The Multiplication Formulas of Weighted Quantum Cluster Functions

By applying the property of Ext-symmetry and the affine space structure of certain fibers, we introduce the notion of weighted quantum cluster functions and prove their multiplication formulas associated to abelian categories with Ext-symmetry and 2-Calabi-Yau triangulated categories with cluster-ti...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2023
Main Authors:	Chen, Zhimin, Xiao, Jie, Xu, Fan
Format:	Article
Language:	English
Published:	Інститут математики НАН України 2023
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/212034
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Cite this:	The Multiplication Formulas of Weighted Quantum Cluster Functions. Zhimin Chen, Jie Xiao and Fan Xu. SIGMA 19 (2023), 097, 60 pages

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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author	Chen, Zhimin Xiao, Jie Xu, Fan
author_facet	Chen, Zhimin Xiao, Jie Xu, Fan
citation_txt	The Multiplication Formulas of Weighted Quantum Cluster Functions. Zhimin Chen, Jie Xiao and Fan Xu. SIGMA 19 (2023), 097, 60 pages
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container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	By applying the property of Ext-symmetry and the affine space structure of certain fibers, we introduce the notion of weighted quantum cluster functions and prove their multiplication formulas associated to abelian categories with Ext-symmetry and 2-Calabi-Yau triangulated categories with cluster-tilting objects.
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 097, 60 pages The Multiplication Formulas of Weighted Quantum Cluster Functions Zhimin CHEN a, Jie XIAO b and Fan XU c a) Department of Mathematics, Tsinghua University, Beijing 100084, P. R. China E-mail: chen-zm15@mails.tsinghua.edu.cn b) School of Mathematical Sciences, Beijing Normal University, Beijing 100875, P. R. China E-mail: jxiao@bnu.edu.cn c) Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P. R. China E-mail: fanxu@mail.tsinghua.edu.cn Received June 02, 2022, in final form November 26, 2023; Published online December 13, 2023 https://doi.org/10.3842/SIGMA.2023.097 Abstract. By applying the property of Ext-symmetry and the affine space structure of certain fibers, we introduce the notion of weighted quantum cluster functions and prove their multiplication formulas associated to abelian categories with Ext-symmetry and 2- Calabi–Yau triangulated categories with cluster-tilting objects. Key words: weighted quantum cluster functions; cluster categories; 2-Calabi–Yau triangu- lated categories; preprojective algebras 2020 Mathematics Subject Classification: 17B37; 16G20; 17B20 1 Introduction, notation and main results Cluster algebras were introduced by Fomin and Zelevinsky [9] in order to find an algebraic framework for understanding total positivity in algebraic groups and canonical bases in quantum groups. They are commutative algebras over Z generated by certain elements called cluster variables. There is an iterative procedure called mutation to obtain new cluster variables from initial cluster variables. For an cluster algebra of finite type, the cluster variables are in bijection with the almost positive roots of the corresponding simple Lie algebra. Cluster algebras have close connections to representation theory of algebras via cluster cat- egories and cluster characters. The cluster category was constructed by Buan–Marsh–Reineke– Reiten–Todorov [3] as a quotient of the bounded derived category of the module category of a finite-dimensional hereditary algebra. Given an acyclic quiver Q, the cluster category CQ as- sociated with Q is the orbit category Db(Q)/τ ◦ [−1]. Indecomposable objects in CQ correspond to the almost positive roots in the root system of the Lie algebra gQ of type Q. The cluster category has the 2-Calabi–Yau property, i.e., HomCQ(M,N [1]) ∼= DHomCQ(N,M [1]) for any M,N ∈ CQ. For a Dynkin quiver Q with n vertices, Caldero and Chapoton [4] defined a map X? from the set of objects in the cluster category CQ to Q(x1, . . . , xn) such that XM = ∑ e χ(Gre(M))x−Be−(In−R)m for a CQ-module M and XP [1] = xdimP/radP for a projective CQ-module P (See [6] for the nota- tion in the above formula.) In particular, it gives a bijection between the indecomposable rigid mailto:chen-zm15@mails.tsinghua.edu.cn mailto:jxiao@bnu.edu.cn mailto:fanxu@mail.tsinghua.edu.cn https://doi.org/10.3842/SIGMA.2023.097 2 Z. Chen, J. Xiao and F. Xu objects and cluster variables and then the mutation rule of cluster variables is written as the fol- lowing property. For indecomposable rigid objects M , N in CQ with dimCHomCQ(M,N [1]) = 1, we have XM · XN = XL + XL′ , where L, L′ are the middle terms of two non-split triangles induced by ε ∈ HomCQ(M,N [1]) and ε′ ∈ HomCQ(N,M [1]), respectively. This map is called the cluster character. Then the cluster character XM for an indecomposable rigid object M can be viewed as the general formula of a cluster variable, computed from the initial variables XP1[1], . . . , XPn[1] using the mutation rule. Caldero and Keller [5] generalized these results to any acyclic quiver and then categorified all cluster algebras with acyclic initial clusters. As a corollary, the cluster algebra A(Q) is generated by XM for all rigid objects M ∈ CQ. Palu [16] generalized the definition of cluster character to a 2-Calabi–Yau triangulated category C with a cluster tilting object T . His cluster character, denoted by XT ? , is given by the formula XT M = ∑ e χ(Gre(HomC(T,M)))xindT M−ι(e) for M ∈ C (see Section 4.1 for more details). In [6], Caldero and Keller proved a higher-dimensional multiplication formula between clus- ter characters for the cluster category CQ of a Dynkin quiver Q. They showed that for any indecomposable objects M,N ∈ CQ, χ(PHomCQ(M,N [1]))XM ·XN = ∑ [L] (χ(PHomCQ(M,N [1])L[1]) + χ(PHomCQ(N,M [1])L[1]))XL, where χ is the Euler–Poincaré characteristic. The proof heavily depends on the 2-Calabi–Yau property of CQ. This multiplication formula was generalized firstly to acyclic quivers [22, 24] and then to any 2-Calabi–Yau triangulated category with a cluster-tilting object [17]. Inspired by the results in [6], Geiss, Leclerc and Schröer [10, Theorem 3] proved an analogous formula for nilpotent module categories of preprojective algebras. Let ΛQ be the preprojective algebra of an acyclic quiver Q and nil(ΛQ) be the category of finite dimensional nilpotent ΛQ- modules. Let n be the positive part of gQ. Given two sequences i = (i1, . . . , im), a = (a1, . . . , am) where ij ∈ {1, . . . , n} and aj ∈ {0, 1} for 1 ⩽ j ⩽ m, Lusztig introduced the constructible functions di,a : nil(ΛQ) → C defined by di,a(M) = χ(Fi,a(M)), where Fi,a(M) is the set of flags of type (i,a) (see Section 2.4 for the definition). He proved that the enveloping algebra U(n) is isomorphic to M = ⊕ Md, where the Md are certain vector spaces generated by di,a for proper pairs (i,a). Geiss, Leclerc and Schröer showed that the category nil(ΛQ) has the Ext-symmetry, i.e., Ext1ΛQ (M,N) ∼= DExt1ΛQ (N,M) for M,N ∈ nil(ΛQ). Let M∗ be the graded dual of M. For M ∈ nil(ΛQ), they defined an evaluation form δM ∈ M∗, which is the map from M to Z mapping di,a to di,a(M) for any pair (i,a). We denote by Λd the variety of nilpotent Λ-modules of dimension vector d and set ⟨M⟩ := {X ∈ Λd \| δX = δM}. Then there exists a finite subset S(d) of Λd such that Λd = ⊔ M∈S(d)⟨M⟩. For M ∈ Λd and N ∈ Λd′ , they obtained the following multiplication formula: χ ( PExt1ΛQ (M,N) ) δM · δN = ∑ L∈S(d) ( χ ( PExt1ΛQ (M,N)⟨L⟩ ) + χ ( PExt1ΛQ (N,M)⟨L⟩ )) δL. The Multiplication Formulas of Weighted Quantum Cluster Functions 3 The similarity between CQ and nil(ΛQ) was further studied by many authors [2, 11]. Let W be the Weyl group of gQ. For w ∈W, Buan, Iyama, Reiten, and Scott [2] constructed a 2-Calabi– Yau Frobenius subcategory Cw of nil(ΛQ). When w = (sn · · · s1)2, the stable category Cw of Cw is just CQ. Given a reduced expression of w = sir · · · si1 and set i := (ir, . . . , i1). ForM ∈ Cw, δM can be reformulated into the following form: ∆M = ∑ a∈Nr δM (di,a)t a = ∑ a∈Nr χ(Fi,a(M))ta. Geiss, Leclerc and Schröer made explicit correspondences between the two multiplication for- mulas as follows. They defined a cluster tilting object Wi in Cw and set A := EndCw(Wi) op. For X ∈ Cw, F (X) = Ext1Λ(Wi, X) is an A-module. There is a bijiection di,M between {a ∈ Nr \| Fi,a(M) ̸= ∅} and {d ∈ Nn \| Grd(F (M)) ̸= ∅}. Furthermore, there is an isomorphism of algebraic varieties between Fi,a(M) and Grdi,M (a)(F (M)). These lead to a correspondence between ∆M and XT M for M ∈ Cw where T = Wi and M are respectively the image of Wi and M under the natural functor from Cw to Cw. Quantum cluster algebras were introduced by Berenstein and Zelevinsky [1] as the quanti- zation of cluster algebras. They are defined to be certain noncommutative algebras over Z[q±] generated by quantum cluster variables for a compatible pair (B,Λ) of two matrices B and Λ. Rupel [21] defined a quantum cluster character as a quantum analogue of the Caldero–Chapoton map. Let Q be an acyclic valued quiver over a finite field F = Fq. For M ∈ RepFQ, XF M = ∑ e q− 1 2 ⟨e,m−e⟩\|Gre(M)\|x−Be−(I−R)m (see Section 5 for the explicit definition). Qin [19] gave an alternative definition for the quantum cluster characters of rigid objects in the cluster category of an acyclic equally valued quiver. The definition does not involve the choice of the ground finite field. Then he proved that the definition is consistent with the mutation rules between quantum cluster variables, i.e., for indecomposable rigid objects M , N in CQ with dimCHomCQ(M,N [1]) = 1, we have XF M ·XF N = q 1 2 Λ(indM,indL)XF L + q 1 2 Λ(indM,indL′)XF L′ . The multiplication formula of quantum cluster characters was generalized in [8]. More generally, we give a quantum analogue of Caldero–Keller’s multiplication formula in Section 4. Recently, another quantum analogue was proved in [7]. The exact relation between these two formulas will be discovered in the near future. In contrast to the case of the cluster character, the general definition of the quantum cluster character is still unknown. The aim of this paper is to define the weighted quantum cluster functions for both an abelian category A with the Ext-symmetry and a 2-Calabi–Yau triangu- lated category C with a cluster tilting object. A weighted quantum cluster function is of the form f ∗[ϵ] δL or f ∗ϵ XL with a weight function f . So far we do not know whether or not weighted quantum cluster functions can give quantum cluster characters which compute the quantum cluster variables in the sense of Berenstein–Zelevinsky by taking the proper weights. In this paper, we use the following notation: given a finite set S, and a function g on S, we define∫ x∈S g(x) := ∑ x∈S g(x). The first main result in this paper is the following theorem (cf. Theorem 3.25 for details). 4 Z. Chen, J. Xiao and F. Xu Theorem 1.1. Let A be a Hom-finite, Ext-finite abelian category with Ext-symmetry over a finite field k = Fq such that the iso-classes of objects form a set. For any weighted quantum cluster functions f ∗[ϵ′] δM and g ∗[ϵ′′] δN such that Ext1A(M,N) ̸= 0, we have∣∣PExt1A(M,N) ∣∣(f ∗[ϵ′] δM ) ∗ (g ∗[ϵ′′] δN ) = ∫ Pϵ∈PExt1A(M,N) ( f+ext ∗[ϵ] Sfg ) ∗[ϵ] δmtϵ + ∫ Pη∈PExt1A(N,M) (fhom ∗[η] Sgf ) ∗[η] δmtη = ∫ Pϵ∈PExt1A(M,N) Sfg ∗[ϵ] δmtϵ + ∫ Pη∈PExt1A(N,M) (f−ext ∗[η] fhom ∗[η] Sgf ) ∗[η] δmtη. This theorem gives the explicit multiplication between two weighted quantum cluster func- tions over an abelian category A with the Ext-symmetry. The second main result is as follows (cf. Theorem 4.23 for details). Theorem 1.2. Let C be a Hom-finite, 2-Calabi–Yau, Krull–Schmidt triangulated category over a finite field k = Fq with a cluster tilting object T . For any weighted quantum cluster functions f ∗ϵ′ XM and g ∗ϵ′′ XN such that HomC(M,ΣN) ̸= 0, we have \|PHomC(M,ΣN)\|(f ∗ϵ′ XM ) · (g ∗ϵ′′ XN ) = ∫ Pϵ∈PHomC(M,ΣN) ( g+ext ∗ϵ fspec ∗ϵ Tfg ) ∗ϵ Xmtϵ + ∫ Pη∈PHomC(N,ΣM) (gskew ∗η fspec ∗η Tgf ) ∗η Xmtη = ∫ Pϵ∈PHomC(M,ΣN) (fspec ∗ϵ Tfg) ∗ϵ Xmtϵ + ∫ Pη∈PHomC(N,ΣM) (g−ext ∗η gskew ∗η fspec ∗η Tgf ) ∗η Xmtη. This theorem gives the explicit multiplication between two weighted quantum cluster func- tions over a 2-Calabi–Yau triangulated category C with a cluster tilting object T . It can be viewed as the quantum analogue of Caldero–Keller’s multiplication formula. The third main result is the quantum analogue of the above Geiss–Leclerc–Schröer’s multi- plication formula (cf. Theorem 5.14 for details). Theorem 1.3. For any weighted quantum cluster functions f ∗ϵ′ ∆i,M and g ∗ϵ′′ ∆i,N such that Ext1Cω(M,N) ̸= 0, in Am,λ we have∣∣PExt1Cω(M,N) ∣∣(f ∗ϵ′ ∆i,M ) · (g ∗ϵ′′ ∆i,N ) = ∫ Pϵ∈PExt1Cω (M,N) ( f+ext ∗ϵ fspec ∗ϵ Sfg ) ∗ϵ ∆i,mtϵ + ∫ Pη∈PExt1Cω (N,M) (fskew ∗η fspec ∗η Sgf ) ∗η ∆i,mtη = ∫ Pϵ∈PExt1Cω (M,N) (fspec ∗ϵ Sfg) ∗ϵ ∆i,mtϵ + ∫ Pη∈PExt1Cω (N,M) (f−ext ∗η fskew ∗η fspec ∗η Sgf ) ∗η ∆i,mtη. The fourth main result focuses on weighted quantum cluster functions from hereditary alge- bras. By choosing a particular weight function, we recover Qin’s result [18, Proposition 5.4.1] (cf. Theorem 6.22 for details). The Multiplication Formulas of Weighted Quantum Cluster Functions 5 Theorem 1.4. In the cluster category C = Db ( Ã ) /τ−1Σ of a hereditary algebra Ã, given two indecomposable coefficient-free rigid objects M , N ∈ Ã with dimk HomC(M,ΣN) = 1 and two non-split triangles N → L→M ϵ−→ ΣN and M → L′ → N η−→ ΣM, where L′ is located in the fundamental domain, then we have X̃M · X̃N = q 1 2 λ(indM,indN)− 1 2 · X̃L + q 1 2 λ(indM,indN) · X̃L′ . Recently, Keller, Plamondon and Qin gave a refined multiplication formula for cluster char- acters over 2-Calabi–Yau triangulated category with tilting objects [15]. Its quantum analogue has been obtained in [23] via a motivic version of weighted quantum cluster functions. The paper is organized as follows. In Section 2, we first define a chain of monomorphisms as a generalization of flag and then define a set of chains of monomorphisms of certain type. In Section 3, we introduce the quantum cluster function attached to an abelian category with the Ext-symmetry in a general context. Corollaries 3.3 and 3.7 play an important role in char- acterizing the quantum cluster function. Then we define the weight function and the weighted quantum cluster function. By choosing appropriate pair of weight functions, we prove the mul- tiplication formula of weighted quantum cluster functions in Section 3.9. Section 4 is devoted to defining the weighted quantum cluster function for a 2-Calabi–Yau triangulated category with a cluster tilting object. Lemma 4.8 helps to describe the structure of the quantum cluster func- tion. We prove the multiplication formula of weighted quantum cluster functions in Section 4.9. Section 5 establishes the explicit connection between the two multiplication formulas as stated in Theorem 1.1 and Theorem 1.2. We make a correspondence between weighted quantum clus- ter functions in Section 3 and Section 4 by applying Geiss–Leclerc–Schröer’s correspondence as above. In Section 6, we show that Theorem 4.23 recovers the multiplcation formula in [18] by assigning proper weight functions. 