Caldero-Chapoton algebras

Motivated by the representation theory of quivers with potentials introduced by Derksen, Weyman and Zelevinsky and by work of Caldero and Chapoton, who gave explicit formulae for the cluster variables of Dynkin quivers, we associate a Caldero-Chapoton algebra to any (possibly infinite dimensional) basic algebra. By definition, the Caldero-Chapoton algebra is (as a vector space) generated by the Caldero-Chapoton functions of the decorated representations of the basic algebra. The Caldero-Chapoton algebra associated to the Jacobian algebra of a quiver with potential is closely related to the cluster algebra and the upper cluster algebra of the quiver. The set of generic Caldero-Chapoton functions, which conjecturally forms a basis of the Caldero-Chapoton algebra) is parametrized by the strongly reduced components of the varieties of representations of the Jacobian algebra and was introduced by Geiss, Leclerc and Schr\"oer. Plamondon parametrized the strongly reduced components for finite-dimensional basic algebras. We generalize this to arbitrary basic algebras. Furthermore, we prove a decomposition theorem for strongly reduced components. Thanks to the decomposition theorem, all generic Caldero-Chapoton functions can be seen as generalized cluster monomials. As another application, we obtain a new proof for the sign-coherence of g-vectors. Caldero-Chapoton algebras lead to several general conjectures on cluster algebras.

1. Introduction 1 2. Basic algebras and decorated representations 4 3. E-invariants and g-vectors of decorated representations 8 4. Caldero-Chapoton algebras 11 5. Strongly reduced components of representation varieties 14 6. Component graphs and CC-clusters 22 7. Caldero-Chapoton algebras and cluster algebras 25 8. Sign-coherence of generic g-vectors 26 9. Examples 27 References 34 1. Introduction 1.1. Let A Q be the Fomin-Zelevinsky cluster algebra [FZ1,FZ2] associated to a finite 2-acyclic quiver Q. By definition A Q is generated by an inductively defined set of rational functions, called cluster variables. The cluster variables are contained in the set M Q of cluster monomials, which are by definition certain monomials in the cluster variables. Now let W be a non-degenerate potential for Q, and let Λ = P(Q, W ) be the associated Jacobian algebra introduced by Derksen, Weyman and Zelevinsky [DWZ1,DWZ2]. The category of decorated representations of Λ is denoted by decrep(Λ). To any M ∈ decrep(Λ) Date: 03.12.2012Date: 03.12. . 2010 Mathematics Subject Classification. Primary 13F60; Secondary 16G10, 16G20. one can associate a Laurent polynomial C Λ (M), the Caldero-Chapoton function of M. It follows from [DWZ1,DWZ2] that the cluster monomials form a subset of the set C Λ of Caldero-Chapoton functions.
1.2. The generic basis conjecture. One of the main problems in cluster algebra theory is to find a basis of A Q with favourable properties. As an important requirement, this basis should contain the set M Q of cluster monomials in a natural way.
The concept of strongly reduced irreducible components of varieties of decorated representations of a Jacobian algebra Λ was introduced in [GLS]. To each strongly reduced component Z one can associate a generic Caldero-Chapoton function C Λ (Z), see Sections 4.1 and 6.4. It was conjectured in [GLS] that the set B Λ of generic Caldero-Chapoton functions forms a C-basis of A Q . Using a non-degenerate potential defined by Labardini [La1,La2], Plamondon [P2] found a counterexample and then conjectured that B Λ is a basis of the upper cluster algebra A up Q . This conjecture should also be wrong in general. We replace it by yet another conjecture.
We study the Caldero-Chapoton algebra generated by all Caldero-Chapoton functions. We do not restrict ourselves to Jacobian algebras, but work with algebras Λ defined as arbitrary quotients of completed path algebras. In particular, we generalize the notion of a Caldero-Chapoton function to this general setup. One easily checks that the functions C Λ (M) do not only generate A Λ as an algebra but also as a vector space over the ground field C.
