai x bj ]] . By Lemma 3.6, we have Q jµ ( x ) a = Q iµ ( x ) b . It then follows that F µ ( x ) a = Q iµ ( x ) and F µ ( x ) b = Q jµ ( x ) . We can factor F µ ( x ) into an infinite product F µ ( x ) = Y k ≥ (1 + ( x ai x bj ) k ) c ( µ,k ) for integers c ( µ, k ) ∈ Z . These two identities allow us to write T µ in the form T µ : (cid:26) x i x i · Q k ≥ (1 + x kai x kbj ) − mbc ( µ,k ) ,x j x j · Q k ≥ (1 + x kai x kbj ) mac ( µ,k ) (cid:27) . T ka,kb and defining d ( ka, kb ) = k c ( µ, k ), the theorem follows. (cid:3) It is now easy to re-derive the examples of [4, Section 1.4]. For m = 1, thereare three isomorphism classes of stable representations of K , namely the twosimple representations S i , S j and their non-trivial extension X . Thus, therelevant smooth models reduce to Grassmannians as in the case of the quiver Q , yielding the factorization T , ◦ T , = T , ◦ T , ◦ T , , which can also be verified directly.For m = 2, we have a unique stable representation for any dimension vector ki +( k +1) j or ( k +1) i + kj . Moreover, we have a P -family of stable representationsof dimension vector i + j , and no stables of dimension vector ki + kj for k ≥ M Θ ki + kj,n ( K ) ≃ Hilb k ( P ) ≃ P k for n = i, j , and thus Q i / ( x ) = Q j / ( x ) = X k ≥ ( k + 1)( x i x j ) k = (1 − x i x j ) − . We arrive at the following factorization in B : T , ◦ T , = ( T , ◦ T , ◦ T , ◦ . . . ) ◦ T ◦ ( . . . ◦ T , ◦ T , ◦ T , ) , where T : (cid:26) x i x i · (1 − x i x j ) x j x j · (1 − x i x j ) − (cid:27) . Remark: In [3, Theorem 0.1], the automorphisms T µ , given by generatingfunctions of Euler characteristics of quiver moduli, are calculated in terms ofgenerating function of genus 0 Gromov-Witten invariants of toric surfaces. Thisalternative interpretation hints towards a link between the underlying geome-tries. Let Q be a quiver of Dynkin type. Then the isomorphism classes of indecom-posable representations U α correspond bijectively to the positive roots α ∈ ∆ + of the corresponding root system. The Harder-Narasimhan filtration of section4.2 can be replaced by a more explicit filtration (we refer to [5] for details and arelated application of this filtration). Namely, the positive roots can be orderedas α , . . . , α ν in such a way that Hom Q ( U α l , U α k ) = 0 = Ext Q ( U α k , U α l ) for k < l . Now every representation M of Q is of the form M = L k U m k α k , and thusadmits a unique filtration0 = M ν ⊂ M ν − ⊂ . . . ⊂ M ⊂ M = M M k − /M k ≃ U m k α k for all k = 1 , . . . , ν . The arguments of section 4 canbe applied to this filtration, yielding a factorization P i · . . . · P i r = P α ν · . . . · P α .Applying the map Φ : S → Aut( B ), we conclude T i ◦ . . . ◦ T i r = T α ν ◦ . . . ◦ T α . 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