2 Chains of monomorphisms 2.1 Chains of morphisms Let A be a Hom-finite, Ext-finite abelian category with Ext-symmetry over a finite field k = Fq such that the isoclasses of objects form a set. We assume that A has finitely many simple objects up to isomorphism and fix a complete set of simple objects {S1, . . . , Sn} up to isomorphism in A. For a fixed m ∈ N, we denote by F̃mor m the set of all chains of morphisms c = ( Lm ιc,m−−→ Lm−1 −→ · · · −→ L1 ιc,1−−→ L0 ) , where L0, . . . , Lm ∈ A and ιc,1, . . . , ιc,m ∈ MorA. Using the isomorphism relations L ∼= L′ in A, we can induce the isomorphism relations in F̃mor m . Definition 2.1. Two chains of morphisms c = ( Lm ιc,m−−→ Lm−1 −→ · · · −→ L1 ιc,1−−→ L0 ) and c′ = ( L′ m ιc′,m−−−→ L′ m−1 −→ · · · −→ L′ 1 ιc′,1−−→ L′ 0 ) in F̃mor m are called isomorphic if there are isomorphisms Li fi−→ L′ j in MorA such that fi−1 ◦ ιc,i = ιc′,i ◦ fi, i.e., we have the following commutative diagram: Lm Lm−1 · · · L1 L0 L′ m L′ m−1 · · · L′ 1 L′ 0. ιc,m fm ιc,m−1 fm−1 ιc,1 f1 f0 ιc′,m ιc′,m−1 ιc′,1 6 Z. Chen, J. Xiao and F. Xu We denote the set of isomorphism classes in F̃mor m by Fmor m and we still write c for its isomor- phism class in Fmor m . 2.2 Exact structure Given a short exact sequence ϵ : 0 // N // L //M // 0 in A, write [ϵ] for its equivalence class in Ext1A(M,N). Recall that two short exact sequences 0 // N f1 // L g1 //M // 0 and 0 // N f2 // L′ g2 //M // 0 correspond to the same element in Ext1A(M,N) if and only if there is an A-isomorphism a : L −→ L′ with f2 = af1 and g2 = g1a −1, i.e., we have the following commutative diagram: 0 // N f2 // L′ g2 //M // 0 0 // N f1 // L g1 // a OO M // 0. Then any morphism λ ∈ Mor(N,N ′) induces an equivalence class λ ◦ [ϵ] in Ext1A(M,N ′) which is called the pushout of [ϵ] along λ. Similarly, for any ρ ∈ Mor(M ′′,M), there is an equivalence class [ϵ] ◦ ρ in Ext1A(M ′′, N) which is called the pullback of [ϵ] along ρ. We have the following commutative diagram: λ ◦ [ϵ] : 0 // N ′ // L′ //M // 0 [ϵ] : 0N // λ OO // L // OO M // 0 [ϵ] ◦ ρ : 0 // N // L′′ // OO M ′′ // ρ OO 0. Conversely, given [ϵ′] ∈ Ext1A(M ′, N ′), [ϵ′′] ∈ Ext1A(M ′′, N ′′), λ ∈ Mor(N ′′, N ′), ρ ∈ Mor(M ′′,M ′) as in the diagram [ϵ′] : 0 // N ′ // L′ //M ′ // 0 [ϵ′′] : 0 // N ′′ // λ OO L′′ //M ′′ // ρ OO 0, we can complete the commutative diagram by adding an appropriate morphism in Mor(L′′, L′) if and only if the pushout and pullback coincide, i.e., λ ◦ [ϵ′′] = [ϵ′] ◦ ρ. The Multiplication Formulas of Weighted Quantum Cluster Functions 7 Moreover, in this case, the morphism in Mor(L′′, L′) is given by g ◦ f as shown in the following diagram: [ϵ′] : 0 // N ′ // L′ //M ′ // 0 λ ◦ [ϵ′′] = [ϵ′] ◦ ρ : 0 // N ′ // L // g OO M ′′ // ρ OO 0 [ϵ′′] : 0 // N ′′ // λ OO L′′ // f OO M ′′ // 0. The equality [ϵ′] ◦ ρ = λ ◦ [ϵ′] can be illustrated by the following commutative diagram: [ϵ′] : 0 // N ′ // L′ //M ′ // 0 [ϵ′] ◦ ρ : 0 // N ′ // L̃′ // OO ∼= �� M ′′ // ρ OO 0 λ ◦ [ϵ′] : 0 // N ′′ // L̃′′ //M ′′ // 0 [ϵ′′] : 0 // N ′′ // λ OO L′′ // OO M ′′ // 0. Since every [ϵ] ∈ Ext1A(M,N) can be represented by ϵ : 0 // N // L //M // 0 with M , N , L being unique up to isomorphism, we denote qtϵ :=M , stϵ := N and mtϵ := L in the following sections. Now, more generally, let [ϵi] ∈ Ext1A(Mi, Ni) for 0 ⩽ i ⩽ m, and take two isomorphism classes of chains of morphisms c′ = ( Mm ιc′,m−−−→Mm−1 −→ · · · −→M1 ιc′,1−−→M0 =M ) , c′′ = ( Nm ιc′′,m−−−→ Nm−1 −→ · · · −→ N1 ιc′′,1−−−→ N0 = N ) in Fmor m as in the diagram [ϵ0] : 0 // N0 // L0 //M0 // 0 [ϵ1] : 0 // N1 // ιc′′,1 OO L1 //M1 // ιc′,1 OO 0 ... OO ... ... OO [ϵm−1] : 0 // Nm−1 // OO Lm−1 //Mm−1 // OO 0 [ϵm] : 0 // Nm // ιc′′,m OO Lm //Mm // ιc′,m OO 0. (2.1) 8 Z. Chen, J. Xiao and F. Xu In order to decide whether there exists a chain of morphisms c = ( Lm ιc,m−−→ Lm−1 −→ · · · −→ L1 ιc,1−−→ L0 = L ) to complete the above commutative diagram, we introduce a linear map βc′,c′′ : m⊕ j=0 Ext1(Mj , Nj) −→ m−1⊕ j=0 Ext1(Mj+1, Nj), ([ϵ0], . . . , [ϵm]) 7−→ ([ϵj−1] ◦ ιc′,j − ιc′′,j ◦ [ϵj ], 1 ⩽ j ⩽ m). The definition and some properties of maps βc′,c′′ and β ′ c′′,c′ , which will be introduced later as the linear dual of βc′,c′′ , were firstly given in [6] for module categories. We generalize these two maps to abelian categories and prove some necessary properties here. Lemma 2.2. There exists a chain of morphisms c = ( Lm ιc,m−−→ Lm−1 −→ · · · −→ L1 ιc,1−−→ L0 = L ) , which can complete the commutative diagram (2.1) if and only if ([ϵ0], . . . , [ϵm]) ∈ Kerβc′,c′′ . Proof. A family of elements ([ϵ0], . . . , [ϵm]) ∈ Kerβc′,c′′ is exactly one which makes pushout and pullback coincide at every level. ■ In this case, we get the morphism ιL,i by composing two vertical morphisms in the following diagram: [ϵi−1] : 0 // Ni−1 // Li−1 //Mi−1 // 0 ιc′′,i ◦ [ϵi] = [ϵi−1] ◦ ιc′,i : 0 // Ni−1 // L′ i // g OO Mi // ιc′,i OO 0 [ϵi] : 0 // Ni // ιc′′,i OO Li // f OO Mi // 0. We denote this assignment by Bc′,c′′ : Kerβc′,c′′ −→ Fmor m , ([ϵ0], . . . , [ϵm]) 7−→ (Lm ιc,m−−→ Lm−1 −→ · · · −→ L1 ιc,1−−→ L0 = L). Lemma 2.3. For any chains of morphisms c′, c′′, the map Bc′,c′′ is well defined and injective. Proof. The proof is similar to the discussion in the case when m = 1. The choice of short exact sequences representing ([ϵ0], . . . , [ϵm]) is not unique, but is unique up to the following commutative diagram: ϵi : 0 // Ni // Li //Mi // 0 ϵ′i : 0 // Ni // L′ i // ai OO Mi // 0, The Multiplication Formulas of Weighted Quantum Cluster Functions 9 where ai is an isomorphism. Then we get two chains of morphisms Bc′,c′′([ϵ0], . . . , [ϵm]) and Bc′,c′′([ϵ ′ 0], . . . , [ϵ ′ m]). One can check they are isomorphic in F̃mor m through the family of isomor- phisms ai. Conversely, given two choices of exact sequences which induce the same chain of morphisms in Fmor m , then the isomorphism Li → L′ i also give the equivalence between the two exact se- quences: Ni−1 L′ i−1 Mi−1 Ni L′ i Mi Ni−1 Li−1 Mi−1. Ni Li Mi ≃ ≃ ■ Since all extension groups considered here are finite dimensional over a finite field, we have Corollary 2.4. Given chains of morphisms c′, c′′, the map Bc′,c′′ : Kerβc′,c′′ −→ ImBc′,c′′ is a bijection and \| ImBc′,c′′ \| = \|Kerβc′,c′′ \|. 2.3 Chains of monomorphisms Definition 2.5. Given a chain of morphisms c = ( Lm ιc,m−−→ Lm−1 −→ · · · −→ L1 ιc,1−−→ L0 ) in F̃mor m , it is called a chain of monomorphisms if all ιc,i are monomorphisms and Lm = 0. One can easily check this definition is independent of choice of chain in an isomorphism class and we denote by Fmono m the subset of Fmor m consisting of all isomorphism classes of chains of monomorphisms. We also denote by Fmono m,L the subset of Fmono m consisting of all isomorphism classes of chains of monomorphisms with L0 ∼= L. Lemma 2.6. If c′, c′′ ∈ Fmono m , then ImBc′,c′′ ⊆ Fmono m . Proof. By the snake lemma, all rows in the following commutative diagram are exact: 0 Nj−1 Lj−1 Mj−1 0 0 Nj Lj Mj 0. Ker ιc′′,j Ker ιc,j Ker ιc′,j Since c′, c′′ are chains of monomorphisms, Ker ιc′′,j = 0 and Ker ιc′,j = 0. So is Ker ιc,j . ■ 10 Z. Chen, J. Xiao and F. Xu 2.4 Type Recall that A admits a complete set simple objects {S1, . . . , Sn} up to isomorphism, so we can consider composition factors of an object in A with finite length. Definition 2.7. Given a chain of monomorphisms c = ( Lm ιc,m−−→ Lm−1 −→ · · · −→ L1 ιc,1−−→ L0 ) and two sequences i = (i1, . . . , im), a = (a1, . . . , am), where ij ∈ {1, . . . , n}, aj ∈ {0, 1}, c is called of type (i,a) if Coker ιc,j ∼= Sij when aj = 1 and 0 otherwise for 1 ⩽ j ⩽ m. We denote the set of all chains of monomorphisms of type (i,a) by Fmono i,a and the set of all chains of monomorphisms of type (i,a) with L0 ∼= L by Fmono i,a,L . Lemma 2.8. If (c′, c′′) ∈ Fmono i,a′,M ×Fmono i,a′′,N and a′j + a′′j ⩽ 1 for 1 ⩽ j ⩽ m, then ImBc′,c′′ ⊆ Fmono i,a′+a′′ . Proof. Since a′j + a′′j ⩽ 1 for 1 ⩽ j ⩽ m, Fmono i,a′+a′′ is well defined. By the snake lemma, we have an exact sequence Ker ιc′,j → Coker ιc′′,j → Coker ιc,j → Coker ιc′,j → 0. Since c′ is a chain of monomorphisms, Ker ιc′,j = 0. So the composition factors of Coker ιc,j are the union of composition factors of Coker ιc′,j and Coker ιc′′,j , which are one copy of Sij . ■ If a = (1, . . . , 1), we simply write Fmono i instead of Fmono i,a . 3 Abelian categories with Ext-symmetry and the multiplication formula 3.1 Quantum cluster function Now we introduce the concept of the quantum cluster functions over A. Let di,a be a formal notation representing a type of chains of monomorphisms and Mq := ⊕ (i,a) Qdi,a be the Q-space spanned by all di,a. For each object L in A, we define a Q-valued linear function δL on Mq by δL(di,a) = \|Fmono i,a,L \| and call such a function the quantum cluster function of L. The core purpose of this section is to study the relationship between quantum cluster functions of objects in A related by a short exact sequence. 3.2 Mappings with affine fibers We denote by EFi,a(M,N) := { ([ϵ], c) \| [ϵ] ∈ Ext1A(M,N), c ∈ Fmono i,a,mtϵ } the set of all pairs of extensions of M , N and chain of monomorphisms of the middle term of type (i,a). The Multiplication Formulas of Weighted Quantum Cluster Functions 11 From ([ϵ], c) ∈ EFi,a(M,N), we can induce two chains of monomorphisms ending with M , N respectively. More precisely, we have 0 N L M 0 0 N1 L1 M1 0 ... ... ... 0 Nm−1 Lm−1 Mm−1 0 0 Nm Lm Mm 0, i p ιc′′,1 i1 ιc,1 p1 ιc′,1 im−1 pm−1 im ιc′′,m pm ιc,m ιc′,m where Nj = i−1(Im ιc,1 ◦ · · · ◦ ιc,j), Mj = p(Im ιc,1 ◦ · · · ◦ ιc,j), ιc′′,j , ιc′,j are natural embeddings and ij , pj are naturally induced by i and p respectively for 1 ⩽ j ⩽ m. We write this assignment as ϕMN : EFi,a(M,N) → ∐ a′+a′′=a Fmono i,a′,M ×Fmono i,a′′,N . The coproduct runs over all pairs (a′,a′′) with a′ + a′′ = a since the composition factor at each level is fixed by a. Given two chains of monomorphisms c′, c′′ ending with M and N respectively, we are inter- ested in the fiber ϕ−1 MN (c ′, c′′). Lemma 3.1. Consider chains of monomorphisms c′, c′′ ending with M , N respectively. There exists a bijection between Kerβc′,c′′ and ϕ −1 MN (c ′, c′′), given by ([ϵ0], . . . , [ϵm]) 7−→ ([ϵ0], Bc′,c′′([ϵ0], . . . , [ϵm])). Proof. Consider chains of monomorphisms c′, c′′ ending with M , N respectively and [ϵ0] ∈ Ext1A(M,N) with mtϵ0 = L as in the diagram ϵ0 : 0 N L M 0. N1 M1 ... ... Nm−1 Mm−1 Nm Mm ιc′′,1 ιc′,1 ιc′′,m ιc′,m For any c ∈ Fmono m,L , the following two statements are equivalent: 12 Z. Chen, J. Xiao and F. Xu (1) there are extensions ([ϵ1], . . . , [ϵm]) satisfying ([ϵ0], [ϵ1], . . . , [ϵm]) ∈ Kerβc′,c′′ such that Bc′,c′′([ϵ0], [ϵ1], . . . , [ϵm]) = c. By definition of Bc′,c′′ , this means that there is a commuta- tive diagram ϵ0 : 0 N L M 0 ϵ1 : 0 N1 L1 M1 0 ... ... ... ϵm−1 : 0 Nm−1 Lm−1 Mm−1 0 ϵm : 0 Nm Lm Mm 0, i p i1 ιc,1 p1 im−1 pm−1 im pm ιc,m where ιc,i are given by composing the middle vertical morphisms in diagrams of the form 0 // Ni−1 // Li−1 //Mi−1 // 0 0 // Ni−1 // L′ i // g OO Mi // ιc′,i OO 0 0 // Ni // ιc′′,i OO Li // f OO Mi // 0; (2) ϕMN ([ϵ0], c) = (c′, c′′). By definition of ϕMN , this means that there is a commutative diagram ϵ0 : 0 N L M 0, N1 L1 M1 ... ... ... Nm−1 Lm−1 Mm−1 Nm Lm Mm i p i1 ιc,1 p1 im−1 pm−1 im pm ιc,m where Nj = i−1 j (Lj), Mj = pj(Lj), and ij and pj are restrictions of i and p. Assume (1) is true. Since all rows in (1) are short exact sequences, we have Nj = i−1 j (Lj) and Mj = pj(Lj), i.e., (2) holds. Conversely, given (2), by definitions of Nj andMj , we know that Nj ij−→ Lj pj−→Mj is a short The Multiplication Formulas of Weighted Quantum Cluster Functions 13 exact sequence for 1 ⩽ j ⩽ m. Then we obtain a commutative diagram ϵ0 : 0 N L M 0 ϵ1 : 0 N1 L1 M1 0 ... ... ... ϵm−1 0 Nm−1 Lm−1 Mm−1 0 ϵm : 0 Nm Lm Mm 0, i p i1 ιc,1 p1 im−1 pm−1 im pm ιc,m which consists of chains of monomorphisms as columns and short exact sequences as rows. From the uniqueness of pushouts and pullbacks, the middle column c must be isomorphic to Bc′,c′′([ϵ0], [ϵ1], . . . , [ϵm]) where ([ϵ1], . . . , [ϵm]) is as shown in the diagram. ■ Let [ϵ] ∈ Ext1A(M,N) with mtϵ = L, and let c′ ∈ Fmono m,M , c′′ ∈ Fmono m,N . Then the chains c ∈ Fmono m,L such that ϕMN ([ϵ], c) = (c′, c′′) are precisely those of the form Bc′,c′′([ϵ], . . . , [ϵm]) with ([ϵ], . . . , [ϵm]) ∈ Kerβc′,c′′ . On the other hand, note that ϕMN describes the relationship among Fmono m,L , Fmono m,M and Fmono m,N , and hence the relationship among characters δM , δN and δL. In order to describe the structure more clearly, we need to refine this map. Fix [ϵ] ∈ Ext1A(M,N), and define ϕ[ϵ] := ϕMN ([ϵ],−) : Fmono i,a,L → ∐ a′+a′′=a Fmono i,a′,M ×Fmono i,a′′,N , where L = mtϵ and write the preimage as Fmono i,a,L ([ϵ],a′,a′′) = ϕ−1 [ϵ] (F mono i,a′,M ×Fmono i,a′′,N ). Since the coproduct∐ a′+a′′=a Fmono i,a′,M ×Fmono i,a′′,N is a disjoint union, we naturally have Fmono i,a,L = ∐ a′+a′′=a Fmono i,a,L ([ϵ],a′,a′′). However, the structure of a fiber of ϕ[ϵ] : Fmono i,a,L ([ϵ],a′,a′′) → Fmono i,a′,M ×Fmono i,a′′,N is heavily dependent on the relation between [ϵ] and specific chains (c′, c′′). The following corollary describes the image of ϕ[ϵ] restricted to Fmono i,a,L ([ϵ],a′,a′′). Corollary 3.2. Given (c′, c′′) in Fmono i,a′,M ×Fmono i,a′′,N , (c′, c′′) ∈ ϕ[ϵ](Fmono i,a,L ([ϵ],a′,a′′)) if and only if [ϵ] ∈ p0Kerβc′,c′′ , where p0 is the projection from ⊕m j=0 Ext 1 A(Mj , Nj) to Ext1A(M,N). 