We show that the set B Λ of generic Caldero-Chapoton functions is linearly independent provided the kernel of the skew-symmetric incidence matrix B Q of Q does not contain any non-zero element in Q n ≥0 . This generalizes [P2,Proposition 3.19]. For Λ = P(Q, W ) a Jacobian algebra associated to a quiver Q with non-degenerate potential W we have where A Q is the cluster algebra and A up Q is the upper cluster algebra associated to Q. (We refer to [BFZ,DWZ1,FZ1] for missing definitions.) For this special case, we give a list of open problems, which hopefully will lead to a better understanding of the rather mysterious relation between A Q and A up Q .
1.3. Parametrization of strongly reduced components. Plamondon [P2,Theorem 1.2] parametrized the strongly reduced components for finite-dimensional basic algebras. (For our (non-standard) definition of a basic algebra we refer to Section 2.1.) We generalize Plamondon's result to arbitrary basic algebras. Let Λ = C Q /I be a basic algebra, where the quiver Q has n vertices. Let decIrr(Λ) be the set of irreducible components of all varieties decrep d,v (Λ) of decorated representations of Λ, where (d, v) runs through N n × N n . By decIrr s.r. (Λ) we denote the subset of strongly reduced components. (The definition is in Section 5.) Recall that decIrr s.r. (Λ) parametrizes the elements in B Λ .
Let G s.r. Λ : decIrr s.r. (Λ) → Z n be the map sending Z ∈ decIrr s.r. (Λ) to the generic g-vector g Λ (Z) of Z. (The definition of a g-vector is in Section 3.) Using Plamondon's result for finite-dimensional algebras, and a long-path truncation argument, we get the following parametrization of strongly reduced components for arbitrary Λ.
(ii) The following are equivalent: (a) G s.r. Λ is surjective. (b) Λ := C Q /I is finite-dimensional, where I is the m-adic closure of I.
1.4. A decomposition theorem for strongly reduced components. The notion of a direct sum of irreducible components of representation varieties was introduced in [CBS]. The Zariski closure Z := Z 1 ⊕ · · · ⊕ Z t of a direct sum of irreducible components Z 1 , . . . , Z t of varieties of representations of Λ is always irreducible, but in general Z is not an irreducible component. It was shown in [CBS] that Z is an irreducible component provided the dimension of the first extension group between the components is generically zero. The following decomposition theorem is an analogue for strongly reduced components. Instead of extension groups, we work with a generalization E Λ (−, ?) of the Derksen-Weyman-Zelevinsky E-invariant [DWZ2]. (We define E Λ (−, ?) in Section 3.) Theorem 1.3. For Z 1 , . . . , Z t ∈ decIrr(Λ) the following are equivalent: (i) Z 1 ⊕ · · · ⊕ Z t is a strongly reduced irreducible component. (ii) Each Z i is strongly reduced and E Λ (Z i , Z j ) = 0 for all i = j.
Based on Theorem 1.3, we show that all elements of B Λ can be seen as CC-cluster monomials. (The CC-cluster monomials generalize Fomin and Zelevinsky's notion of cluster monomials.) 1.5. Sign-coherence of g-vectors. A subset U of Z n is called sign-coherent if for each 1 ≤ i ≤ n we have either a i ≥ 0 for all (a 1 , . . . , a n ) ∈ U , or we have a i ≤ 0 for all (a 1 , . . . , a n ) ∈ U .
Theorem 1.4. Let Λ be a basic algebra, and let Z 1 , . . . , Z t ∈ decIrr s.r. (Λ) be strongly reduced components. Assume that 1.6. The paper is organized as follows. In Section 2 we recall definitions and basic properties of basic algebras and their (decorated) representations. We also introduce truncations of basic algebras, which play a crucial role in some of our proofs. In Section 3 we introduce and study g-vectors and E-invariants of decorated representations. Caldero-Chapoton functions and Caldero-Chapoton algebras are defined in Section 4. Our main results Theorem 1.2 and 1.3 are proved in Section 5. In Section 6 we introduce component graphs, component clusters and CC-clusters, and we show that the cardinality of loopcomplete subgraphs of a component graph is bounded by the number of simple modules. Section 7 explains the relation between Caldero-Chapoton algebras and cluster algebras. Section 8 contains the proof of Theorem 1.4. Finally, in Section 9 we discuss several examples of Caldero-Chapoton algebras. 1.7. Notation. We denote the composition of maps f : M → N and g : N → L by gf = g • f : M → L. We write |U | for the cardinality of a set U .