14 Z. Chen, J. Xiao and F. Xu Now, in the case that [ϵ] ∈ p0Kerβc′,c′′ , we calculate the fiber ϕ−1 [ϵ] (c ′, c′′). From Lemma 2.3, we know that the assignment Bc′,c′′ which maps a family of extensions ([ϵ0], . . . , [ϵm]) to a chain of monomorphisms ending with L = mtϵ0 is injective. We denote K(c′, c′′, [ϵ]) := {([ϵ0], . . . , [ϵm]) ∈ Kerβc′,c′′ \|[ϵ0] = [ϵ]}. Then we have Kerβc′,c′′ = ∐ [ϵ]∈p0 Kerβc′,c′′ K(c′, c′′, [ϵ]). Then by Lemma 3.1, we have ϕ−1 [ϵ] (c ′, c′′) ̸= ∅ only if [ϵ] ∈ p0Kerβc′,c′′ and obtain the immediate corollary Corollary 3.3. Given (c′, c′′) in Fmono i,a′,M ×Fmono i,a′′,N , (1) Bc′,c′′ : K(c′, c′′, [ϵ]) → ϕ−1 [ϵ] (c ′, c′′) is bijective; (2) If [ϵ] /∈ p0Kerβc′,c′′, ϕ −1 [ϵ] (c ′, c′′) = ∅. 3.3 Dual case Recall that the abelian category A considered here admits Ext-symmetry. That is to say, for any two objects M , N in A, there is a natural isomorphism EM,N : Ext1A(M,N) ∼= DExt1A(N,M), where D = Homk(−, k) is the linear dual. That is Proposition 3.4. Given λ ∈ Hom(N,N ′), ρ ∈ Hom(M ′,M), [ϵ] ∈ Ext1A(M,N) and [η] ∈ Ext1A(N ′,M ′), we have EM ′,N ′(λ ◦ [ϵ] ◦ ρ)([η]) = EM,N ([ϵ])(ρ ◦ [η] ◦ λ). Consider the linear map βc′,c′′ : m⊕ j=0 Ext1(Mj , Nj) −→ m−1⊕ j=0 Ext1(Mj+1, Nj), ([ϵ0], . . . , [ϵm]) 7−→ ([ϵj−1] ◦ ιc′,j − ιc′′,j ◦ [ϵj ], 1 ⩽ j ⩽ m). Since it is a linear map between finite dimensional spaces, we can calculate the dual of βc′,c′′ explicitly using Proposition 3.4 β′c′′,c′ = Dβc′,c′′ : m−1⊕ j=0 Ext1(Nj ,Mj+1) −→ m⊕ j=0 Ext1(Nj ,Mj), ([η0], . . . , [ηm−1]) 7−→ (ιc′,j+1 ◦ [ηj ]− [ηj−1] ◦ ιc′′,j , 0 ⩽ j ⩽ m), where [η−1] := 0 and [ηm] := 0. To decide whether a pair of chains (c′′, c′) is located in the image of ϕ[η], we need Lemma 3.5. Given (c′′, c′) in Fmono i,a′′,N ×Fmono i,a′,M , (c′′, c′) ∈ ϕ[η](Fmono i,a,L ([η],a′′,a′)) if and only if [η] ∈ Imβ′c′′,c′ ∩ Ext1A(N,M), where the intersection is realized through regarding Ext1A(N,M) as a linear subspace of ⊕m j=0 Ext1A(Nj ,Mj). The Multiplication Formulas of Weighted Quantum Cluster Functions 15 Proof. By definition of β′c′′,c′ , [η] ∈ Imβ′c′′,c′ ∩ Ext1A(N,M) if and only if there exists a family of extensions ([η0], . . . , [ηm−1]) such that (1) ιc′,1 ◦ [η0] = [η]; (2) ιc′,j+1 ◦ [ηj ] = [ηj−1] ◦ ιc′′,j for 1 ⩽ j ⩽ m− 1; (3) [ηm−1] ◦ ιc′′,m = 0. The third condition is always satisfied since Ext1A(Nm,Mm) = Ext1A(0, 0) = 0. Since a′ + a′′ = a is a 0-1 sequence, either ιc′,j or ιc′′,j is an isomorphism. In either case, one can check that the first and second condition are equivalent to the condition that the pushout of [ηj ] along ιc′,j coincides with the pullback of [ηj−1] along ιc′′,j and the pushout of extension betweenMj and Nj along ιM,j coincides with the pullback of extension betweenMj−1 and Nj−1 along ιN,j Nj−1 Mj−1 Nj−1 Mj Nj , Mj Nj . Mj+1 pullback pushout [ηj−1] ιc′,j+1◦[ηj ] [ηj−1]◦ιc′′,j ιc′′,j ιc′,j ιc′′,j [ηj ] ιc′,j+1 According to Lemma 3.1, the condition that the pullback and the pushout coincide ensures the preimage ϕ−1 [η] (c ′′, c′) is non-empty. ■ Remark 3.6. In fact, we can check that (c′′, c′) ∈ ϕ[η](Fmono i,a,L ([η],a′′,a′)) if and only if [η] ∈ p0Kerβc′′,c′ as in Corollary 3.2. By Lemmas 3.1 and 3.5, we obtain the dual of Corollary 3.3. Corollary 3.7. Given (c′′, c′) in Fmono i,a′′,N ×Fmono i,a′,M , (1) Bc′′,c′ : K(c′′, c′, [η]) → ϕ−1 [η] (c ′′, c′) is bijective; (2) If [η] /∈ Imβ′c′′,c′ ∩ Ext1A(N,M), ϕ−1 [η] (cN , cM ) = ∅. 3.4 Cardinality Based on several lemmas above, we can refine the calculation of δL. Recall that all extension groups are finite dimensional over a finite field. Since Fmono i,a,L = ∐ a′+a′′=a Fmono i,a,L ([ϵ],a′,a′′), 16 Z. Chen, J. Xiao and F. Xu we can write \|Fmono i,a,L \| = ∫ a′+a′′=a \|Fmono i,a,L ([ϵ],a′,a′′)\|. Note that the decomposition of Fmono i,a,L depends on the choice of the short exact sequence ϵ. By linear algebra, \|K(c′, c′′, [ϵ])\| = \|K(c′, c′′,0)\| if [ϵ] ∈ p0Kerβc′,c′′ . Note that K(c′, c′′,0) is a vector space over k so we can define k(c′, c′′) := dimkK(c′, c′′,0). Moreover, Corollary 3.3 shows that if [ϵ] /∈ p0Kerβc′,c′′ , then \|ϕ−1 [ϵ] (c ′, c′′)\| = 0, and otherwise \|ϕ−1 [ϵ] (c ′, c′)\| = \|K(c′, c′′, [ϵ])\| = qk(c ′,c′′). So if mtϵ = L, we have δL(di,a) = \|Fmono i,a,L \| = ∫ a′+a′′=a \|Fmono i,a,L ([ϵ],a′,a′′)\| = ∫∫ a′+a′′=a,(c′,c′′)∈Fmono i,a′,M×Fmono i,a′′,N ∣∣ϕ−1 [ϵ] (c ′, c′′) ∣∣ = ∫∫ a′+a′′=a,(c′,c′′)∈ϕ[ϵ](Fmono i,a,L ([ϵ],a′,a′′)) qk(c ′,c′′), where all integrals are finite sums. Remark 3.8. Since different [ϵ], [ϵ′] in Ext1A(M,N) may admit the same middle term mtϵ = mtϵ′ = L, the refinement of δL(di,a) in the above equation depends on the choice of [ϵ]. Moreover, Corollary 3.7 shows if [η] /∈ Imβ′c′′,c′ ∩ Ext1A(N,M), \|ϕ−1 [η] (c ′′, c′′)\| = 0 and if [η] ∈ Imβ′c′′,c′ ∩ Ext1A(N,M), \|ϕ−1 [η] (c ′′, c′)\| = \|K(c′′, c′, [η])\| = qk(c ′′,c′). So if mtη = L, we have δL(di,a) = \|Fmono i,a,L \| = ∫ a′′+a′=a \|Fmono i,a,L ([η],a′′,a′)\| = ∫∫ a′′+a′=a,(c′′,c′)∈Fmono i,a′′,N×Fmono i,a′,M ∣∣ϕ−1 [η] (c ′′, c′) ∣∣ = ∫∫ a′′+a′=a,(c′′,c′′)∈ϕ[η](Fmono i,a,L ([η],a′′,a′)) qk(c ′′,c′). 3.5 Weight Based on the calculation of δL, we introduce the notion of weight functions and weighted quan- tum cluster functions. Denote Fmono := ⋃ m∈NFmono m . We define MF := Fmono ×Fmono = {(c′, c′′)\|c′ ∈ Fmono m,M , c′′ ∈ Fmono m,N , M,N ∈ A, m ∈ N} and set ZMF := { f : MF×Ext1A → Z\|f(c′, c′′, [ϵ]) = 0 unless c′0 = qtϵ, c′′0 = stϵ } , where Ext1A = ∐ M,N∈A Ext1A(M,N) and Z = { n 2 \| n ∈ Z } is the set of all half integers. The functions in ZMF are called weight functions. Given [ϵ] ∈ Ext1A(M,N), we define ZMF[ϵ] := {f ∈ ZMF \| f(c′, c′′, [ρ]) = 0 if [ρ] ̸= [ϵ]}. Given f ∈ ZMF[ϵ], we write f(c′, c′′, [ϵ]) instead as f(c′, c′′). The Multiplication Formulas of Weighted Quantum Cluster Functions 17 Definition 3.9 (weighted quantum cluster function). Given a weight function f ∈ ZMF[ϵ], the weighted quantum cluster function f ∗[ϵ] δL is the linear function on Mq defined by f ∗[ϵ] δL(di,a) = ∫∫ a′+a′′=a,(c′,c′′)∈Fmono i,a′,M×Fmono i,a′′,N ∣∣ϕ−1 [ϵ] (c ′, c′′) ∣∣ · qf(c′,c′′) = ∫∫ a′+a′′=a,(c′,c′′)∈ϕ[ϵ](Fmono i,a,L ([ϵ],a′,a′′)) qk(c ′,c′′) · qf(c′,c′′), where M = qtϵ, N = stϵ and L = mtϵ. If f ∈ ZMF is the zero function, we have f ∗[ϵ] δL = δL. So weight functions provide q- deformations of δL. 3.6 Multiplication We now introduce the multiplication of weighted quantum cluster functions. Definition 3.10 (multiplication of quantum cluster functions). Given M , N in A, we define the multiplication of quantum cluster functions as δM ∗ δN = δM⊕N . By definition, we have Proposition 3.11. The multiplication ∗ is associative and commutative. We denote the zero element in Ext1A(M,N) by 0MN . Remark 3.12. From the refinement of the quantum cluster function, we have δM⊕N (di,a) = ∫ a′+a′′=a \|Fmono i,a,M⊕N (0MN ,a ′,a′′)\| = ∫∫ a′+a′′=a,(c′,c′′)∈Fmono i,a′,M×Fmono i,a′′,N qk(c ′,c′′). Notice that the image of ϕ0MN is the whole of Fmono i,a′,M × Fmono i,a′′,N since any two chains c′, c′′ can be assembled into a chain ending with M ⊕N through direct sum. On the other hand, the convolution of the two functions δM and δN can be formally defined as∫ a′+a′′=a δM (di,a′) · δN (di,a′′) = ∫ a′+a′′=a \|Fmono i,a′,M \| · \|Fmono i,a′′,N \| = ∫∫ a′+a′′=a,(c′,c′′)∈Fmono i,a′,M×Fmono i,a′′,N 1. So the definition of multiplication is a q-deformation of conventional convolution. Definition 3.13 (multiplication of weighted quantum cluster functions). Given weighted quan- tum cluster functions f ∗[ϵ′] δM and g ∗[ϵ′′] δN , we define the multiplication as (f ∗[ϵ′] δM ) ∗ (g ∗[ϵ′′] δN ) = h ∗0MN δM⊕N , where h(c′, c′′, [ϵ]) = 0 unless [ϵ] = 0MN and h(c′, c′′) = f(ϕ[ϵ′](c ′)) + g(ϕ[ϵ′′](c ′′)). Proposition 3.14. The multiplication operation from Definition 3.13 is associative. 18 Z. Chen, J. Xiao and F. Xu Proof. Given fi ∈ ZMF[ϵi] such that mt ϵi =Mi for i = 1, 2, 3, we set ((f1 ∗[ϵ1] δM1) ∗ (f2 ∗[ϵ2] δM2)) ∗ (f3 ∗[ϵ3] δM3) = f(12)3 ∗0M1⊕M2,M3 δM1⊕M2⊕M3 , (f1 ∗[ϵ1] δM1) ∗ ((f2 ∗[ϵ2] δM2) ∗ (f3 ∗[ϵ3] δM3)) = f1(23) ∗0M1,M2⊕M3 δM1⊕M2⊕M3 , where f(12)3 ∈ ZMF0M1⊕M2,M3, f1(23) ∈ ZMF0M1,M2⊕M3. Note that the non-weighted parts of the two compositions are the same. We have (f1 ∗[ϵ1] δM1) ∗ (f2 ∗[ϵ2] δM2) = f(12) ∗0M1⊕M2 δM1⊕M2 , where f(12) ∈ ZMF0MN and f(12)(c1, c2) = f1(ϕ[ϵ1](c1)) + f2(ϕ[ϵ2](c2)). Then f(12)3(c1 ⊕ c2, c3) = f(12)(c1, c2) + f3(ϕ[ϵ3](c3)) = f1(ϕ[ϵ1](c1)) + f2(ϕ[ϵ2](c2)) + f3(ϕ[ϵ3](c3)). Similarly, we have that f1(23)(c1, c2 ⊕ c3) = f1(ϕ[ϵ1](c1)) + f(23)(c2, c3) = f1(ϕ[ϵ1](c1)) + f2(ϕ[ϵ2](c2)) + f3(ϕ[ϵ3](c3)). Then we have f(12)3 ∗0M1⊕M2⊕M3 δM1⊕M2⊕M3(di,a) = ∫∫ a′+a′′+a′′′=a,(c1,c2,c3) qk(12)3(c1,c2,c3) · qf1(ϕ[ϵ1](c1))+f2(ϕ[ϵ2](c2))+f3(ϕ[ϵ3](c3)) and f1(23) ∗0M1⊕M2⊕M3 δM1⊕M2⊕M3(di,a) = ∫∫ a′+a′′+a′′′=a,(c1,c2,c3) qk1(23)(c1,c2,c3) · qf1(ϕ[ϵ1](c1))+f2(ϕ[ϵ2](c2))+f3(ϕ[ϵ3](c3)), where the second integral in both equations runs over Fmono i,a′,M1 ×Fmono i,a′′,M2 ×Fmono i,a′′′,M3 and the functions k(12)3 and k1(23) compute the dimensions of fibers of the diagonal maps ϕ(12)3 and ϕ1(23) respectively in the diagram Fmono i,a,M1⊕M2⊕M3 ∐ Fmono i,a′+a′′,M1⊕M2 ×Fmono i,a′′,M3 ∐ Fmono i,a′,M1 ×Fmono i,a′′+a′′′,M2⊕M3 ∐ Fmono i,a′,M1 ×Fmono i,a′′,M2 ×Fmono i,a′′′,M3 . ϕM1,M2⊕M3,0M1,M2⊕M3 ϕM1⊕M2,M3,0M1⊕M2,M3 ϕ(12)3 ϕ1(23) ∐ ϕM1,M2,0M1,M2 ×id ∐ id×ϕM2,M3,0M2,M3 From the associativity of δM ∗ δN , we know that the diagram commutes. So k(12)3 = k1(23). ■ The Multiplication Formulas of Weighted Quantum Cluster Functions 19 Remark 3.15. By definition, δM ∗ δN = δN ∗ δM . This implies∫∫ a′+a′′=a,(c′,c′′)∈Fmono i,a′,M×Fmono i,a′′,N qk(c ′,c′′) = ∫∫ a′+a′′=a,(c′,c′′)∈Fmono i,a′,M×Fmono i,a′′,N qk(c ′′,c′). However, it need not be the case that k(c′, c′′) = k(c′′, c′), since one value is computed using 0MN and the other using 0NM . So even though the weight functions h and h′ in (f ∗[ϵ′] δM ) ∗ (g ∗[ϵ′′] δN ) = h ∗0MN δM⊕N , (g ∗[ϵ′′] δN ) ∗ (f ∗[ϵ′] δM ) = h′ ∗0NM ∗δM⊕N satisfy h(c′, c′′) = h′(c′′, c′) by definition, we still can not deduce that∫∫ a′+a′′=a,(c′,c′′)∈Fmono i,a′,M×Fmono i,a′′,N qk(c ′,c′′) · qh(c′,c′′) = ∫∫ a′+a′′=a,(c′,c′′)∈Fmono i,a′,M×Fmono i,a′′,N qk(c ′′,c′) · qh′(c′′,c′). 3.7 The projectivization of Ext1A(M,N) Since Ext1A(M,N) is a finite dimensional vector space, we can consider PExt1A(M,N). We denote the equivalence class of [ϵ] in PExt1A(M,N) by P[ϵ]. The core aim in this subsection is to check that the multiplication of weighted quantum cluster functions is still well defined if we replace [ϵ] by P[ϵ]. In this subsection, we fix a non-zero parameter λ in the field k. Recall the mapping ϕMN : EFi,a(M,N) → ∐ a′+a′′=a Fmono i,a′,M ×Fmono i,a′′,N with affine fibers. By Lemma 3.1 and the linearity of βc′,c′′ , [ϵ] ∈ p0Kerβc′,c′′ if and only if λ[ϵ] ∈ p0Kerβc′,c′′ . So in this case,∣∣ϕ−1 [ϵ] (c ′, c′′) ∣∣ = \|K(c′, c′′, [ϵ])\| = \|K(c′, c′′, λ[ϵ])\| = ∣∣ϕ−1 λ[ϵ](c ′, c′′) ∣∣. Otherwise, they are all zero. So we have Proposition 3.16. If f, f ′ ∈ ZMF satisfy f(c′, c′′, [ϵ]) = f ′(c′, c′′, λ[ϵ]) for all (c′, c′′) ∈ Fmono i,a′,M ×Fmono i,a′′,N , then f ∗[ϵ] δL = f ∗λ[ϵ] δL. Proof. f ∗[ϵ] δL(di,a) = ∫∫ a′+a′′=a,(c′,c′′)∈Fmono i,a′,M×Fmono i,a′′,N \|ϕ−1 [ϵ] (c ′, c′′)\| · qf(c′,c′′,[ϵ]) = ∫∫ a′+a′′=a,(c′,c′′)∈Fmono i,a′,M×Fmono i,a′′,N \|ϕ−1 λ[ϵ](c ′, c′′)\| · qf(c′,c′′,λ[ϵ]) = f ∗λ[ϵ] δL(di,a). ■ Recall that given f ∗[ϵ′] δM and g ∗[ϵ′′] δN , we define (f ∗[ϵ′] δM ) ∗ (g ∗[ϵ′′] δN ) = h ∗0MN δM⊕N , where h(c′, c′′) = f(ϕ[ϵ′](c ′)) + g(ϕ[ϵ′′](c ′′)). Since ϕ[ϵ](c) = ϕλ[ϵ](c) for the corresponding [ϵ] and c, we have (f ∗[ϵ′] δM ) ∗ (g ∗[ϵ′′] δN ) = (f ∗λ[ϵ′] δM ) ∗ (g ∗µ[ϵ′′] δN ) for any non-zero λ and µ in k. 20 Z. Chen, J. Xiao and F. Xu 3.8 Multiplication of weight functions Definition 3.17. Given weight functions f, g ∈ ZMF, f ∗[η] g is defined by f ∗[η] g(c′, c′′, [ϵ]) = { f(c′, c′′, [ϵ]) + g(c′, c′′, [ϵ]), if [ϵ] = [η], 0, otherwise. In particular, f ∗[η] g ∈ ZMF[η]. Definition 3.18. Given (f ∗[ϵ′] δM ) ∗ (g ∗[ϵ′′] δN ) = h ∗0M⊕N δM⊕N and [ϵ] ∈ Ext1A(M,N), we define Sfg ∈ ZMF[ϵ] by Sfg(c′, c′′, [ϵ]) = h(c′, c′′) = f(ϕ[ϵ′](c ′)) + g(ϕ[ϵ′′](c ′′)). Corollary 3.19. Given f ∗[ϵ]′ δM and g∗[ϵ]′′ δN , for any [ϵ] ∈ Ext1A(M,N) and [η] ∈ Ext1A(N,M), we have Sfg(c′, c′′, [ϵ]) = Sgf (c′′, c′, [η]). Proof. Both sides are equal to f(ϕ[ϵ′](c ′)) + g(ϕ[ϵ′′](c ′′)). ■ 3.9 Multiplication formula and balanced pairs Recall the linear map βc′,c′′ : m⊕ j=0 Ext1(Mj , Nj) −→ m−1⊕ j=0 Ext1(Mj+1, Nj), ([ϵ0], . . . , [ϵm]) 7−→ ([ϵj−1] ◦ ιc′,j − ιc′′,j ◦ [ϵj ], 1 ⩽ j ⩽ m) and its dual β′c′′,c′ = Dβc′,c′′ : m−1⊕ j=0 Ext1(Nj ,Mj+1) −→ m⊕ j=0 Ext1(Nj ,Mj) ([η0], . . . , [ηm−1]) 7−→ (ιc′,j+1 ◦ [ηj ]− [ηj−1] ◦ ιc′′,j , 0 ⩽ j ⩽ m), where [η−1] := 0 and [ηm] := 0. So we have Lemma 3.20. For any chains of morphisms c′, c′′ ending with M , N respectively, Kerβc′,c′′ = (Imβ′c′′,c′) ⊥. In particular, dimk p0Kerβc′,c′′ + dimk ( Imβ′c′′,c′ ∩ Ext1A(N,M) ) = dimk Ext 1 A(M,N). Based on this lemma, we introduce several special weight functions. Definition 3.21. There are three weight functions in ZMF defined as (1) fhom(c ′′, c′, [η]) = k(c′, c′′)− k(c′′, c′); (2) f+ext(c ′, c′′, [ϵ]) = dimk ( Imβ′c′′,c′ ∩ Ext1A(N,M) ) ; (3) f−ext(c ′′, c′, [η]) = dimk p0Kerβc′,c′′ for any M , N in A, c′ ∈ Fmono m,M , c′′ ∈ Fmono m,N , [ϵ] ∈ Ext1A(M,N) and [η] ∈ Ext1A(N,M). Note that all of these functions are constant in the extension. The Multiplication Formulas of Weighted Quantum Cluster Functions 21 Definition 3.22. Given a pair of weight functions (f+, f−), set σ1 ( f+ ) := ∫ Pϵ∈Pp0 Kerβc′,c′′ qk(c ′,c′′)+f+(c′,c′′,[ϵ]), σ2(f −) := ∫ Pη∈P Imβ′ c′,c′′∩Ext 1 A(N,M) qk(c ′′,c′)+f−(c′′,c′,[η]). This pair is called pointwise balanced if qdimk Ext1A(M,N) − 1 q − 1 · qk(c′,c′′) = σ1 ( f+ ) + σ2(f −) holds for any M,N ∈ A and (c′, c′′) ∈ Fmono i,a′,M ×Fmono i,a′′,N . Proposition 3.23. The following two pairs of weighted functions (1) ( f+ext, fhom ) ; (2) (0, f−ext + fhom) are pointwise balanced. Proof. We first consider σ2(fhom) = ∫ Pη∈P Imβ′ c′′,c′∩Ext 1 A(N,M) qk(c ′′,c′)+fhom((c′′,c′),[η]) = ∫ Pη∈P Imβ′ c′′,c′∩Ext 1 A(N,M) qk(c ′,c′′). Notice that k(c′, c′′) and k(c′′, c′) are independent of [ϵ] and [η] as long as they are located in the domain of integration as follows. Since dimk p0Kerβc′,c′′ + dimk ( Imβ′c′′,c′ ∩ Ext1A(N,M) ) = dimk Ext 1 A(M,N), we have two equalities qdimk Ext1A(M,N) − 1 q − 1 = q dimk(Imβ′ c′′,c′∩Ext 1 A(N,M)) · q dimk p0 Kerβc′,c′′ − 1 q − 1 + q dimk Imβ′ c′′,c′∩Ext 1 A(N,M) − 1 q − 1 = qdimk p0 Kerβc′,c′′ − 1 q − 1 + qdimk p0 Kerβc′,c′′ · q dimk Imβ′ c′′,c′∩Ext 1 A(N,M) − 1 q − 1 . So by definition, σ1 ( f+ext ) + σ2(fhom) = ∫ Pϵ∈Pp0 Kerβc′,c′′ qk(c ′,c′′) · qdimk(Imβ′ c′′,c′∩Ext 1 A(N,M)) + ∫ Pη∈P Imβ′ c′′,c′∩Ext 1 A(N,M) qk(c ′,c′′) = qk(c ′,c′′) · ( qdimk p0 Kerβc′,c′′ − 1 q − 1 · qdimk(Imβ′ c′′,c′∩Ext 1 A(N,M)) + q dimk Imβ′ c′′,c′∩Ext 1 A(N,M) − 1 q − 1 ) 22 Z. Chen, J. Xiao and F. Xu = qk(c ′,c′′) · q dimk Ext1A(M,N) − 1 q − 1 and σ1(0) + σ2(f − ext + fhom) = ∫ Pϵ∈Pp0 Kerβc′,c′′ qk(c ′,c′′) + ∫ Pη∈P Imβ′ c′′,c′∩Ext 1 A(N,M) qk(c ′,c′′) · qdimk p0 Kerβc′,c′′ = qk(c ′,c′′) · ( qdimk p0 Kerβc′,c′′ − 1 q − 1 + q dimk Imβ′ c′′,c′∩Ext 1 A(N,M) − 1 q − 1 · qdimk p0 Kerβc′,c′′ ) = qk(c ′,c′′) · q dimk Ext1A(M,N) − 1 q − 1 . ■ Theorem 3.24. If a pair of weight functions ( f+, f− ) in ZMF is pointwise balanced, then for any weighted quantum cluster functions f ∗[ϵ′] δM and g ∗[ϵ′′] δN , we have∣∣PExt1A(M,N) ∣∣(f ∗[ϵ′] δM ) ∗ (g ∗[ϵ′′] δN ) = ∫ Pϵ∈PExt1A(M,N) ( f+ ∗[ϵ] Sfg ) ∗[ϵ] δmtϵ + ∫ Pη∈PExt1A(N,M) (f− ∗[η] Sgf ) ∗[η] δmtη. Proof. We simplify the equality in the theorem to l.h.s. = Σ1 ( f+ ) +Σ2(f −). Just for simplicity, we omit some independent variables of functions which are obvious in following calculation. For example, Sfg(c′, c′′,0MN ) is simplified to Sfg. Direct calculation: l.h.s.(di,a) = ∣∣PExt1A(M,N) ∣∣(f ∗[ϵ′] δM ) ∗ (g ∗[ϵ′′] δN )(di,a) = ∣∣PExt1A(M,N) ∣∣ ∫∫ a′+a′′=a,(c′,c′′)∈Fmono i,a′,M×Fmono i,a′′,N qk(c ′,c′′)+Sfg = ∫∫ a′+a′′=a,(c′,c′′)∈Fmono i,a′,M×Fmono i,a′′,N qSfg · q dimk Ext1A(M,N) − 1 q − 1 · qk(c′,c′′), Σ1 ( f+ ) (di,a) = ∫ Pϵ∈PExt1A(M,N) ( f+ ∗[ϵ] Sfg ) ∗[ϵ] δmtϵ(di,a) = ∫∫∫ Pϵ∈PExt1A(M,N),a′+a′′=a,(c′,c′′)∈ϕ[ϵ](Fmono i,a,L ([ϵ],a′,a′′)) qk(c ′,c′′)+Sfg+f+(c′,c′′,[ϵ]) = ∫∫ a′+a′′=a,(c′,c′′)∈Fmono i,a′,M×Fmono i,a′′,N qSfg ∫ Pϵ∈Pp0 Kerβc′,c′′ qk(c ′,c′′)+f+(c′,c′′,[ϵ]) = ∫∫ a′+a′′=a,(c′,c′′)∈Fmono i,a′,M×Fmono i,a′′,N qSfg · σ1 ( f+ ) , Σ2(f −)(di,a) = ∫ Pη∈PExt1A(N,M) (f− ∗[η] Sgf ) ∗[η] δmtη(di,a) = ∫∫∫ Pη∈PExt1A(N,M),a′+a′′=a,(c′′,c′)∈ϕ[η](Fmono i,a,L ([η],a′′,a′)) qk(c ′′,c′)+Sgf+f−(c′′,c′,[η]) = ∫∫ a′+a′′=a,(c′,c′′)∈Fmono i,a′,M×Fmono i,a′′,N qSfg ∫ Pη∈P Imβ′ c′′,c′∩Ext 1 A(N,M) qk(c ′′,c′)+f−(c′′,c′,[η]) The Multiplication Formulas of Weighted Quantum Cluster Functions 23 = ∫∫ a′+a′′=a,(c′,c′′)∈Fmono i,a′,M×Fmono i,a′′,N qSfg · σ2(f−). By Definition 3.22, we have l.h.s. = Σ1(f +) + Σ2(f −). ■ From Proposition 3.23 and Theorem 3.24, we have Theorem 3.25 (multiplication formula of weighted quantum cluster functions). Let A be a Hom-finite, Ext-finite abelian category with Ext-symmetry over a finite field k = Fq such that the iso-classes of objects form a set. For any weighted quantum cluster functions f ∗[ϵ′] δM and g ∗[ϵ′′] δN such that Ext1A(M,N) ̸= 0, we have∣∣PExt1A(M,N) ∣∣(f ∗[ϵ′] δM ) ∗ (g ∗[ϵ′′] δN ) = ∫ Pϵ∈PExt1A(M,N) ( f+ext ∗[ϵ] Sfg ) ∗[ϵ] δmtϵ + ∫ Pη∈PExt1A(N,M) (fhom ∗[η] Sgf ) ∗[η] δmtη = ∫ Pϵ∈PExt1A(M,N) Sfg ∗[ϵ] δmtϵ + ∫ Pη∈PExt1A(N,M) (f−ext ∗[η] fhom ∗[η] Sgf ) ∗[η] δmtη. Theorem 3.24 shows that every pointwise balanced pair is a balanced pair as in the following definition. Definition 3.26. A pair of weight functions ( f+, f− ) in ZMF is called a balanced pair if for any weighted quantum cluster functions f ∗[ϵ′] δM and g ∗[ϵ′′] δN such that Ext1A(M,N) ̸= 0,∣∣PExt1A(M,N) ∣∣(f ∗[ϵ′] δM ) ∗ (g ∗[ϵ′′] δN ) = ∫ Pϵ∈PExt1A(M,N) ( f+ ∗[ϵ] Sfg ) ∗[ϵ] δmtϵ + ∫ Pη∈PExt1A(N,M) (f− ∗[η] Sgf ) ∗[η] δmtη holds. Thus by definition, any balanced pair satisfies a multiplication formula as in Theorem 3.25. 3.10 The case dimk Ext 1 A(M,N) = 1 Finally, we simplify the formula in the case that dimk Ext 1 A(M,N) = dimk Ext 1 A(N,M) = 1, so that PExt1A(M,N) and PExt1A(N,M) are both singleton sets. Moreover, since p0Kerβc′,c′′ and Imβ′c′′,c′ ∩ Ext1A(N,M) are orthogonal, one of them is of dimension 1 and the other is zero. Lemma 3.27. Assume dimk Ext 1 A(M,N) = 1, and that [ϵ] and [η] are non-zero in Ext1A(M,N) and Ext1A(N,M), respectively. We have (1) If (c′, c′′) ∈ ϕ[ϵ](Fmono i,a,L ([ϵ],a′,a′′)), then f+ext(c ′, c′′, [ϵ]) = 0; (2) If (c′′, c′) ∈ ϕ[η](Fmono i,a,L ([η],a′′,a′)), then f−ext(c ′′, c′, [η]) = 0. Proof. By Corollary 3.2, (c′, c′′) ∈ ϕ[ϵ](Fmono i,a,L ([ϵ],a′,a′′)) if and only if [ϵ] ∈ p0Kerβc′,c′′ . In this case, p0Kerβc′,c′′ contains a non-zero element, so dimk p0Kerβc′,c′′ = 1. By definition, f+ext((c ′, c′′), [ϵ]) = dimk ( Imβ′c′′,c′ ∩ Ext1A(N,M) ) = 0. Dually, by Lemma 3.5, (c′′, c′) ∈ ϕ[η](Fmono i,a,L ([η],a′′,a′)) if and only if [η] ∈ Imβ′c′′,c′ ∩ Ext1A(N,M). In this case, dimk p0Kerβc′,c′′ = 0. ■ 24 Z. Chen, J. Xiao and F. Xu Corollary 3.28. Assume that dimk Ext 1 A(M,N) = 1, and that [ϵ] and [η] are non-zero in Ext1A(M,N) and Ext1A(N,M), respectively. Then (c′, c′′) ∈ ϕ[ϵ] ( Fmono i,a,L ([ϵ],a′,a′′) ) if and only if (c′′, c′) /∈ ϕ[η] ( Fmono i,a,L ([η],a′′,a′) ) . Proof. Since dimk Ext 1 A(M,N) = 1, dimk p0Kerβc′,c′′ is either 0 or 1. Both conditions are equivalent to dimk p0Kerβc′,c′′ = 1. ■ Theorem 3.29. Assume dimk Ext 1 A(M,N) = 1 and there are nonsplit short exact sequences ϵ : 0 → N → L→M → 0, and η : 0 →M → L′ → N → 0. Then we have (f ∗[ϵ′] δM ) ∗ (g ∗[ϵ′′] δN ) = Sfg ∗ [ϵ]δL + (fhom ∗[η] Sgf ) ∗[η] δL′. In particular, δM ∗ δN = δL + (fhom ∗[η] f) ∗[η] δL′, where f is the zero function in ZMF. Proof. Recall the multiplication formula∣∣PExt1A(M,N) ∣∣(f ∗[ϵ′] δM ) ∗ (g ∗[ϵ′′] δN ) = ∫ Pϵ∈PExt1A(M,N) ( f+ext ∗[ϵ] Sfg ) ∗[ϵ] δmtϵ + ∫ Pη∈PExt1A(N,M) (fhom ∗[η] Sgf ) ∗[η] δmtη = ∫ Pϵ∈PExt1A(M,N) Sfg ∗[ϵ] δmtϵ + ∫ Pη∈PExt1A(N,M) (f−ext ∗[η] fhom ∗[η] Sgf ) ∗[η] δmtη from Theorem 3.25. Since dimk Ext 1 A(M,N) = 1, ∣∣PExt1A(M,N) ∣∣ = 1 and both integrals in the formula degener- ate into a pair of terms indexed by [ϵ] and [η] respectively. Then we have (f ∗[ϵ′] δM ) ∗ (g ∗[ϵ′′] δN ) = ( f+ext ∗[ϵ] Sfg ) ∗ϵ] δL + (fhom ∗[η] Sgf ) ∗[η] δL′ = Sfg ∗[ϵ] δL + (f−ext ∗[η] fhom ∗[η] Sgf ) ∗[η] δL′ . From Lemma 3.27, ( f+ext ∗[ϵ] Sfg ) ∗[ϵ] δL(di,a) = ∫∫ a′+a′′=a,(c′,c′′)∈ϕ[ϵ](Fmono i,a,L ([ϵ],a′,a′′)) qk(c ′,c′′)+Sfg+f+ext(c′,c′′,[ϵ]) = ∫∫ a′+a′′=a,(c′,c′′)∈ϕ[ϵ](Fmono i,a,L ([ϵ],a′,a′′)) qk(c ′,c′′)+Sfg = Sfg ∗[ϵ] δL(di,a) and (f−ext ∗[η] fhom ∗[η] Sgf ) ∗[η] δL′(di,a) = ∫∫ a′+a′′=a,(c′′,c′′)∈ϕ[η](Fmono i,a,L ([η],a′′,a′)) qk(c ′′,c′)+Sgf+f−ext(c′′,c′,[η])+fhom(c′′,c′,[η]) = ∫∫ a′+a′′=a,(c′′,c′)∈ϕ[η](Fmono i,a,L ([η],a′′,a′)) qk(c ′′,c′)+Sgf+fhom(c′′,c′,[η]) = (fhom ∗[η] Sgf ) ∗[η] δL′(di,a). ■ The Multiplication Formulas of Weighted Quantum Cluster Functions 25 4 2-Calabi–Yau triangulated categories and multiplication formula 4.1 2-Calabi–Yau triangulated categories Let C be a Hom-finite, 2-Calabi–Yau, Krull–Schmidt triangulated category over a finite field k = Fq, which admits a cluster tilting object T . The shift functor on C is denoted by Σ. Let B be the endomorphism algebra of T , so there is a functor F := Hom(T,−) : C −→modB, X 7−→Hom(T,X) from C to the abelian category modB. This induces an equivalence of categories C/(ΣT ) ≃−→ modB, where (ΣT ) denotes the ideal of morphisms of C which factor through a direct sum of copies of ΣT and C/(ΣT ) is the corresponding quotient category (see [16]). Let {S1, . . . , Sn} be a complete set of isomorphism classes of simple objects in modB. For any two objects M , N in modB, define ⟨M,N⟩ = dimk HomB(M,N)− dimk Ext 1 B(M,N), ⟨M,N⟩a = ⟨M,N⟩ − ⟨N,M⟩. The form ⟨−,−⟩a can be reduced to the Grothendieck group K0(modB) (see [16, Lemma 1.3]). For any object X in C, there are two triangles T2 → T1 → X → ΣT2 and ΣT4 → X → Σ2T3 → Σ2T4 with T1, T2, T3, T4 in addT . So we can define index and coindex of X indX := [FT1]− [FT2] and coindX := [FT3]− [FT4], where [−] represents the equivalence class of an object in the Grothendieck group K0(projB). 4.2 Exact structure An element ϵ ∈ HomC(M,ΣN) induces a triangle N i−→ L p−→M ϵ−→ ΣN in C. In this case, we denote qtϵ = M , stϵ = N and mtϵ = L. Applying F , we get an exact sequence in modB FN Fi−→ FL Fp−−→ FM Fϵ−→ FΣN. Assume M0 ⊆ FM and N0 ⊆ FN are submodules, and consider the diagram FN FL FM FΣN. N0 L0 M0 We denote GrϵM0,N0 (FL) = { L0 ⊆ FL\|Fi−1(L0) = N0, Fp(L0) =M0 } . 26 Z. Chen, J. Xiao and F. Xu We now calculate \|GrϵM0,N0 (FL)\|. Recall that C is 2-Calabi–Yau, so for any two objectsM , N in C, there are natural isomorphisms HomC(M,ΣN) → DHomC(N,ΣM) and HomC ( Σ−1M,N ) → HomC(M,ΣN), which induce an isomorphism DM,N : HomC ( Σ−1M,N ) −→ DHomC(N,ΣM) and then a bilinear form D̃M,N : HomC(Σ −1M,N)×HomC(N,ΣM) −→ k, (a, b) 7−→ DM,N (a)(b). Via the composition of functors C −→ C/(ΣT ) ≃−→ modB, any two inclusions M0 ⊆ FM and N0 ⊆ FN in modB can be lifted to two morphisms M̃0 ιM−−→ M and Ñ0 ιN−→ N in C, respectively. Then we can define two linear maps αM0,N0 : HomC ( Σ−1M, Ñ0 ) ⊕HomC ( Σ−1M,N ) −→ HomC/(T ) ( Σ−1M̃0, Ñ0 ) ⊕HomC ( Σ−1M̃0, N ) ⊕HomC/(ΣT ) ( Σ−1M,N ) , (a, b) 7−→ ( a ◦ Σ−1ιM , ιN ◦ a ◦ Σ−1ιM − b ◦ Σ−1ιM , ιN ◦ a− b ) and α′ N0,M0 : HomΣT ( Ñ0,ΣM̃0 ) ⊕HomC ( N,ΣM̃0 ) ⊕HomΣ2T (N,ΣM) −→ HomC ( Ñ0,ΣM ) ⊕HomC(N,ΣM), (a, b, c) 7−→ (ΣιM ◦ a+ c ◦ ιN +ΣιM ◦ b ◦ ιN ,−c− ΣιM ◦ b), where HomΣT (−,−) and HomΣ2T (−,−) represent the homomorphisms of C which factor through a direct sum of copies of ΣT and Σ2T , respectively. The following properties of αM0,N0 and α′ N0,M0 are given in [16, Lemma 4.2, Proposition 4.3]. Lemma 4.1. For any M , N in C, let ϵ ∈ HomC(M,ΣN) and η ∈ HomC(N,ΣM) and choose submodules M0 ⊆ FM , N0 ⊆ FN . (1) GrϵM0,N0 (F (mtϵ)) is non-empty if and only if Σ−1ϵ ∈ pKerαM0,N0 where p is the projection from HomC ( Σ−1M, Ñ0 ) ⊕HomC ( Σ−1M,N ) to HomC ( Σ−1M,N ) ; (2) GrηN0,M0 (F (mtη)) is non-empty if and only if η ∈ Imα′ N0,M0 ∩ HomC(N,ΣM) where the intersection is realized through regarding HomC(N,ΣM) as a linear subspace of HomC ( Ñ0,ΣM ) ⊕HomC(N,ΣM). Moreover, note that αM0,N0 is the dual of α′ N0,M0 with respect to the pairing D̃M,N [10, Lemma 7.3.1], so Lemma 4.2. For any M , N in C and submodules M0 ⊆ FM , N0 ⊆ FN , we have KerαM0,N0 = (Imα′ N0,M0 )⊥. In particular, dimk pKerαM0,N0 + dimk(Imα′ N0,M0 ∩HomC(N,ΣM)) = dimk HomC(M,ΣN). The Multiplication Formulas of Weighted Quantum Cluster Functions 27 Then, to calculate \|GrϵM0,N0 (FL)\| when it is non-zero, we use the following result. Lemma 4.3. If GrϵM0,N0 (F (mtϵ)) is non-empty, then HomB(M0, FN/N0) acts freely and tran- sitively on GrϵM0,N0 (F (mtϵ)). Moreover, if we set l(M,N,M0, N0) := dimk HomB(M0, FN/N0), then \|GrϵM0,N0 (FL)\| = ql(M,N,M0,N0). Proof. We sketch the proof. Consider the commutative diagram in modB FN/N0 FN FL FM FΣN N0 L0 M0, Fi π Fp Fϵ iN Fi iL Fp iM where mtϵ = L and M0, N0, L0 are submodules with Fi−1(L0) = N0, Fp(L0) =M0. We can define an action of HomB(M0, FN/N0) on GrϵM0,N0 (FL) as follows. For any f ∈ HomB(M0, FN/N0) and L0 ∈ GrϵM0,N0 (FL), Lf0 := {Fi(n) + x \| n ∈ FN, x ∈ L0, f(Fp(x)) = π(n)} is a linear subspace of FL. Since Fi, Fp, f , π are B-module homomorphisms, B(Fi(n) + x) = B(Fi(n)) +B(x) = Fi(B(n)) +B(x) and f(Fp(B(x))) = B(f(Fp(x))) = B(π(n)) = π(B(n)). So Lf0 is a submodule of FL. For Fi(n) + x ∈ Lf0 , since Fp(Fi(n) + x) = Fp(x) ∈ M0 and Fi−1(Fi(n)) = Fi−1(n0) for some n0 ∈ N0, we have Fi −1(Fi(n)+x) = Fi−1(n0)+Fi −1(x) ∈ N0 and then Fi−1 ( Lf0 ) = N0, Fp ( Lf0 ) =M0. Thus L f 0 ∈ GrϵM0,N0 (FL). If f = 0, then f(Fp(x)) = 0 and π(n) = 0 implies n ∈ N0, so L0 0 = L0. Moreover,( Lf0 )g = Lf+g0 . Since Fi−1(L0) = N0, Fi(n) ∈ L0 if and only if n ∈ N0 and in this case, π(n) = 0. So if Lf0 = L0, then f(Fp(L0)) = f(M0) = 0. That is to say, the action is free. For any L′ 0 ∈ GrϵM0,N0 (FL), define f as f(m) = π(x′ − x) where x ∈ L0 and x′ ∈ L′ 0 with Fp(x) = Fp(x′) = m. Then Lf0 = L′ 0, so the action is transitive. ■ Moreover, according to the proof, we can define a linear structure on GrϵM0,N0 (FL) with respect to a fixed L0 in GrϵM0,N0 (FL) by λLf0 = Lλf0 , Lf0 + Lg0 = Lf+g0 . Then the map HomB(M0, FN/N0) → GrϵM0,N0 (F (mtϵ)) which sends f to Lf0 is a linear isomor- phism. Remark 4.4. Notice that HomB(M0, FN/N0) just depends on M , N , M0, N0 and is indepen- dent of mtϵ. So as long as Σ−1ϵ is chosen from pKerαM0,N0 , the dimension l(M,N,M0, N0) is invariant. That is to say, all ϵ in ΣpKerαM0,N0 lead to the same quantity \|GrϵM0,N0 (FL)\|. 28 Z. Chen, J. Xiao and F. Xu Corollary 4.5. All submodules in GrϵM0,N0 (F (mtϵ)) have the same dimension vector. Proof. By Lemma 4.3, any submodule in GrϵM0,N0 (F (mtϵ)) has the form Lf0 for some f ∈ HomB(M0, FN/N0). Since L f 0 ∈ GrϵM0,N0 (FL), we have the commutative diagram FN FL FM N0 Lf0 M0. Fi Fp iN Fi iL Fp iM Both rows are exact at the middle term, therefore Lf0/F i(N0) ∼=M0 and then dimLf0 = dimFi(N0) + dimM0. ■ Summarizing all the analysis in this subsection, we have Proposition 4.6. For anyM , N in C, let ϵ ∈ HomC(M,ΣN) and choose submodulesM0 ⊆ FM , N0 ⊆ FN . (1) The set GrϵM0,N0 (F (mtϵ)) is non-empty if and only if ϵ ∈ ΣpKerαM0,N0 if and only if ϵ ∈ Imα′ M0,N0 ∩HomC(M,ΣN); (2) If GrϵM0,N0 (F (mtϵ)) is non-empty, then there is a bijection HomB(M0, FN/N0) −→ GrϵM0,N0 (F (mtϵ)) and \|GrϵM0,N0 (F (mtϵ))\| = ql(M,N,M0,N0). 4.3 Quantum cluster functions Let An,λ be the Q-algebra generated by x±1 , . . . , x ± n with defining relations: for any e = (e1, . . . , en), f = (f1, . . . , fn) ∈ Zn, Xe ·Xf = q 1 2 λ(e,f)Xe+f , where Xe = xe11 . . . xenn and Xf = xf11 . . . xfnn are monomials in An,λ and λ(−,−) is a skew- symmetric bilinear form. Such An,λ is called a skew-polynomial algebra and elements in An,λ are called skew-polynomials. On the other hand, for any L ∈ C and d ∈ Nn, the set of all submodules F0 of FL with dimension vector d is denoted by Grd(FL), called a quiver Grassmannian. Definition 4.7. For each object L in C, we assign a skew-polynomial in An,λ as XL := ∫ d \|Grd(FL)\| ·Xp(L,d), where the integral runs over all d in Nn and p(L,d) ∈ Nn with p(L,d)i = −(coindL)i + ⟨dimkSi,d⟩a. The skew-polynomial is called a quantum cluster function of L. Notice that the set of all d such that Grd(FL) is non-empty is finite, so the integral is just a finite sum. The Multiplication Formulas of Weighted Quantum Cluster Functions 29 4.4 Mappings with affine fibers Given M,N ∈ C, define EG(M,N) := { (ϵ, L0) \| ϵ ∈ HomC(M,ΣN), L0 ∈ ∐ d Grd(F (mtϵ)) } . Then from (ϵ, L0) ∈ EG(M,N), we can induce two submodules M0, N0 of FM and FN respec- tively as N0 = i−1(L0) andM0 = p(L0). Let L := mtϵ. Then we naturally have the commutative diagram FN FL FM FΣN. N0 L0 M0 i p Fϵ We denote this assignment by ψMN : EG(M,N) −→ ∐ e,f Gre(FM)×Grf (FN), where the coproduct runs over Nn × Nn. Fix ϵ, and set ψϵ := ψMN (ϵ,−) : ∐ d Grd(FL) −→ ∐ e,f Gre(FM)×Grf (FN). To separate all pieces indexed by d, we set ψϵ,d := ψϵ\|Grd(FL) : Grd(FL) −→ ∐ e,f Gre(FM)×Grf (FN). Moreover, denote Grϵd(FL, e, f) := ψ−1 ϵ,d(Gre(FM)×Grf (FN)) and Grϵe,f (FM,FN,d) := ψϵ,d(Grϵd(FL, e, f)). Notice that Grϵe,f (FM,FN,d) may not be the whole of Gre(FM) × Grf (FN) because ψϵ,d is not surjective in general. Rewriting Proposition 4.6 in this notation, we get the following. Lemma 4.8. Given (M0, N0) ∈ Gre(FM) × Grf (FN) and ϵ ∈ HomC(M,ΣN), If (M0, N0) ∈ Grϵe,f (FM,FN,d), then HomB(M0, FN/N0) acts freely and transitively on ψ−1 ϵ,d(M0, N0). 4.5 Cardinality In this subsection, we calculate cardinalities of sets involved in several lemmas above and refine the calculation of XL. Considering ψMN , since Grϵd(FL, e, f) = ψ−1 ϵ,d(Gre(FM)×Grf (FN)) and the Grassmannian is a finite set here, we have \|Grd(FL)\| = ∫ e,f \|Grϵd(FL, e, f)\| 30 Z. Chen, J. Xiao and F. Xu for any ϵ ∈ HomC(M,ΣN) with mtϵ = L, and \|Grϵd(FL, e, f)\| = ∫ (M0,N0)∈Gre(FM)×Grf (FN) ∣∣ψ−1 ϵ,d(M0, N0) ∣∣. Moreover, based on Lemma 4.8, we have the following. Lemma 4.9. Let (M0, N0) ∈ Gre(FM)×Grf (FN) and ϵ ∈ HomC(M,ΣN). (1) If (M0, N0) /∈ Grϵe,f (FM,FN,d), ∣∣ψ−1 ϵ,d(M0, N0) ∣∣ = 0; (2) If (M0, N0) ∈ Grϵe,f (FM,FN,d), ∣∣ψ−1 ϵ,d(M0, N0) ∣∣ = \|HomB(M0, FN/N0)\| = ql(M,N,M0,N0). Given these calculations, if mtϵ = L, we can refine the formula for XL to XL = ∫ d \|Grd(FL)\| ·Xp(L,d) = ∫ d ∫ e,f \|Grϵd(FL, e, f)\| ·Xp(L,d) = ∫ d ∫ e,f ∫ (M0,N0)∈Gre(FM)×Grf (FN) ∣∣ψ−1 ϵ,d(M0, N0) ∣∣ ·Xp(L,d) = ∫ d ∫ e,f ∫ (M0,N0)∈Grϵe,f (FM,FN,d) ql(M,N,M0,N0) ·Xp(L,d). Remark 4.10. Similar to the case in Section 3, the refinement of XL still depends on the choice of ϵ which is not unique. But, although different ϵ lead to different refinements, they all evaluate to XL in the end. 4.6 Weight Based on the refinements of XL, we introduce weight functions and weighted quantum cluster functions. Define MG := { (M0, N0) \| (M0, N0) ∈ ∐ e,f Gre(FM)×Grf (FN), M,N ∈ C } and set ZMG := { f : MG×Ext1C → Z \| f(M0, N0, ϵ) = 0 unless M = qtϵ, N = stϵ } , where Ext1C = ∐ M,N∈C HomC(M,ΣN) and Z = { n 2 \| n ∈ Z } is the set of all half integers. The functions in ZMG are called weight functions. Given ϵ ∈ Hom1 C(M,ΣN), we define ZMG(ϵ) := {f ∈ ZMG \| f(M0, N0, ρ) = 0 if ρ ̸= ϵ}. Given f ∈ ZMG(ϵ), we write f(M0, N0, ϵ) instead as f(M0, N0). Definition 4.11 (weighted quantum cluster function). Given a weight function f ∈ ZMG(ϵ), the weighted quantum cluster function f ∗ϵ XL is a skew-polynomial in An,λ defined as f ∗ϵ XL = ∫ d ∫ e,f ∫ (M0,N0)∈Gre(FM)×Grf (FN) ∣∣ψ−1 ϵ,d(M0, N0) ∣∣ · qf(M0,N0) ·Xp(L,d) = ∫ d ∫ e,f ∫ (M0,N0)∈Grϵe,f (FM,FN,d) ql(M,N,M0,N0) · qf(M0,N0) ·Xp(L,d), where ϵ ∈ HomC(M,ΣN) with M = qtϵ, N = stϵ and L = mtϵ. Similarly, we can take f to be the zero function to check that this definition is a q-deformation of the quantum cluster function. For the zero function f ∈ ZMG(ϵ) with mtϵ = L, f ∗ϵ XL = XL. The Multiplication Formulas of Weighted Quantum Cluster Functions 31 4.7 Multiplication We denote the zero morphism in HomC(M,ΣN) by 0MN . First, we consider the multiplication of quantum cluster functions. Recall that the multiplication of skew-polynomials follows the generating relation Xe ·Xf = q 1 2 λ(e,f)Xe+f . So given M , N in C, we can multiply the quantum cluster functions as XM ·XN = ∫ e \|Gre(FM)\| ·Xp(M,e) · ∫ f \|Grf (FN)\| ·Xp(N,f) = ∫ e,f \|Gre(FM)\| · \|Grf (FN)\| · q 1 2 λ(p(M,e),p(N,f)) ·Xp(M,e)+p(N,f) = ∫ e,f ∫ (M0,N0)∈Gre(FM)×Grf (FN) q 1 2 λ(p(M,e),p(N,f)) ·Xp(M,e)+p(N,f). On the other hand, we calculate XM⊕N as XM⊕N = ∫ d ∫ e,f ∫ (M0,N0)∈Gre(FM)×Grf (FN) ∣∣ψ−1 0MN ,d (M0, N0) ∣∣ ·Xp(M⊕N,d) = ∫ d ∫ e,f ∫ (M0,N0)∈Gr 0MN e,f (FM,FN,d) ql(M,N,M0,N0) ·Xp(M⊕N,d). Comparing these two equalities, there are three differences: the power of X, the domain of integration, and the power of q. We analyze them successively. Firstly, given ϵ ∈ HomC(M,ΣN) with mtϵ = L and submodules FN FL FM FΣN, N0 L0 M0 Fi Fp Fϵ where N0 = Fi−1(L0), M0 = Fp(L0), we denote the dimension vectors of M0, N0 and L0 by e, f and d respectively. Lemma 4.12 ([16, Lemma 5.1]). With the notation above, p(M, e) + p(N, f) = p(L,d). This lemma allows us to identify the powers of X in XM ·XN and XM⊕N , providing we are careful about the indices e, f and d of the terms in these sums. Secondly, recall the map ψϵ = ψMN (ϵ,−) : ∐ d Grd(FL) −→ ∐ e,f Gre(FM)×Grf (FN) and pieces ψϵ,d = ψϵ\|Grd(FL) : Grd(FL) −→ ∐ e,f Gre(FM)×Grf (FN). Although GrϵM0,N0 (FL) can contain many submodules, they all have the same dimension vector by Corollary 4.5. That is, the sets Grϵe,f (FM,FN,d) do not intersect for different values of d. Thus the data (e, f ,M0, N0) indexing a term in the expression for XM ·XN uniquely determines the additional datum d needed to index a term in the expression for XM⊕N . Moreover, for 0MN , ψ0MN : ∐ d Grd(FL) −→ ∐ e,f Gre(FM)×Grf (FN) 32 Z. Chen, J. Xiao and F. Xu is surjective because any two submodules of FM and FN can be assembled into a submodule of F (M ⊕N) through direct sum. So in this case, for any e, f , Gre(FM)×Grf (FN) = ∐ d Gr0MN e,f (FM,FN,d). Then we have XM ·XN = ∫ e,f ∫ (M0,N0)∈Gre(FM)×Grf (FN) q 1 2 λ(p(M,e),p(N,f)) ·Xp(M,e)+p(N,f) = ∫ d ∫ e,f ∫ (M0,N0)∈Gr 0MN e,f (FM,FN,d) q 1 2 λ(p(M,e),p(N,f)) ·Xp(L,d). Finally, we introduce a special family of weight functions. Definition 4.13. For any M , N in C, define a weight function fspec ∈ ZMG by fspec(M0, N0, ϵ) = 1 2 λ(p(M, e), p(N, f))− l(M,N,M0, N0) if M0 ∈ ∐ eGre(FM), N0 ∈ ∐ f Grf (FN), ϵ ∈ HomC(M,ΣN), and 0 otherwise. Note that fspec is constant in ϵ. The following is then immediate. Proposition 4.14. For any M , N in C, in An,λ XM ·XN = fspec ∗0MN XM⊕N . Definition 4.15. Given weight functions f, g ∈ ZMG, f ∗η g is defined by f ∗η g(M0, N0, ϵ) = { f(M0, N0, ϵ) + g(M0, N0, ϵ), if ϵ = η, 0, otherwise, for M0 ∈ ∐ eGre(FM) and N0 ∈ ∐ f Grf (FN). That is to say, f ∗η g ∈ ZMG(η). Now we consider the multiplication (f ∗ϵ′ XM ) · (g ∗ϵ′′ XN ). Obviously, the product must contain information about the weight functions f ∈ ZMG(ϵ ′) and g ∈ ZMG(ϵ ′′). To record these, we define a corresponding weight function for the middle term. Definition 4.16. Given weighted quantum cluster functions f ∗ϵ′XM and g ∗ϵ′′XN , and a mor- phism ϵ ∈ HomC(M,ΣN), define a weight function Tfg ∈ ZMG(ϵ) by Tfg(M0, N0, ϵ) = f(ψϵ′(M0)) + g(ψϵ′′(N0)). By Definition 4.16, given weighted quantum cluster functions f ∗ϵ′ XM and g ∗ϵ′′ XN , for any ϵ ∈ HomC(M,ΣN) and η ∈ HomC(N,ΣM), we have Tfg(M0, N0, ϵ) = Tgf (N0,M0, η) since both sides are equal to f(ψϵ′(M0)) + g(ψϵ′′(N0)). Proposition 4.17. For any weighted quantum cluster functions f ∗ϵ′ XM and g ∗ϵ′′ XN , in An,λ we have (f ∗ϵ′ XM ) · (g ∗ϵ′′ XN ) = (fspec ∗0MN Tfg) ∗0MN XM⊕N . The Multiplication Formulas of Weighted Quantum Cluster Functions 33 Proof. For simplicity, without causing ambiguity, we omit some variables of weight functions in following calculation. For example, Tfg(M0, N0,0MN ) is simplified to Tfg. The key step is to calculate the fibers of the following composition of mappings∐ e1 Gre1(F (qtϵ ′)) ∐ eGre(FM) × ∐ e2 Gre2(F (stϵ ′)) ∐ dGrd(F (M ⊕N)) × ∐ f1 Grf1(F (qtϵ ′′)) ∐ f Grf (FN) × ∐ f1 Grf1(F (stϵ ′′). p2◦ψϵ′ p1◦ψϵ′ p2◦ψ0MN p1◦ψ0MN p2◦ψϵ′′ p1◦ψϵ′′ Notice that ψ0MN is surjective, but ψϵ′ and ψϵ′′ may not be surjective. We have∐ d Gr0MN e,f (FM,FN,d) = Gre(FM)×Grf (FN),∐ e Grϵ ′ e1,e2(F (qtϵ ′), F (stϵ′′), e) ⊆ Gre1(F (qtϵ ′))×Gre2(F (stϵ ′′)),∐ f Grϵ ′′ f1,f2(F (qtϵN ), F (stϵN ), f) ⊆ Grf1(F (qtϵ ′′))×Grf2(F (stϵ ′′)). By Definitions 4.11 and 4.13, and Proposition 4.14, we obtain (fspec ∗0MN Tfg) ∗0MN XM⊕N = ∫ d ∫ e,f ∫ (M0,N0)∈Gr 0MN e,f (FM,FN,d) ql(M,N,M0,N0)+Tfg+fspec(M0,N0) ·Xp(M⊕N,d) = ∫ e,f ∫ (M0,N0)∈Gre(FM)×Grf (FN) qTfg ·Xp(M,e) ·Xp(M,f) = ∫ e ∫ M0∈Gre(FM) Xp(M,e) ∫ f ∫ N0∈Grf (FN) Xp(M,f) · qTfg . Recall from Lemma 4.9 Gre(FM) = ∐ e1,e2 Grϵ ′ e (FM, e1, e2) = ∐ e1,e2 ψ−1 ϵ′,e ( Grϵ ′ e1,e2(F (qtϵ ′), F (stϵ′), e) ) with the dimension of the fiber being l(qtϵ′, stϵ′,M1,M2) and Grf (FN) = ∐ f1,f2 Grϵ ′′ f (FN, f1, f2) = ∐ f1,f2 ψ−1 qtϵ′′,stϵ′′,ϵ′′,f ( Grϵ ′′ f1,f2(F (qtϵ ′′), F (stϵ′′), f) ) with the dimension of the fiber being l(qtϵ′′, stϵ′′, N1, N2). 34 Z. Chen, J. Xiao and F. Xu Then we have (fspec ∗0MN Tfg) ∗0MN XM⊕N = ∫ e ∫ M0∈Gre(FM) Xp(M,e) ∫ f ∫ N0∈Grf (FN) Xp(M,f) · qTfg = ∫ e ∫ e1,e2 ∫ (M1,M2)∈Grϵ ′ e1,e2 (F (qtϵ′),F (stϵ′),e) ql(qtϵ ′,stϵ′,M1,M2) ·Xp(M,e) · ∫ f ∫ f1,f2 ∫ (N1,N2)∈Grϵ ′′ f1,f2 (F (qtϵ′′),F (stϵ′′),f) ql(qtϵ ′′,stϵ′′,N1,N2) ·Xp(N,f) ·qf(ψϵ′ (M0))+g(ψϵ′′ (N0)) = ∫ e ∫ e1,e2 ∫ (M1,M2)∈Grϵ ′ e1,e2 (F (qtϵ′),F (stϵ′),e) ql(qtϵ ′,stϵ′,M1,M2) · qf(ψϵ′ (M0)) ·Xp(M,e) · ∫ f ∫ f1,f2 ∫ (N1,N2)∈Grϵ ′′ f1,f2 (F (qtϵ′′),F (stϵ′′),f) ql(qtϵ ′′,stϵ′′,N1,N2) · qg(ψϵ′′ (N0)) ·Xp(N,f) = (f ∗ϵ′ XM ) · (g ∗ϵ′′ XN ). ■ 4.8 The projectivization of HomC(M,ΣN) Since HomC(M,ΣN) is a finite dimensional vector space, we can consider PHomC(M,ΣN). We denote the equivalence class of ϵ in PHomC(M,ΣN) by Pϵ. In this subsection, we check that multiplication of weighted quantum cluster functions is still well defined if we replace ϵ by Pϵ. We assume the parameter λ is a non-zero element in k. Recall the mapping ψMN : EG(M,N) −→ ∐ e,f Gre(FM)×Grf (FN) with affine fibers. By Lemma 4.8 and the linearity of αM0,N0 , ϵ∈ΣpKerαM0,N0 if and only if λϵ∈ΣpKerαM0,N0 . So in this case,∣∣ψ−1 ϵ,d(M0, N0) ∣∣ = ql(M,N,M0,N0) = ∣∣ψ−1 λϵ,d(M0, N0) ∣∣. Otherwise, they are both zero. So we have the following. Proposition 4.18. Given f ∗ϵ XL, set f ∈ ZMG(λϵ) with f(M0, N0, ϵ) = f(M0, N0, λϵ). Then f ∗ϵ XL = f ∗λϵ XL. Proof. f ∗ϵ XL = ∫ d ∫ e,f ∫ (M0,N0)∈Gre(FM)×Grf (FN) ∣∣ψ−1 ϵ,d(M0, N0) ∣∣ · qf(M0,N0,ϵ) ·Xp(L,d) = ∫ d ∫ e,f ∫ (M0,N0)∈Gre(FM)×Grf (FN) ∣∣ψ−1 λϵL,d (M0, N0) ∣∣ · qf(M0,N0,λϵ) ·Xp(L,d) = f ∗λϵ XL. ■ Note that given f ∗ϵ′ XM and g ∗ϵ′′ XN , by Proposition 4.17, we have (f ∗ϵ′ XM ) · (g ∗ϵ′′ XN ) = (fspec ∗0MN Tfg) ∗0MN XM⊕N , where fspec ∈ ZMG and Tfg(M0, N0, ϵ) = f(ψϵ′(M0)) + g(ψϵ′′(N0)). If we replace ϵ′ and ϵ′′ by their scalar multiple on the left-hand side, since ψϵ′(M0) = ψλϵ′(M0) and ψϵ′′(N0) = ψµϵ′′(N0) for any non-zero λ and µ in k, we have (f ∗ϵ′ XM ) · (g ∗ϵ′′ XN ) = (f ∗λϵ′ XM ) · (g ∗µϵ′′ XN ). The Multiplication Formulas of Weighted Quantum Cluster Functions 35 If we replace ϵ′ and ϵ′′ by their scalar multiple on the right-hand side, since fspec ∈ ZMG is constant in any ϵ and λµ0MN = 0MN , the right-hand side also remains the same. 4.9 Multiplication formula and balanced pairs Firstly, we introduce several special weight functions and an important property. Definition 4.19. There are three weight functions in ZMG defined as (1) gskew(N0,M0, η) = λ(p(M, e), p(N, f)), where λ(−,−) is the skew-symmetric bilinear form defined in Section 4.3; (2) g+ext(M0, N0, ϵ) = dimk(Imα′ N0,M0 ∩HomC(N,ΣM)); (3) g−ext(N0,M0, η) = dimk ΣpKerαM0,N0 for any M , N in C, (M0, N0) ∈ ∐ e,f Gre(FM) × Grf (FN), ϵ ∈ HomC(M,ΣN) and η ∈ HomC(N,ΣM). Note that the three weight functions are all independent of the extension η or ϵ. Definition 4.20. Given a pair of weight functions (g+, g−), set σ1(g +) := ∫ Pϵ∈PΣpKerαM0,N0 qg +(M0,N0,ϵ) and σ2(g −) := ∫ Pη∈P(Imα′ N0,M0 ∩HomC(N,ΣM)) qg −(N0,M0,η). This pair is called pointwise balanced if qdimk HomC(M,ΣN) − 1 q − 1 = σ1(g +) + σ2(g −) holds for any M,N ∈ C and (M0, N0) ∈ ∐ e,f Gre(FM)×Grf (FN). Proposition 4.21. The following two pairs of weight functions (1) ( g+ext, 0 ) ; (2) (0, g−ext) are pointwise balanced. Proof. Recall from Lemma 4.2 that for any M,N ∈ C and (M0, N0) ∈ ∐ e,f Gre(FM) × Grf (FN), we have dimk ΣpKerαM0,N0 + dimk(Imα′ N0,M0 ∩HomC(N,ΣM)) = dimk HomC(M,ΣN). After projectivization, we have qdimk HomC(M,ΣN) − 1 q − 1 = q dimk(Imα′ N0,M0 ∩HomC(N,ΣM)) · q dimk ΣpKerαM0,N0 − 1 q − 1 36 Z. Chen, J. Xiao and F. Xu + q dimk(Imα′ N0,M0 ∩HomC(N,ΣM)) − 1 q − 1 = qdimk ΣpKerαM0,N0 − 1 q − 1 + qdimk ΣpKerαM0,N0 · q dimk(Imα′ N0,M0 ∩HomC(N,ΣM)) − 1 q − 1 . So by definition, σ1(g + ext) + σ2(0) = ∫ Pϵ∈PΣpKerαM0,N0 qg + ext(M0,N0,ϵ) + ∫ Pη∈P(Imα′ N0,M0 ∩HomC(N,ΣM)) 1 = ∫ Pϵ∈PΣpKerαM0,N0 q dimk(Imα′ N0,M0 ∩HomC(N,ΣM)) + ∫ Pη∈P(Imα′ N0,M0 ∩HomC(N,ΣM)) 1 = q dimk(Imα′ N0,M0 ∩HomC(N,ΣM)) · q dimk ΣpKerαM0,N0− 1 q − 1 + q dimk(Imα′ N0,M0 ∩HomC(N,ΣM))− 1 q − 1 = qdimk HomC(M,ΣN) − 1 q − 1 and σ1(0) + σ2(g − ext) = ∫ Pϵ∈PΣpKerαM0,N0 1 + ∫ Pη∈P(Imα′ N0,M0 ∩HomC(N,ΣM)) qg − ext(N0,M0,η) = ∫ Pϵ∈PΣpKerαM0,N0 1 + ∫ Pη∈P(Imα′ N0,M0 ∩HomC(N,ΣM)) qdimk ΣpKerαM0,N0 = qdimk ΣpKerαM0,N0 − 1 q − 1 + qdimk ΣpKerαM0,N0 · q dimk(Imα′ N0,M0 ∩HomC(N,ΣM)) − 1 q − 1 = qdimk HomC(M,ΣN) − 1 q − 1 . ■ Theorem 4.22. If a pair of weight functions ( g+, g− ) in ZMG is pointwise balanced, then for any weighted quantum cluster functions f ∗ϵ′ XM and g ∗ϵ′′ XN such that HomC(M,ΣN) ̸= 0, we have \|PHomC(M,ΣN)\|(f ∗ϵ′ XM ) · (g ∗ϵ′′ XN ) = ∫ Pϵ∈PHomC(M,ΣN) (g+ ∗ϵ fspec ∗ϵ Tfg) ∗ϵ Xmtϵ + ∫ Pη∈PHomC(N,ΣM) (g− ∗η gskew ∗η fspec ∗η Tgf ) ∗η Xmtη. Proof. We simplify the equality in the theorem to l.h.s. = Σ1(g +) + Σ2(g −). Just for sim- plicity, we omit some independent variables without causing ambiguity in following calculation. For example, Tfg((M0, N0), ϵ) is simplified as Tfg, l(M,N,M0, N0) as l and fspec((M0, N0), ϵ) as fspec. Direct calculation: l.h.s. = \|PHomC(M,ΣN)\|(f ∗ϵ′ XM ) · (g ∗ϵ′′ XN ) = \|PHomC(M,ΣN)\|(fspec ∗0MN Tfg) ∗0MN XM⊕N The Multiplication Formulas of Weighted Quantum Cluster Functions 37 = \|PHomC(M,ΣN)\| ∫ d ∫ e,f ∫ (M0,N0)∈Gr 0MN e,f (FM,FN,d) ql+Tfg+fspec ·Xp(M⊕N,d) = \|PHomC(M,ΣN)\| ∫ e,f ∫ (M0,N0)∈Gre(FM)×Grf (FN) qTfg ·Xp(M,e) ·Xp(N,f) = ∫ e,f ∫ (M0,N0)∈Gre(FM)×Grf (FN) qTfg ·Xp(M,e) ·Xp(N,f) · q dimk HomC(M,ΣN) − 1 q − 1 . Recall that the third equality in the above is based on the fact∐ d Gr0MN e,f (FM,FN,d) = Gre(FM)×Grf (FN). For general ϵ ∈ HomC(M,ΣN) and η ∈ HomC(N,ΣM), we only have∐ d Grϵe,f (FM,FN,d) ⊆ Gre(FM)×Grf (FN),∐ d Grηf ,e(FN,FM,d) ⊆ Grf (FN)×Gre(FM), and whether a pair (M0, N0) ∈ Gre(FM)×Grf (FN) belongs to Grϵe,f (FM,FN,d) or Grηf ,e(FN, FM,d) is determined by Proposition 4.6. So we can calculate the right-hand side as Σ1(g +) = ∫ Pϵ∈PHomC(M,ΣN) ( g+ ∗ϵ fspec ∗ϵ Tfg ) ∗ϵ Xmtϵ = ∫ Pϵ∈PHomC(M,ΣN) ∫ d ∫ e,f ∫ (M0,N0)∈Grϵe,f (FM,FN,d) ql+g ++Tfg+fspec ·Xp(L,d) = ∫ Pϵ∈PHomC(M,ΣN) ∫ e,f ∫ (M0,N0)∈ ∐ d Grϵe,f (FM,FN,d) qg ++Tfg ·Xp(M,e) ·Xp(N,f) = ∫ e,f ∫ (M0,N0)∈Gre(FM)×Grf (FN) ∫ Pϵ∈PΣpKerαM0,N0 qg ++Tfg ·Xp(M,e) ·Xp(N,f) = ∫ e,f ∫ (M0,N0)∈Gre(FM)×Grf (FN) qTfg ·Xp(M,e) ·Xp(N,f) · σ1(g+). Notice that in the following calculation of Σ2(g −), we denote l = l(N,M,N0,M0), Σ2(g −) = ∫ Pη∈PHomC(N,ΣM) (g− ∗η gskew ∗η fspec ∗η Tgf ) ∗η Xmtη = ∫ Pη∈PHomC(N,ΣM) ∫ d ∫ f ,e ∫ (N0,M0)∈Grηf ,e(FN,FM,d) ql+g −+Tgf+fspec+gskew ·Xp(L,d) = ∫ Pη∈PHomC(N,ΣM) ∫ f ,e ∫ (N0,M0)∈ ∐ g Grηf ,e(FN,FM,d) qg −+Tgf ·Xp(M,e) ·Xp(N,f) = ∫ f ,e ∫ (N0,M0)∈Grf (FN)×Gre(FM) ∫ Pη∈P(Imα′ N0,M0 ∩HomC(N,ΣM)) qg −+Tgf ·Xp(M,e) ·Xp(N,f) = ∫ f ,e ∫ (N0,M0)∈Grf (FN)×Gre(FM) qTgf ·Xp(M,e) ·Xp(N,f) · σ2(g−). Then by Definitions 4.16 and 4.20, we have l.h.s. = Σ1(g +) + Σ2(g −). ■ 38 Z. Chen, J. Xiao and F. Xu From Proposition 4.21 and Theorem 4.22, we have Theorem 4.23 (multiplication formula of weighted quantum cluster functions). Let C be a Hom- finite, 2-Calabi–Yau, Krull–Schmidt triangulated category over a finite field k = Fq with a cluster tilting object T . For any weighted quantum cluster functions f ∗ϵ′ XM and g ∗ϵ′′ XN such that HomC(M,ΣN) ̸= 0, we have \|PHomC(M,ΣN)\|(f ∗ϵ′ XM ) · (g ∗ϵ′′ XN ) = ∫ Pϵ∈PHomC(M,ΣN) (g+ext ∗ϵ fspec ∗ϵ Tfg) ∗ϵ Xmtϵ + ∫ Pη∈PHomC(N,ΣM) (gskew ∗η fspec ∗η Tgf ) ∗η Xmtη = ∫ Pϵ∈PHomC(M,ΣN) (fspec ∗ϵ Tfg) ∗ϵ Xmtϵ + ∫ Pη∈PHomC(N,ΣM) (g−ext ∗η gskew ∗η fspec ∗η Tgf ) ∗η Xmtη. Moreover, we can define a balanced pair as Definition 4.24. A pair of weight functions ( g+, g− ) in ZMG is called a balanced pair if for any weighted quantum cluster functions f ∗ϵ′ XM and g ∗ϵ′′ XN such that HomC(M,ΣN) ̸= 0, \|PHomC(M,ΣN)\|(f ∗ϵ′ XM ) · (g ∗ϵ′′ XN ) = ∫ Pϵ∈PHomC(M,ΣN) ( g+ ∗ϵ fspec ∗ϵ Tfg ) ∗ϵ Xmtϵ + ∫ Pη∈PHomC(N,ΣM) (g− ∗η gskew ∗η fspec ∗η Tgf ) ∗η Xmtη holds. 4.10 The case dimk HomC(M,ΣN) = 1 In this subsection, we assume dimk HomC(M,ΣN) = dimk HomC(N,ΣM) = 1 and the triangles N → L→M ϵ−→ ΣN and M → L′ → N η−→ ΣM are non-split. Thus PHomC(M,ΣN) and PHomC(N,ΣM) are both singleton sets represented by Pϵ and Pη respectively. Since ΣpKerαM0,N0 and Imα′ N0,M0 ∩HomC(N,ΣM) are orthogonal, one of them is of dimension 1 and the other is zero. Lemma 4.25. With assumptions above: (1) If (M0, N0) ∈ Imψϵ, g + ext(M0, N0, ϵ) = 0; (2) If (N0,M0) ∈ Imψη, g − ext(N0,M0, η) = 0. Proof. If (M0, N0) ∈ Imψϵ, then ϵ ∈ ΣpKerαM0,N0 . Since ϵ is non-zero, ΣpKerαM0,N0 is of dimension 1. Thus g+ext(M0, N0, ϵ) = dimk(Imα′ N0,M0 ∩HomC(N,ΣM)) = 0. On the other hand, if (N0,M0) ∈ Imψη, then η ∈ Imα′ N0,M0 ∩ HomC(N,ΣM). Since η is non-zero, we have g−ext(N0,M0, η) = dimk ΣpKerαM0,N0 = 0. ■ The Multiplication Formulas of Weighted Quantum Cluster Functions 39 Corollary 4.26. With assumptions above, (M0, N0) ∈ Imψϵ if and only if (N0,M0) /∈ Imψη. Proof. Both statements are equivalent to dimk ΣpKerαM0,N0 = 1. ■ By Lemma 4.2, we have dimk ΣpKerαM0,N0 + dimk(Imα′ N0,M0 ∩HomC(N,ΣM)) = dimk HomC(M,ΣN), and qdimk HomC(M,ΣN) − 1 q − 1 = q dimk(Imα′ N0,M0 ∩HomC(N,ΣM)) · q dimk ΣpKerαM0,N0 − 1 q − 1 + q dimk(Imα′ N0,M0 ∩HomC(N,ΣM)) − 1 q − 1 = qdimk ΣpKerαM0,N0 − 1 q − 1 + qdimk ΣpKerαM0,N0 · q dimk(Imα′ N0,M0 ∩HomC(N,ΣM)) − 1 q − 1 . So in the case dimk HomC(M,ΣN) = 1, one of dimk ΣpKerαM0,N0 and dimk(Imα′ N0,M0 ∩HomC(N,ΣM)) is 1 and the other is 0. Then g+ext and g−ext are zero in the relevant domain of integration and the two balanced pairs given in Proposition 4.21 both degenerate to (0, gskew). Theorem 4.27. Assume dimk HomC(M,ΣN) = 1 and the triangles N → L→M ϵ−→ ΣN and M → L′ → N η−→ ΣM are non-split triangles. Then we have (f ∗ϵ′ XM ) · (g ∗ϵ′′ XN ) = (fspec ∗ϵ Tfg) ∗ϵ XL + (gskew ∗η fspec ∗η Tgf ) ∗η XL′ . 5 Connection with preprojective algebras In this section, we provide a quantum analogue of the connections in [12] between Palu’s multi- plication formula for cluster characters [17] and Geiss–Leclerc–Schröer’s multiplication formula for evaluation forms [10]. 5.1 Preprojective algebra and nilpotent modules Let k be a finite field and Q = (Q0, Q1, s, t) be a finite quiver where Q0 = {1, . . . , n} is the vertex set, and for an arrow α : i → j in Q1, set s(α) = i and t(α) = j. We can obtain a new quiver Q̃ from Q by adding a new arrow ᾱ : j → i for each arrow α : i→ j in Q1. Define c := ∑ α∈Q1 αᾱ− ᾱα, and let Λ := kQ̃/⟨c⟩ be the preprojective algebra of Q. For each 1 ⩽ i ⩽ n, let Si be the simple Λ-module associated to the vertex i. Denote the category of all nilpotent Λ-modules 40 Z. Chen, J. Xiao and F. Xu by nil Λ. Let Îi be the injective envelope of Si for 1 ⩽ i ⩽ n. Given an element ω in the Weyl group associated to Q, Buan, Iyama, Reiten, and Scott [2] have attached to ω a 2-Calabi–Yau Frobenius subcategory Cω ⊂ nil Λ. One can refer to [11, Section 2.4] for a detailed description of Cω. We fix the element ω and a reduced expression i. In nil Λ, there is a classical definition of a flag. Definition 5.1. A flag L• of L in nil Λ is a series of submodules 0 = Lm ⊆ Lm−1 ⊆ · · · ⊆ L1 ⊆ L0 = L. Moreover, a flag L• is called of type (i,a) if Lj−1/Lj is isomorphic to S ⊕aj ij for 1 ⩽ j ⩽ m. Denote the set of all flags of L of type (i,a) by Φi,a,L. Note that any flag of type (i,a) can be refined to a flag of type (i′,a′) with a′ ∈ {0, 1}m. As in Section 2, set F̃mono i,a,L := { Lm ιL,m−→ Lm−1−→· · ·−→L1 ιL,1−→ L0 = L \| Lj ∈ nil Λ, ιL,j is mono, 1 ⩽ j ⩽ m } . Consider the action of the group ∏m i=0AutLi on F̃mono i,a,L as follows. For any g̃ = (g0, g1, . . . , gm) ∈ m∏ i=0 AutLi and (ιL,m, ιL,m−1, . . . , ιL,1), define g̃.(ιL,m, ιL,m−1, . . . , ιL,1) := ( g−1 m−1ιL,mgm, g −1 m−2ιL,m−1gm−1, . . . , g −1 0 ιL,1g1 ) , which can be illustrated by the commutative diagram Lm Lm−1 · · · L1 L0 Lm Lm−1 · · · L1 L0. ιL,m ιL,m−1 ιL,1 g−1 m−1ιL,mgm gm g−1 m−2ιL,m−1gm−1 gm−1 g−1 0 ιL,1g1 g1 g0 Note that Fmono i,a,L is the set of orbits of F̃mono i,a,L under the action of ∏m i=0AutLi and Φi,a,L is the set of orbits of F̃mono i,a,L under the action of the group{ (g0, g1, . . . , gm) ∈ m∏ i=0 AutLi\|g0 = idL } . Hence Fmono i,a,L can be viewed as the set of orbits of Φi,a,L under the action of the group{ (g0, idL1 , . . . , idLm) ∈ m∏ i=0 AutLi } ≃ AutL. Note that for cL ∈ Fmono i,a,L , the cardinality of the stabilizer of cL is qt for some t ∈ N (for example, see [20, Section 4.1]). The above characterizes the relationship between Φi,a,L and Fmono i,a,L when a ∈ {0, 1}m. In the following, we will consider Φi,a,L instead of Fmono i,a,L as Geiss–Leclerc–Schröer did in [12]. For a short exact sequence ϵ : 0 −→ N i−→ L p−→M −→ 0, define a map ϕ̄ϵ : Φi,a,L −→ ∐ a′+a′′=a Φi,a′,M × Φi,a′′,N The Multiplication Formulas of Weighted Quantum Cluster Functions 41 which maps a flag fL := (0 = Lm ⊆ Lm−1 ⊆ · · · ⊆ L1 ⊆ L0 = L) ∈ Φi,a,L to (fM := (0 =Mm ⊆Mm−1 ⊆ · · · ⊆M1 ⊆M0 =M), fN := (0 = Nm ⊆ Nm−1 ⊆ · · · ⊆ N1 ⊆ N0 = N) ∈ Φi,a′,M × Φi,a′′,N with Mi = p(Li) and Ni = i−1(Li) for 0 ⩽ i ⩽ m. For any (c′, c′′) ∈ ∐ a′+a′′=aΦi,a′,M ×Φi,a′′,N , if ϕ̄−1 ϵ (c′, c′′) ̸= ∅, then ∣∣ϕ̄−1 ϵ (c′, c′′) ∣∣ = ∣∣ϕ̄−1 0MN (c′, c′′) ∣∣ and ϕ̄−1 0MN (c′, c′′) is a vector space (see [10, Lemma 3.3.1]). We set k̄(c′, c′′) := dimk ϕ̄ −1 0MN (c′, c′′). 5.2 Refined socle and top series In this subsection, we recall some notations and definitions from [12, Section 3.4]. Let L be an Λ-module and S be a simple Λ-module. Let socSL be the sum of all submodules of L which are isomorphic to S. If there exists no such U , set socSL = 0. Similarly, let topSL := L/V where V is the intersection of all submodules U of L such that L/U are isomorphic to S. If there exists no such submodule, then V = L and topSL = 0. Define radSL := V . For i = (i1, . . . , im), there exists a unique chain of submodules 0 = Lm ⊆ Lm−1 ⊆ · · · ⊆ L1 ⊆ L0 ⊆ L such that Lj−1/Lj = socSij L/Lj . We define sociL := L0 and L+ j := L+,i j := Lj . Moreover, the chain is denoted by L+ • := (L+ m ⊆ · · · ⊆ L+ 0 ). In particular, if sociL = L, L+ • is called the refined socle series of type i of L. Similarly, there exists a unique chain of submodules 0 ⊆ Lm ⊆ Lm−1 ⊆ · · · ⊆ L1 ⊆ L0 = L such that Lj−1/Lj = topSij Lj−1. We define topiL := L/Lm, radiL := Lm and L− j := L−,i j := Lj . Moreover, the chain is denoted by L− • := (L− m ⊆ · · · ⊆ L− 0 ). In particular, if radiL = 0, L− • is called the refined top series of type i of L. Lemma 5.2 ([12, Lemma 3.5]). For any L ∈ Cω, we have sociL = L and radiL = 0. Lemma 5.3 ([12, Lemma 3.8]). For any L ∈ Cω and any flag L• = (0 = Lm ⊆ · · · ⊆ L0 = L) ∈ Φi,a,L, we have L− j ⊆ Lj ⊆ L+ j for 0 ⩽ j ⩽ m. The above lemma implies that the refined socle series and the refined top series are the maximal flag and the minimal flag, respectively. 42 Z. Chen, J. Xiao and F. Xu 5.3 Construction of cluster tilting objects Assume for each i ∈ {1, . . . , n}, there exists a j ∈ {1, . . . ,m}, such that i = ij . We define j− := max{0, 1 ⩽ s ⩽ j − 1 \| is = ij}, j+ := min{j + 1 ⩽ s ⩽ m,m+ 1 \| is = ij}, jmax := max{1 ⩽ s ⩽ r \| is = ij}, jmin := min{1 ⩽ s ⩽ r \| is = ij}, ji := max{1 ⩽ s ⩽ m \| is = i}. We define V0 := 0 and Vj := soc(i1,...,ij)Îij for 1 ⩽ j ⩽ m. Moreover, let Vi := ⊕m l=1 Vl. For 1 ⩽ i ⩽ n, let Ii,i := Vji and Ii := ⊕n l=1 Ii,l. Remark 5.4 ([11, Theorems 2.9 and 2.10]). The module Vi is a Cω-cluster-tilting object. Since Cω is a Frobenius category, we can consider its stable category [11, Theorem 2.8]. Let I(M,N) be the subspace of HomΛ(M,N) consisting of all morphisms factoring through objects in addIi and define HomΛ(M,N) = HomΛ(M,N)/I(M,N). Then we have Proposition 5.5 ([12, Proposition 3.24]). For any L ∈ Cω and 1 ⩽ j ⩽ m, DHomΛ(L, Vj) ∼= eij ( L+ j /L − j ) , where ei is the primitive idempotent in Λ associated to the vertex i. 5.4 Quiver Grassmannians Define Ei := EndΛ(Vi) op and E i := EndCω(Vi) op. If k is algebraically closed, then Ei is a finite- dimensional basic algebra and the corresponding quiver can be constructed explicitly. In this section, we make the assumption that over our fixed finite field k, the algebra Ei is presented by the same quiver with relations as in the case that k is algebraically closed. Define the quiver Qi as follows: the vertex set is {1, . . . ,m}; for each pair of subscripts 1 ⩽ k, j ⩽ m satisfying k+ ⩾ j+ ⩾ k > j and each arrow α : ij → ik, there is an arrow γk,jα : j → k called the ordinary arrow; for each 1 ⩽ j ⩽ m, there is an arrow γj : j → j− if j− ̸= 0 called the horizontal arrow. Proposition 5.6 ([12, Proposition 3.25]). There is an isomorphism of quivers Qi → QEi which maps j to Vj. Moreover, for L ∈ Cω, the Ei-module DHomA(L, Vi) can be realized as follows: the vector space at the vertex j is DHomA(L, Vj) = eij ( L+ j /L − j ) ; for the ordinary arrow γk,jα : j → k, the linear map is given by eij ( L+ j /L − j ) α·−→ eik ( L+ k /L − k ) ; for the horizontal arrow γj : j → j−, the linear map is given by eij ( L+ j /L − j ) eij ·−−→ eij ( L+ j−/L − j− ) . 5.5 Bijection Given L ∈ Cω, define the map di,L : {a ∈ Nn \| Φi,a,L ̸= ∅} −→ { g ∈ Nn \| GrEig (FL) ̸= ∅ } , (a1, . . . , am) 7−→ (g1, . . . gm), The Multiplication Formulas of Weighted Quantum Cluster Functions 43 where gj = (a−j − aj) + (a− j− − aj−) + · · ·+ (a−jmin − ajmin) and the map FGi,a,L : Φi,a,L −→ Gr Ei di,L(a) (FL), where FL = DHomA(L, Vi) ∼= Ext1Λ(Wi, L). For a given L• = (0 = Lm ⊆ · · · ⊆ L0 = L) ∈ Φi,a,L, the image FGi,a,L(L•) is a submodule of FL which can be realized as follows: the vector space at the vertex j is eij (Lj/L − j ); for the ordinary arrow γk,jα : j → k, the linear map is given by eij (Lj/L − j ) α·−→ eik(Lk/L − k ); for the horizontal arrow γj : j → j−, the linear map is given by eij (Lj/L − j ) eij ·−−→ eij (Lj−/L − j−). Theorem 5.7 ([12, Theorem 3.27]). For any L ∈ Cω and a ∈ Nn such that Φi,a,L ̸= ∅, the maps di,L and FGi,a,L are bijective. For the proof of this theorem, one can refer to [12, Theorem 1] where Geiss, Leclerc and Schröer proved an isomorphism of algebraic varieties from Φi,a,L to a certain quiver Grassman- nian over the complex field C. This result degenerates to a bijection between finite sets on a finite field k, therefore here we state the result without providing a detailed proof. The above theorem provides the relation between flags and Grassmannians. Furthermore, in order to discover relations between multiplication formulas, we need the following commutative diagram. Theorem 5.8. Let M,N,L ∈ Cω where L = mtϵ for some [ϵ] ∈ Ext1Λ(M,N). The short exact sequence ϵ provides a triangle N → L→M ϵ−→ ΣN after stabilization which we still denote by ϵ. Then there is a commutative diagram Φi,a,L Gr Ei di,L(a) (FL) ∐ a′+a′′=a Φi,a′,M × Φi,a′′,N ∐ a′+a′′=a GrEidi,M (a′)(FM)×GrEidi,N (a′′)(FN). ϕ̄ϵ FGi,a,L ψϵ ∐ FGi,a′,M×FGi,a′′,N Proof. Consider the following diagram: Φi,a,L GrEidi,L(a)(FL) ∐ a′+a′′=a Φi,a′,M × Φi,a′′,N ∐ a′,a′′ GrEidi,M (a′)(FM)×GrEidi,N (a′′)(FN). ϕ̄ϵ FGi,a,L ψϵ ∐ FGi,a′,M×FGi,a′′,N Denote FGi,a,L, FGi,a′,M and FGi,a′′,N by F̄L, F̄M and F̄N respectively. Given L• ∈ Φi,a,L, we claim that( F̄M × F̄N )( ϕ̄ϵ(L•) ) = ψϵ ( F̄L ) . For ϵ : N p−→ L q−→M−→ΣN , we have ϕ̄ϵ(L•) = ( q(L•), p −1(L•) ) , F (ϵ) : FN Fp−→ FL Fq−→ FM −→, eik(F (ϵ)) : eik ( N+ k /N − k ) p−→ eik ( L+ k /L − k ) q−→ eik ( M+ k /M − k ) −→. 44 Z. Chen, J. Xiao and F. Xu Denoted by p̄ the induced map of p and p̄k := eik p̄. Consider the following commutative diagram eik(N + k /N − k ) eik(L + k /L − k ) eik(M + k /M − k ) eik(p −1(Lk)/N − k ) eik(Lk/L − k ) eik(q(Lk)/M − k ) . p̄k q̄k p̄\|p−1(Lk) q̄\|Lk To prove ( F̄M × F̄N )( ϕ̄ϵ(L•) ) = ψϵ ( F̄L ) , it suffices to show that (a) q̄\|Lk is surjective which follows from the surjectivity of q\|Lk : Lk −→ q(Lk); (b) since p : N−→ L is injective, Ker p̄k = p−1(L− k )∩N + k /N − k = p−1(L− k )/N − k = Ker(p̄\|p−1(Lk)); (c) Ker(q̄\|Lk ) = Im(p̄\|p−1(Lk)) which follows by chasing the above diagram. Therefore, the above diagram is commutative. We can obtain the commutative diagram de- scribed in the theorem. ■ By Theorem 5.8, we have Corollary 5.9. For any (M•, N•) ∈ ∐ a′+a′′=aΦi,a′,M × Φi,a′′,N , under the same assumptions as in Theorem 5.8, the map FGi,a,L : ϕ̄−1 ϵ (M•, N•) −→ ψ−1 ϵ (FGi,a′,M (M•),FGi,a′′,N (N•)) is a bijection. In particular,∣∣ϕ̄−1 ϵ (M•, N•) ∣∣ = ∣∣ψ−1 ϵ (FGi,a′,M (M•),FGi,a′′,N (N•)) ∣∣. In order to keep compatibility with Sections 2 and 3, throughout the rest of this section, we only handle Φi,a,L with a ∈ {0, 1}m. 5.6 The skew-polynomial corresponding to ∆L In this subsection, we introduce a variant of the quantum cluster function defined in Section 3. Notice that Cω is a 2-Calabi–Yau Frobenius subcategory of nil Λ with Ext-symmetry, so that after stabilization, Cω is a 2-Calabi–Yau triangulated category. We define FL := { (c′, c′′) \| (c′, c′′) ∈ ∐ a′,a′′ Φi,a′,M × Φi,a′′,N , M,N ∈ Cω, a′,a′′,a′ + a′′ ∈ {0, 1}m } and set ZFL := {f : FL×ExactCω → Z \| f(c′, c′′, ϵ) = 0 unless c′0 = qtϵ, c′′0 = stϵ}, where ExactCω = {ϵ : 0 → N → L → M → 0 \| ϵ is a short exact sequence, N,L,M ∈ Cω}. The functions in ZFL are called weight functions. Given ϵ ∈ ExactCω , we define ZFL[ϵ] := {f ∈ ZFL \| f(c′, c′′, ρ) = 0 if ρ ̸= ϵ}. For f ∈ ZFL[ϵ], we write f(c′, c′′, ϵ) instead as f(c′, c′′). The Multiplication Formulas of Weighted Quantum Cluster Functions 45 Definition 5.10. Given a weight function f ∈ ZFL[ϵ], f ∗ϵ∆i,L is the skew-polynomial in Am,λ defined by f ∗ϵ ∆i,L := ∫ a ∫ a′+a′′=a ∫ (c′,c′′)∈ϕ̄ϵ(Φi,a,L) qk̄(c ′,c′′) · qf(c′,c′′) ·Xp(L,di,L(a)), where p(L, di,L(a)) is defined as in Definition 4.7 with L being treated as an object in Cω. For the convenience of calculation in the following, we use the notation f ∗ϵ δL(Φi,a) : = ∫ a′+a′′=a ∫ (c′,c′′)∈ϕ̄ϵ(Φi,a,L) qk̄(c ′,c′′) · qf(c′,c′′) ·Xp(L,di,L(a)). Notice that Am,λ is defined in Section 4.3 and we take the same skew polynomial algebra for f ∗ϵ ∆i,L as the one where f ∗ϵ XL is located. The skew-polynomial f ∗ϵ ∆i,L is also called the weighted quantum cluster function of L here. To state the corresponding multiplication formula of f ∗ϵL ∆i,L, we need two specific weight functions. Definition 5.11. For any objects M , N in Cω, define the weight function fspec by fspec(M•, N•, ϵ) = 1 2 λ(p(M,di,M (a′)), p(N, di,N (a ′′)))− k̄(M•, N•) if M• ∈ ∐ a′ Φi,a′,M , N• ∈ ∐ a′′ Φi,a′′,N , and 0 otherwise. Definition 5.12. For any objects M , N in Cω, define the weight function fskew as fskew(N•,M•, ϵ) = λ(p(M,di,M (a′)), p(N, di,N (a ′′))) if M• ∈ ∐ a′ Φi,a′,M , N• ∈ ∐ a′′ Φi,a′′,N , and 0 otherwise. Then we have Proposition 5.13. For any objects M,N ∈ Cω and any weighted quantum cluster functions f ∗ϵ′ ∆i,M and g ∗ϵ′′ ∆i,N , in Am,λ we have (f ∗ϵ′ ∆i,M ) · (g ∗ϵ′′ ∆i,N ) = (fspec ∗0MN Sfg) ∗0MN ∆i,M⊕N . Proof. For simplicity, without causing ambiguity, we omit some variables of weight functions in following calculation. For example, fspec((M•, N•), ϵ) is simplified as fspec. Direct calculation shows (f ∗ϵ′ ∆i,M ) · (g ∗ϵ′′ ∆i,N ) = ∫ a′,a′′ (f ∗ϵ′ δM (Φi,a′)) · (g ∗ϵ′′ δN (Φi,a′′)) ·Xp(M,di,M (a′)) ·Xp(N,di,N (a′′)) = ∫ a ∫ a′+a′′=a (f ∗ϵ′ δM (Φi,a′)) · (g ∗ϵ′′ δN (Φi,a′′)) · q 1 2 λ(p(M,di,M (a′)),p(N,di,N (a′′))) ·Xp(M,di,M (a′))+p(N,di,N (a′′)) = ∫ a ∫ a′+a′′=a (f ∗ϵ′ δM (Φi,a′)) · (g ∗ϵ′′ δN (Φi,a′′)) · q 1 2 λ(p(M,di,M (a′)),p(N,di,N (a′′))) ·Xp(M⊕N,di,M⊕N (a)). On the other hand, (fspec ∗0MN Sfg) ∗0MN ∆i,M⊕N 46 Z. Chen, J. Xiao and F. Xu = ∫ a (fspec ∗0MN Sfg) ∗0MN δM⊕N (Φi,a) ·Xp(M⊕N,di,M⊕N (a)) = ∫ a ∫ a′+a′′=a ∫ (M•,N•)∈Φi,a′,M×Φi,a′′,N qk̄(M•,N•) · qf+g+fspec ·Xp(M⊕N,di,M⊕N (a)) = ∫ a ∫ a′+a′′=a ∫ (M•,N•)∈Φi,a′,M×Φi,a′′,N qf+gq 1 2 λ(p(M,di,M (a′)),p(N,di,N (a′′))) ·Xp(M⊕N,di,M⊕N (a)). Then by definitions of f ∗ϵ′ δM and g ∗ϵ′′ δN , we can obtain the equality. ■ Theorem 5.14. For any weighted quantum cluster functions f ∗ϵ′ ∆i,M and g ∗ϵ′′ ∆i,N such that Ext1Cω(M,N) ̸= 0, in Am,λ we have∣∣PExt1Cω(M,N) ∣∣(f ∗ϵ′ ∆i,M ) · (g ∗ϵ′′ ∆i,N ) = ∫ Pϵ∈PExt1Cω (M,N) (f+ext ∗ϵ fspec ∗ϵ Sfg) ∗ϵ ∆i,mtϵ + ∫ Pη∈PExt1Cω (N,M) (fskew ∗η fspec ∗η Sgf ) ∗η ∆i,mtη = ∫ Pϵ∈PExt1Cω (M,N) (fspec ∗ϵ Sfg) ∗ϵ ∆i,mtϵ + ∫ Pη∈PExt1Cω (N,M) (f−ext ∗η fskew ∗η fspec ∗η Sgf ) ∗η ∆i,mtη, where f+ext, f − ext and fskew are defined as in Definition 4.19. Proof. We only prove the first equality. The calculation for the second one is similar. Recall that fspec(M•, N•, ϵ) = 1 2 λ(p(M,di,M (a′)), p(N, di,N (a ′′)))− k̄(M•, N•), fspec(N•,M•, η) = 1 2 λ(p(N, di,N (a ′′)), p(M,di,M (a′)))− k̄(N•,M•), fskew(N•,M•, η) = λ(p(M,di,M (a′)), p(N, di,N (a ′′))), fhom(N•,M•, η) = k̄(M•, N•)− k̄(N•,M•). So we have fskew(N•,M•, η) + fspec(N•,M•, η) = fhom(N•,M•, η) + fspec(M•, N•, ϵ). Then we have∣∣PExt1Cω(M,N) ∣∣(f ∗ϵ′ ∆i,M ) · (g ∗ϵ′′ ∆i,N ) = ∣∣PExt1Cω(M,N) ∣∣(fspec ∗0MN Sfg) ∗0MN ∆i,M⊕N = ∫ a ∣∣PExt1Cω(M,N) ∣∣(fspec ∗0MN Sfg) ∗0MN δM⊕N (Φi,a) ·Xp(M⊕N,di,M⊕N (a)),∫ Pϵ∈PExt1Cω (M,N) ( f+ext ∗ϵ fspec ∗ϵ Sfg ) ∗ϵ ∆i,mtϵ = ∫ a ∫ Pϵ∈PExt1Cω (M,N) ( f+ext ∗ϵ fspec ∗ϵ Sfg ) ∗ϵ δmtϵ(Φi,a) ·Xp(mtϵ,di,mtϵ(a)), The Multiplication Formulas of Weighted Quantum Cluster Functions 47 and ∫ Pη∈PExt1Cω (N,M) (fskew ∗η fspec ∗η Sgf ) ∗η ∆i,mtη = ∫ a ∫ Pη∈PExt1Cω (N,M) (fskew ∗η fspec ∗η Sgf ) ∗η δmtη(Φi,a) ·Xp(mtη,di,mtη(a)). We can rewrite∫ a ∫ Pη∈PExt1Cω (N,M) (fskew ∗η fspec ∗η Sgf ) ∗η δmtη(Φi,a) ·Xp(mtη,di,mtη(a)) as ∫ a ∫ Pη∈PExt1Cω (N,M) (fhom ∗η fspec ∗η Sgf ) ∗η δmtη(Φi,a) ·Xp(mtη,di,mtη(a)). But in the calculation of q-powers, we can replace q(fskew∗(Sgf ,η)∗fspec) by q(fhom∗(Sgf ,η)∗fspec). Then the desired equality is a direct consequence of Theorem 4.23. ■ Definition 5.15. We denote the Q-algebra generated by all weighted quantum cluster functions f ∗ϵ ∆i,L, where L runs over Cω and f ∈ ZFL[ϵ] with mtϵ = L, by Apq(Cω). Definition 5.16. We denote the Q-algebra generated by f ∗ϵ XL, where L runs over Cω and f ∈ ZMG(ϵ), by Aq(Cω). 5.7 Connection between two multiplication formulas Now we have introduced multiplication rules and proved multiplication formulas for both f∗ϵ∆i,L and g ∗ϵ XL. To end this section, we give the relationship between weight functions. Given a weight function f ∈ ZFL[ϵ], we define a weight function FG(f) ∈ ZMG(ϵ) by FG(f)(M0, N0, ϵ) = f ( FG−1 i,a′,M (M0),FG −1 i,a′′,N (N0) ) for any (M0, N0) ∈ ∐ e,f Gre(FM)×Grf (FN), where a′ = d−1 i,M (e) and a′′ = d−1 i,N (f). Similarly, given a weight function g ∈ ZMG(ϵ), we define a weight function FG∗(g) ∈ ZFL[ϵ] by FG∗(g)(M•, N•, ϵ) = g(FGi,a′,M (M•),FGi,a′′,N (N•), ϵ) for any (M•, N•) ∈ ∐ a′,a′′ Φi,a′,M × Φi,a′′,N . Moreover, we can extend the definitions of FG and FG∗ to Apq(Cω) and Aq(Cω). Definition 5.17. Define FG as a map from Apq(Cω) to Aq(Cω) by FG(f ∗ϵ ∆i,L) = FG(f) ∗ϵ XL and FG∗ as a map from Aq(Cω) to A p q(Cω) by FG∗(g ∗ϵ XL) = FG∗(g) ∗ϵ ∆i,L. 48 Z. Chen, J. Xiao and F. Xu Theorem 5.18. The map FG is an isomorphism of algebras with the inverse FG∗. Proof. In fact, if we consider Apq(Cω) and Aq(Cω) as subalgebras of a fixed Am,λ, FG and FG∗ are identities. Set FG(f ∗ϵ ∆i,L) = g ∗ϵ XL. By Definition 5.10, f ∗ϵ ∆i,L = ∫ a f ∗ϵ δL(Φi,a) ·Xp(L,di,L(a)) = ∫ a ∫ a′+a′′=a ∫ (M•,N•)∈ϕ̄ϵ(Φi,a,L) qk̄(M•,N•) · qf(M•,N•) ·Xp(L,di,L(a)) and g ∗ϵ XL = ∫ d ∫ e,f ∫ (M0,N0)∈Grϵe,f (FM,FN,d) ql(M,N,M0,N0) · qg(M0,N0) ·Xp(L,d). By Corollary 5.9, k̄(M•, N•) = l(M,N,M0, N0) and by Definition 5.17, qf(M•,N•) = qFG(f)(M0,N0) for M0 = FGi,a′,M (M•) and N0 = FGi,a′′,N (N•). Then we have f ∗ϵ ∆i,L = g ∗ϵ XL. ■ Remark 5.19. By Definition 5.10, when L is a Cω-projective-injective object, FG(L) ∈ Q. 6 Special version in hereditary case In this section, we consider the cluster category from a hereditary algebra and the corresponding multiplication formula. 6.1 Cluster category from a hereditary algebra Let m ⩾ n be two positive integers and Q̃ an acyclic quiver with the vertex set {1, . . . ,m}. Let Q be the full subquiver of Q̃ with the vertex set {1, . . . , n}. Given a finite field k, set Ã = kQ̃ and A = kQ. For any vertex i of Q̃ (respectively Q), denote by Si the simple Ã-module (respectively A-module) at i and by Pi the indecomposable projective Ã-module corresponding to i. Let Ã be the category of finite-dimensional left Ã-modules and the cluster category of Ã introduced by Buan–Marsh–Reineke–Reiten–Todorov [3], is defined as C : = CÃ = Db ( Ã ) /τ−1Σ where Db ( Ã ) is the bounded derived category of Ã, τ is the Auslander–Reiten translation and Σ is the shift functor. Respectively, one can define A and CA = Db(A)/τ−1Σ. Let B̃ = (bij) be an m× n-matrix where bij = dimk Ext 1 Ã(Si, Sj)− dimk Ext 1 Ã(Sj , Si) for 1 ⩽ i ⩽ m, 1 ⩽ j ⩽ n. Assume there exists a skew-symmetric m×m-matrix Λ such that Λ(−B̃) = [ In 0 ] m×n , where In is the n× n identity matrix. In the following, the bilinear form λ is always given by λ(e, f) = eTΛf . The cluster category C is a 2-Calabi–Yau triangulated category [14] with a cluster tilting object T = ΣÃ. There is a natural functor F : = HomC(Ã,−) : C −→ Ã which induces an equivalence of categories C/ addΣÃ ≃−→ Ã. The Multiplication Formulas of Weighted Quantum Cluster Functions 49 Moreover, all iso-classes of indecomposable objects in C can be classified by ind C = ind Ã ∪ {ΣP1, . . . ,ΣPm}, where ind Ã is the iso-classes of all indecomposable objects in Ã. We say L ∈ C is located in the fundamental domain if L ∈ Ã. An object M ∈ C is called coefficient-free if M does not contain a direct summand Pi[1], i > n. In particular, ifM,N ∈ indA and dimHomC(M,ΣN) = 1, there exist two non-split triangles N → L→M → ΣN and M → L′ → N → ΣM, where one of L and L′ is located in the fundamental domain and the other is not [13]. Without loss of generality, in the following, we always assume L′ is located in the fundamental domain. Now we can introduce weighted quantum cluster functions and corresponding multiplication formulas given in Section 4 to the cluster category C. 6.2 A special weighted quantum cluster function Let L be an object with ℓ := dimkFL in C and consider the trivial triangle L =−→ L→ 0 σL−−→ ΣL. Definition 6.1. Given L as above, define a special weight function fL as fL((0, L0), σL) := −1 2 ⟨dimkL0, ℓ− dimkL0⟩ for any L0 ∈ ∐ dGrdFL and 0 otherwise, where ⟨−,−⟩ is the Euler form of Ã. Definition 6.2. With respect to fL, define the weighted quantum cluster function X̃L := fL ∗σL XL. Notice that, for a given dimension vector d of a submodule of FL, the appropriate dimension vectors of qtσL and stσL are unique and obviously 0 and d. So we can simplify the calculation of X̃L to X̃L = ∫ d ∫ (0,d) ∫ (0,L0) qfL((0,L0),σL) ·Xp(L,d) = ∫ d \|GrdFL\| · q− 1 2 ⟨d,l−d⟩ ·Xp(L,d). Note that Rupel [21] firstly gave the above definition of quantum cluster characters over cluster categories of hereditary algebras over finite fields. Qin provided an alternative definition of quantum cluster characters via Serre polynomials [18]. Remark 6.3. Note that f ∗ϵXL = (f ∗ϵ fL) ∗ϵXL, with support area still decided by ϵ instead of σL. Lemma 6.4. For any M,N ∈ C, we have X̃M · X̃N = f̃MN ∗0MN X̃M⊕N in An,λ where f̃MN ∈ ZMG(0MN ) is defined as f̃MN (M0, N0,0MN ) = 1 2 λ(p(M, e), p(N, f))− l(M,N,M0, N0) + fM ((0,M0), σM ) + fN ((0, N0), σN )− fM⊕N ((0,M0 ⊕N0), σM⊕N ) for any M0 ∈ ∐ eGre(FM), N0 ∈ ∐ f Grf (FN) and 0 otherwise. 50 Z. Chen, J. Xiao and F. Xu Proof. We can calculate both sides as X̃M · X̃N = ∫ e \|Gre(FM)\| · qfM ((0,M0),σM ) ·Xp(M,e) · ∫ f \|Grf (FN)\| · qfN ((0,N0),σN ) ·Xp(N,f) = ∫ e,f \|Gre(FM)\|\|Grf (FN)\| · q 1 2 λ(p(M,e),p(N,f))+fM ((0,M0),σM )+fN ((0,N0),σN ) ·Xp(M,e)+p(N,f) = ∫ e,f ∫ (M0,N0)∈Gre(FM)×Grf (FN) q 1 2 λ(p(M,e),p(N,f))+fM ((0,M0),σM )+fN ((0,N0),σN ) ·Xp(M,e)+p(N,f) and X̃M⊕N = ∫ d ∫ e,f ∫ (M0,N0)∈Gre(FM)×Grf (FN) ∣∣ψ−1 0MN ,d (M0, N0) ∣∣ · qf0((0,M0⊕N0),σM⊕N ) = ∫ d ∫ e,f ∫ (M0,N0)∈Gr 0MN e,f (FM,FN,g) ql(M,N,M0,N0)+fM⊕N ((0,M0⊕N0),σM⊕N ) ·Xp(M⊕N,d). Then by definition of weighted quantum cluster functions, one can easily check such de- fined f̃MN is the appropriate weight function to satisfy the lemma. ■ Moreover, if we set dimkFM = m, dimkFN = n, then Lemma 6.5 ([18]). With the notation above, we have 1 2 λ(p(M, e), p(N, f)) = 1 2 λ(indM, indN) + 1 2 ⟨f ,m⟩ − 1 2 ⟨e,n⟩+ 1 2 ⟨e, f⟩ − 1 2 ⟨f , e⟩. By this lemma, we can simplify the calculation of f̃MN to f̃MN (M0, N0,0MN ) = 1 2 λ(p(M, e), p(N, f))− l(M,N,M0, N0) + fM ((0,M0), σM ) + fN ((0, N0), σN )− fM⊕N ((0,M0 ⊕N0), σM⊕N ) = 1 2 λ(indM, indN) + 1 2 ⟨f ,m⟩ − 1 2 ⟨e,n⟩+ 1 2 ⟨e, f⟩ − 1 2 ⟨f , e⟩ − l(M,N,M0, N0)− 1 2 ⟨e,m− e⟩ − 1 2 ⟨f ,n− f⟩ + 1 2 ⟨e+ f ,m+ n− e− f⟩ = 1 2 λ(indM, indN) + ⟨f ,m− e⟩ − l(M,N,M0, N0). 6.3 Special version of multiplication formula Recall that given any balanced pair of weight functions ( g+, g− ) and weighted quantum cluster functions f ∗ϵ′ XM and g ∗ϵ′′ XN such that HomC(M,ΣN) ̸= 0, we have \|PHomC(M,ΣN)\|(f ∗ϵ′ XM ) · (g ∗ϵ′′ XN ) = ∫ Pϵ∈PHomC(M,ΣN) ( g+ ∗ϵ fspec ∗ϵ Tfg ) ∗ϵ Xmtϵ + ∫ Pη∈PHomC(N,ΣM) (g− ∗η gskew ∗η fspec ∗η Tgf ) ∗η Xmtη. The Multiplication Formulas of Weighted Quantum Cluster Functions 51 In particular, we can take X̃M = fM ∗σM XM and X̃N = fN ∗σN XN , then we have \|PHomC(M,ΣN)\|X̃M · X̃N = ∫ Pϵ∈PHomC(M,ΣN) ( g+ ∗ϵ fspec ∗ϵ TfMfN ) ∗ϵ Xmtϵ + ∫ Pη∈PHomC(N,ΣM) (g− ∗η gskew ∗η fspec ∗η TfNfM ) ∗η Xmtη. To express the right-hand side in terms of X̃mtϵ and X̃mtη, we need to extend f̃MN to the non-split case. Definition 6.6. Given M,N ∈ C and ϵ ∈ HomC(M,ΣN) with mtϵ = L, define f̃ϵ ∈ ZMG(ϵ) as f̃ϵ(M0, N0, ϵ) = 1 2 λ(p(M, e), p(N, f))− l(M,N,M0, N0) + fM ((0,M0), σM ) + fN ((0, N0), σN )− fL((0, L0), σL) for (M0, N0) ∈ Imψϵ where L0 satisfies ψϵ(L0) = (M0, N0), and 0 otherwise. Remark 6.7. Note that (1) although the submodule L0 which satisfies ψϵ(L0) = (M0, N0) is not unique, according to Corollary 4.5, the dimension vector of L0 is independent of choice, so f̃ϵ is well defined; (2) by definition, f̃0MN = f̃MN ; (3) if the triangle induced by ϵ is mapped by F to a short exact sequence 0 → FN → FL→ FM → 0, then for any (M0, N0) ∈ Imψϵ, f̃ϵ(M0, N0, ϵ) = f̃MN (M0, N0,0MN ). Lemma 6.8. For any balanced pair ( g+, g− ) and morphisms ϵ ∈ HomC(M,ΣN), η ∈ HomC(N, ΣM) with mtϵ = L,mtη = L′, we have( g+ ∗ϵ fspec ∗ϵ TfMfN ) ∗ϵ XL = ( g+ ∗ϵ f̃ϵ ) ∗ϵ X̃L, (g− ∗η gskew ∗η fspec ∗η TfNfM ) ∗η XL′ = ( g+ ∗η f̃η ) ∗η X̃L′ . Proof. We only prove the first equality. The proof of the second equality is similar. By definition, the right-hand side is equal to ( g+ ∗ϵ f̃ϵ ∗ϵ fL ) ∗ϵ XL. We just need to compare the values that both weight functions take at any ((M0, N0), ϵ) satisfying (M0, N0) ∈ Imψϵ. We have g+ ∗ϵ fspec ∗ϵ TfMfN (M0, N0, ϵ) = g+(M0, N0, ϵ) + fM ((0,M0), σM ) + fN ((0, N0), σN ) + 1 2 (p(M, e), p(N, f))− l(M,N,M0, N0). On the other hand, g+ ∗ϵ f̃ϵ ∗ϵ fL(M0, N0, ϵ) = g+(M0, N0, ϵ) + 1 2 λ(p(M, e), p(N, f))− l(M,N,M0, N0) + fM ((0,M0), σM ) + fN ((0, N0), σN ) − fL((0, L0), σL) + fL((0, L0), σL) 52 Z. Chen, J. Xiao and F. Xu = g+(M0, N0, ϵ) + 1 2 λ(p(M, e), p(N, f))− l(M,N,M0, N0) + fM ((0,M0), σM ) + fN ((0, N0), σN ). Recall the definition of weighted quantum cluster function f ∗ϵ XL = ∫ d ∫ e,f ∫ (M0,N0)∈Grϵe,f (FM,FN,d) ql(M,N,M0,N0) · qf(M0,N0) ·Xp(L,d). Since Imψϵ = ∐ d ∐ e,f Grϵe,f (FM,FN,d), the weight functions g+ ∗ϵ fspec ∗ϵ TfMfN and g+ ∗ϵ f̃ϵ ∗ϵ fL have the same values on the whole domain of integration. Hence the weighted quantum cluster functions are the same. ■ Then we can express the right-hand side of the multiplication formula in terms of X̃mtϵ and X̃mtη. Theorem 6.9. Given a balanced pair of weight functions ( g+, g− ) and two objects M,N ∈ C such that HomC(M,ΣN) ̸= 0, we have \|PHomC(M,ΣN)\|X̃M · X̃N = ∫ Pϵ∈PHomC(M,ΣN) ( g+ ∗ϵ f̃ϵ ) ∗ϵ X̃mtϵ + ∫ Pη∈PHomC(N,ΣM) ( g− ∗η gskew ∗η f̃η ) ∗η X̃mtη. Recall that Proposition 4.21 provides two pointwise balanced pairs of weight functions. Corollary 6.10. Given two objects M , N in C such that HomC(M,ΣN) ̸= 0, we have \|PHomC(M,ΣN)\|X̃M · X̃N = ∫ Pϵ∈PHomC(M,ΣN) ( g+ext ∗ϵ f̃ϵ ) ∗ϵ X̃mtϵ + ∫ Pη∈PHomC(N,ΣM) (gskew ∗η f̃η) ∗η X̃mtη = ∫ Pϵ∈PHomC(M,ΣN) f̃ϵ ∗ϵ X̃mtϵ + ∫ Pη∈PHomC(N,ΣM) ( g−ext ∗ gskew ∗ f̃η ) ∗η X̃mtη. Remark 6.11. As this work was being completed, we became aware of a similar result by Chen, Ding and Zhang [7] for hereditary categories. The exact relationship between these two results will be investigated in the near future. In particular, if dimk HomC(M,ΣN) = 1, both balanced pairs degenerate to (0, gskew). Thus we have Theorem 6.12. Given two objects M,N ∈ C with dimk HomC(M,ΣN) = 1 and two non-split triangles N → L→M ϵ−→ ΣN and M → L′ → N η−→ ΣM, we have X̃M · X̃N = f̃ϵ ∗ϵ X̃L + ( gskew ∗ f̃η ) ∗η X̃L′ . The Multiplication Formulas of Weighted Quantum Cluster Functions 53 6.4 Recalculation and simplification Theorem 6.12 is a special case of Theorem 4.27 which was proved generally in Section 4.10. But in this subsection, we calculate the right-hand side again to obtain a simple expression. We always assumeM , N are indecomposable coefficient-free rigid objects in Ã with dimk HomC(M,ΣN) = 1 and the triangles N → L→M ϵ−→ ΣN and M → L′ → N η−→ ΣM are non-split, where L′ is located in the fundamental domain. By definition of weighted quantum cluster functions and Lemma 6.5, f̃ϵ ∗ϵ X̃L = ∫ d ∫ e,f ∫ (M0,N0)∈Grϵe,f (FM,FN,d) q 1 2 λ(p(M,e),p(N,f))+fM ((0,M0),σM )+fN ((0,N0),σN ) ·Xp(L,d) = ∫ d ∫ e,f ∫ (M0,N0)∈Grϵe,f (FM,FN,d) q 1 2 λ(indM,indN)+⟨f ,m−e⟩+fM⊕N ((0,M0⊕N0),σM⊕N ) ·Xp(L,d). On the other hand,( gskew ∗η f̃η ) ∗η X̃L′ = ∫ d ∫ f ,e ∫ (N0,M0)∈Grηf ,e(FN,FM,d) q 1 2 λ(p(M,e),p(N,f))+fN ((0,N0),σN )+fM ((0,M0),σM ) ·Xp(L′,d) = ∫ d ∫ f ,e ∫ (N0,M0)∈Grηf ,e(FN,FM,d) q 1 2 λ(indM,indN)+⟨f ,m−e⟩+fN⊕M ((0,N0⊕M0),σN⊕M ) ·Xp(L′,d). Remark 6.13. One can check the left-hand side in Theorem 6.12 is X̃M · X̃N = ∫ e,f ∫ (M0,N0)∈GreFM×GrfFN q 1 2 λ(p(M,e),p(N,f))+fM ((0,M0),σM )+fN ((0,N0),σN ) ·Xp(M,e)+p(N,f) = ∫ e,f ∫ (M0,N0)∈GreFM×GrfFN q 1 2 λ(indM,indN)+⟨f ,m−e⟩+fM⊕N ((0,M0⊕N0),σM⊕N ) ·Xp(M,e)+p(N,f). Notice that in the case when dimk HomC(M,ΣN) = 1, Grϵe,f (FM,FN,d) is complementary to Grηf ,e(FN,FM,d). That is to say, GreFM ×GrfFN = ∐ d (Grϵe,f (FM,FN,d) ∪Grηf ,e(FN,FM,d)). Thus the above calculation also provides a direct proof of Theorem 6.12. Now we focus on( gskew ∗η f̃η ) ∗η X̃L′ = ∫ d ∫ f ,e ∫ (N0,M0)∈Grηf ,e(FN,FM,d) q 1 2 λ(indM,indN)+⟨f ,m−e⟩+fN⊕M ((0,N0⊕M0),σN⊕M ) ·Xp(L′,d). 54 Z. Chen, J. Xiao and F. Xu Recall the assumption that L′ is located in the fundamental domain, so 0 → FM → FL′ → FN → 0 is a short exact sequence. The following result is implicitly implied by an argument used by Qin in [18, Proposi- tion 5.4.1]. Lemma 6.14. With the assumptions above, if ψη(L ′ 0) = (N0,M0) as in the following diagram FM/M0 FN/N0 ϵ : FM FL′ FN ϵ0 : M0 L′ 0 N0, pM pN iM iN then dimk Ext 1 Ã(N0, FM/M0) = 0. Proof. Notice that Ã is hereditary, thus Ext2Ã(−,−) = 0. Applying HomÃ(N0,−) on 0 →M0 → FM → FM/M0 → 0, we get Ext1Ã(N0,M0) Ext1 Ã (N0,iM ) −−−−−−−−→ Ext1Ã(N0, FM) ↠ Ext1Ã(N0, FM/M0) → 0. Similarly, applying HomÃ(−, FM) on 0 → N0 → FN → FN/N0 → 0, we get Ext1Ã(FN/N0, FM) → Ext1Ã(FN,FM) Ext1 Ã (iN ,FM) −−−−−−−−−→ Ext1Ã(N0, FM) → 0. Since dimk Ext 1 Ã(FN,FM) = 1, we have that dimk Ext 1 Ã(N0, FM/M0) is at most one. If dimk Ext 1 Ã(N0, FM/M0) = 1, then dimk Ext 1 Ã(N0, FM) = 1, Ext1Ã(N0, iM ) is zero and Ext1Ã(iN , FM) is a linear isomorphism between one dimensional vector spaces. However, considering the construction of α′ N0,M0 , such a commutative diagram implies that Ext1Ã(N0, iM )(ϵ0) and Ext1Ã(iN , FM)(ϵ) coincide when ϵ is non-zero, as we are assuming. This is a contradiction. ■ Recalling Remark 6.7, for a short exact sequence 0 → FM → FL′ → FN → 0 and (N0,M0) ∈ Imψη, we have fN⊕M ((N0,M0), σN⊕M ) = fL′((N0,M0), σL′). Finally we can simplify the calculation of ( gskew ∗η f̃η ) ∗η X̃L′ . Lemma 6.15. We have( gskew ∗η f̃η ) ∗η X̃L′ = q 1 2 λ(indM,indN) · X̃L′ . The Multiplication Formulas of Weighted Quantum Cluster Functions 55 Proof. A direct calculation shows( gskew ∗η f̃η ) ∗η X̃L′ = ∫ d ∫ f ,e ∫ (N0,M0)∈Grηf ,e(FN,FM,d) q 1 2 λ(indM,indN)+⟨f ,m−e⟩+fN⊕M ((0,N0⊕M0),σN⊕M ) ·Xp(L′,d) = ∫ d ∫ f ,e ∫ (N0,M0)∈Grηf ,e(FN,FM,d) q 1 2 λ(indM,indN)+l(N,M,N0,M0)+fL′ ((0,L′ 0),σL′ ) ·Xp(L′,d) = q 1 2 λ(indM,indN) · ∫ d ∫ f ,e ∫ (N0,M0)∈Grηf ,e(FN,FM,d) ql(N,M,N0,M0)+fL′ ((0,L′ 0),σL′ ) ·Xp(L′,d) = q 1 2 λ(indM,indN) · fL′ ∗σL′ XL′ = q 1 2 λ(indM,indN) · X̃L′ . ■ Notice that in the last expression above, the q-power is independent of the specific choice of submodules, and depends only on M and N . Now we analyze f̃ϵ ∗ϵ X̃L = ∫ d ∫ e,f ∫ (M0,N0)∈Grϵe,f (FM,FN,d) q 1 2 λ(indM,indN)+⟨f ,m−e⟩+fM⊕N ((0,M0⊕N0),σM⊕N ) ·Xp(L,d), where the middle term L in the triangle N → L→M ϵ−→ ΣN is not located in the fundamental domain. In this case, there is an explicit construction of FL given as follows. Let U = ImFϵ, then there exists a short exact sequence 0 → V → FM Fϵ−→ U → 0. Then we have an exact sequence 0 → U → FΣN → τW ′ ⊕ I → 0. Let W =W ′ ⊕ P , there exists a sequence FN Fi−→W ⊕ V Fp−−→ FM Fϵ−→ FΣN. Finally, FL =W ⊕ V . Applying F to the triangle N → L→M ϵ−→ ΣN leads to the commutative diagram 0 0 0 FN/N0 FL/L0 FM/M0 CokerFp FN FL FM FΣN. KerFi N0 L0 M0 0 0 0 Fi Fp Fϵ 56 Z. Chen, J. Xiao and F. Xu Since FL =W ⊕ V , we have a commutative diagram 0 0 0 0 V/J FL/L0 W/K 0 0 V W ⊕ V W 0 0 J L0 K 0. 0 0 0 Considering J as a submodule of V , we have the commutative diagram 0 0 J J 0 V FM U 0 V/J FM/J U. 0 0 On the other hand, given submodule K of W , we can obtain an injection τK → τW ′ ⊕ I by applying τ and using the commutative diagram τ(W/K) U FΣN τW ′ ⊕ I 0 U τN0 τK 0, 0 0 where τN0 is given by the pullback which is unique up to isomorphism. The Multiplication Formulas of Weighted Quantum Cluster Functions 57 Lemma 6.16. There is a commutative diagram J J τ(W/K) 0 V FM FΣN τW ′ ⊕ I 0 0 V/J FM/J τN0 τK 0, 0 0 Fϵ Fϵ where Fϵ is induced by Fϵ. Proof. By definition, U = ImFϵ, therefore the composition of morphisms FM U FΣN is exactly Fϵ. Thus we denote the composition of morphisms FM/J U τN0 by Fϵ where Fϵ(m̄) = Fϵ(m). Since J ⊆ V = KerFϵ, Fϵ(J) = 0 and Fϵ(FM/J) = Fϵ(FM) = U ⊆ τN0. Therefore, Fϵ is well defined and the diagram FM FΣN FM/J τN0 Fϵ Fϵ commutes. Moreover, KerFϵ = KerFϵ/J = V/J and ImFϵ = ImFϵ = U . Thus the third row in the above four-row diagram is a long exact sequence. ■ Let d = dimL0, n = dimN , w = dimW , k = dimK and j = dim J . We can construct a correspondence ϕ : ∐ d ∐ k,j Gr0WV k,j (W,V,d) −→ ∐ d ∐ e,f Grϵe,f (FM,FN,d), (K,J) 7−→ (J,N0), where J =M0 and N0 is determined by τN0 in the second row of the above two-row diagram. Lemma 6.17. The map ϕ is bijective. Proof. First we prove ϕ is surjective. Given a long exact sequence in Ã 0 −→ V i−→ FM Fϵ−→ FΣN (π1π2 )−→ τW ′ ⊕ I −→ 0. It gives a triangle in C N −→W ⊕ I[−1]⊕ V −→M −→ τN. 58 Z. Chen, J. Xiao and F. Xu Applying F to the above triangle, we obtain the exact sequence FN W ⊕ V FM. (π0 ) ( 0,i ) Given any (J,N0) ∈ ∐ d ∐ e,f Grϵe,f (FM,FN,d), by definition (see Section 4.4), there exists (W0, V0) ∈ ∐ d ∐ k,jGr0WV k,j (W,V,d) such that the following diagram is commutative: FN W ⊕ V FM N0 W0 ⊕ V0 J. (π0 ) ( 0,i ) iM iN Hence, W0 = π1(N0), V = J and then ϕ(π1(N0), J) = (J,N0). We now prove ϕ is injective. By the definition of ϕ, N0 = π−1 1 (K) andK = π1(N0). Therefore, there is a unique (K,J) mapped to the given (J,N0). ■ Lemma 6.18. We have that dimϕ(K,J) := (dim J, dimN0) = (j,n− (w − k)). Proof. Given K ⊆W , consider the commutative diagram 0 U FΣN τW ′ ⊕ I 0 0 U τN0 τK 0. Since both rows are short exact sequences, we can compute dimension vectors after apply- ing τ−1 as dimkN0 = dimkFN + dimkK − dimkW = n− (w − k). ■ To complete the final calculation, we need an identity given in [18]. Lemma 6.19 ([18]). With the notation above, we have ⟨n− (w − k),m− j⟩ − 1 2 ⟨e+ f ,m+ n− e− f⟩ = ⟨k,v − j⟩ − 1 2 ⟨j+ k,w + v − j− k⟩ − 1 2 . Remark 6.20. Recalling the definitions of fM⊕N and fL, we can rewrite the above identity as ⟨n− (w − k),m− j⟩+ fM⊕N ((0,M0 ⊕N0), σM⊕N ) = ⟨k,v − j⟩+ fL((0, L0), σL)− 1 2 . Finally, we can simplify the calculation of f̃ϵ ∗ϵ X̃L. Lemma 6.21. We have f̃ϵ ∗ϵ X̃L = q 1 2 λ(indM,indN)− 1 2 · X̃L. Proof. A direct calculation shows f̃ϵ ∗ϵ X̃L = ∫ d ∫ e,f ∫ (M0,N0)∈Grϵe,f (FM,FN,d) q 1 2 λ(indM,indN)+⟨f ,m−e⟩+fM⊕N ((0,M0⊕N0),σM⊕N ) ·Xp(L,d) = q 1 2 λ(indM,indN) The Multiplication Formulas of Weighted Quantum Cluster Functions 59 · ∫ d ∫ e,f ∫ (M0,N0)∈Grϵe,f (FM,FN,d) q⟨f ,m−e⟩+fM⊕N ((0,M0⊕N0),σM⊕N ) ·Xp(L,d) = q 1 2 λ(indM,indN) · ∫ d ∫ k,j ∫ (K,J)∈Gr 0WV k,j (K,J,d) q⟨n−(w−k),m−j⟩+fM⊕N ((0,M0⊕N0),σM⊕N ) ·Xp(L,g) = q 1 2 λ(indM,indN)− 1 2 · ∫ d ∫ k,j ∫ (K,J)∈Gr 0WV k,j (K,J,d) q⟨k,v−j⟩+fL((0,L0),σL) ·Xp(L,d). Now we focus on the exponent ⟨k,v − j⟩ of q in the above integration. Since M and N are rigid, so isW⊕V ⊕Σ−1I. Notice that 0 → V →W⊕V →W → 0 is a split short exact sequence. Therefore in this case, we also have dimk Ext 1 A(K,V/J) = 0 for any (K,J) ∈ GrkW ×GrjV and hence ⟨k,v − j⟩ = dimk HomA(K,V/J). On the other hand, we can consider the mapping∐ d GrdFL −→ ∐ k,j GrkW ×GrjV with affine fibers, induced by this split exact sequence, and observe that dimk HomA(K,V/J) is exactly the dimension of the fiber at (K,J). Thus we have f̃ϵ ∗ϵ X̃L = q 1 2 λ(indM,indN)− 1 2 · ∫ d ∫ k,j ∫ (K,J)∈Gr 0WV k,j (K,J,d) q⟨k,v−j⟩+fL((0,L0),σL) ·Xp(L,d) = q 1 2 λ(indM,indN)− 1 2 · ∫ d ∫ k,j ∫ (K,J)∈Gr 0WV k,j (K,J,d) qdimk HomA(K,V/J)+fL((0,L0),σL) ·Xp(L,d) = q 1 2 λ(indM,indN)− 1 2 · ∫ d ∫ L0∈GrdFL qfL((0,L0),σL) ·Xp(L,d) = q 1 2 λ(indM,indN)− 1 2 · ∫ d \|GrdFL\| · q− 1 2 ⟨d,l−d⟩ ·Xp(L,d) = q 1 2 λ(indM,indN)− 1 2 · X̃L. ■ By Theorem 6.12, Lemmas 6.15 and 6.21, we have Theorem 6.22 ([18, Proposition 5.4.1]). In the cluster category C = Db(Ã)/τ−1Σ of a hereditary algebra Ã, given two indecomposable coefficient-free rigid objects M,N ∈ Ã with dimk HomC(M,ΣN) = 1 and two non-split triangles N → L→M ϵ−→ ΣN and M → L′ → N η−→ ΣM, where L′ is located in the fundamental domain, then we have X̃M · X̃N = q 1 2 λ(indM,indN)− 1 2 · X̃L + q 1 2 λ(indM,indN) · X̃L′ . Acknowledgements The research was supported by the National Natural Science Foundation of China (no. 11771217 and no. 12031007). We greatly appreciate the referees’ extraordinarily useful and detailed com- ments and suggestions which helped us to improve our manuscript. We are grateful to Xueqing Chen and Ming Ding for indicating many mistakes in the preliminary version of this paper and many valuable suggestions. 60 Z. Chen, J. Xiao and F. Xu References [1] Berenstein A., Zelevinsky A., Quantum cluster algebras, Adv. Math. 195 (2005), 405–455, arXiv:math.QA/0404446. [2] Buan A.B., Iyama O., Reiten I., Scott J., Cluster structures for 2-Calabi–Yau categories and unipotent groups, Compos. Math. 145 (2009), 1035–1079, arXiv:math.RT/0701557. [3] Buan A.B., Marsh R., Reineke M., Reiten I., Todorov G., Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), 572–618, arXiv:math.RT/0402054. [4] Caldero P., Chapoton F., Cluster algebras as Hall algebras of quiver representations, Comment. Math. Helv. 81 (2006), 595–616, arXiv:math.RT/0410187. 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id	nasplib_isofts_kiev_ua-123456789-212034
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2026-03-16T04:01:53Z
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publisher	Інститут математики НАН України
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spelling	Chen, Zhimin Xiao, Jie Xu, Fan 2026-01-23T10:09:23Z 2023 The Multiplication Formulas of Weighted Quantum Cluster Functions. Zhimin Chen, Jie Xiao and Fan Xu. SIGMA 19 (2023), 097, 60 pages 1815-0659 2020 Mathematics Subject Classification: 17B37; 16G20; 17B20 arXiv:2110.12429 https://nasplib.isofts.kiev.ua/handle/123456789/212034 https://doi.org/10.3842/SIGMA.2023.097 By applying the property of Ext-symmetry and the affine space structure of certain fibers, we introduce the notion of weighted quantum cluster functions and prove their multiplication formulas associated to abelian categories with Ext-symmetry and 2-Calabi-Yau triangulated categories with cluster-tilting objects. The research was supported by the National Natural Science Foundation of China (no. 11771217 and no. 12031007). We greatly appreciate the referees’ extraordinarily useful and detailed comments and suggestions, which helped us to improve our manuscript. We are grateful to Xueqing Chen and Ming Ding for pointing out many mistakes in the preliminary version of this paper and for their many valuable suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Multiplication Formulas of Weighted Quantum Cluster Functions Article published earlier
spellingShingle	The Multiplication Formulas of Weighted Quantum Cluster Functions Chen, Zhimin Xiao, Jie Xu, Fan
title	The Multiplication Formulas of Weighted Quantum Cluster Functions
title_full	The Multiplication Formulas of Weighted Quantum Cluster Functions
title_fullStr	The Multiplication Formulas of Weighted Quantum Cluster Functions
title_full_unstemmed	The Multiplication Formulas of Weighted Quantum Cluster Functions
title_short	The Multiplication Formulas of Weighted Quantum Cluster Functions
title_sort	multiplication formulas of weighted quantum cluster functions
url	https://nasplib.isofts.kiev.ua/handle/123456789/212034
work_keys_str_mv	AT chenzhimin themultiplicationformulasofweightedquantumclusterfunctions AT xiaojie themultiplicationformulasofweightedquantumclusterfunctions AT xufan themultiplicationformulasofweightedquantumclusterfunctions AT chenzhimin multiplicationformulasofweightedquantumclusterfunctions AT xiaojie multiplicationformulasofweightedquantumclusterfunctions AT xufan multiplicationformulasofweightedquantumclusterfunctions

The Multiplication Formulas of Weighted Quantum Cluster Functions

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