A finite-dimensional module M is basic provided it is a direct sum of pairwise nonisomorphic indecomposable modules. For a module M and some m ≥ 1 let M m be the direct sum of m copies of M .
For a finite-dimensional algebra Λ let τ Λ be its Auslander-Reiten translation. For an introduction to Auslander-Reiten theory we refer to the books [ARS] and [ASS].
For n ≥ 1 and a set S, depending on the situation and if no misunderstanding can occur, we identify S n with the set of (n×1)-or (1×n)-matrices with entries in S. By N we denote the natural numbers, including zero. For d = (d 1 , . . . , d n ) ∈ N n let |d| := d 1 + · · · + d n . For n ∈ N let M n (Z) be the set of (n × n)-matrices with integer entries.
For a ring R let R[x ± 1 , . . . , x ± n ] be the algebra of Laurent polynomials over R in n independent variables x 1 , . . . , x n . For a = (a 1 , . . . , a n ) ∈ Z n set x a := x a 1 1 · · · x an n .

Basic algebras and decorated representations
2.1. Basic algebras and quiver representations. Throughout, let C be the field of complex numbers. A quiver is a quadruple Q = (Q 0 , Q 1 , s, t), where Q 0 is a finite set of vertices, Q 1 is a finite set of arrows, and s, t : Q 1 → Q 0 are maps. For each arrow a ∈ Q 1 we call s(a) and t(a) the starting and terminal vertex of a, respectively. If not mentioned otherwise, we always assume that A path in Q is a tuple p = (a m , . . . , a 1 ) of arrows a i ∈ Q 1 such that s(a i+1 ) = t(a i ) for all 1 ≤ i ≤ m − 1. Then length(p) := m is the length of p. Additionally, for each vertex i ∈ Q 0 there is a path e i of length 0. We often just write a m · · · a 1 instead of (a m , . . . , a 1 ).
A path p = (a m , . . . , a 1 ) of length m ≥ 1 is a cycle in Q, or more precisely an m-cycle in Q, if s(a 1 ) = t(a m ). The quiver Q is acyclic if there are no cycles in Q, and for s ≥ 1 the quiver Q is called s-acyclic if there are no m-cycles for 1 ≤ m ≤ s.
A representation of a quiver Q = (Q 0 , Q 1 , s, t) is a tuple M = (M i , M a ) i∈Q 0 ,a∈Q 1 , where each M i is a finite-dimensional C-vector space, and M a : M s(a) → M t(a) is a C-linear map for each arrow a ∈ Q 1 . We call dim(M ) := (dim(M 1 ), . . . , dim(M n )) the dimension vector of M . Let dim(M ) := dim(M 1 ) + · · · + dim(M n ) be the dimension of M . For a path p = (a m , . . . , a 1 ) in Q let M p := M am • · · · • M a 1 . The representation M is called nilpotent provided there exists some N > 0 such that M p = 0 for all paths p in Q with length(p) > N . For m ∈ N let CQ m be a C-vector space with a C-basis labeled by the set Q m of paths of length m in Q. Note that CQ m is finite-dimensional. We do not distinguish between a path p of length m and the corresponding basis vector in CQ m .
The completed path algebra of a quiver Q is denoted by C Q . As a C-vector space we have We write the elements in C Q as infinite sums m≥0 a m with a m ∈ CQ m . The product in C Q is then defined as A potential of Q is an element W = m≥1 w m of C Q , where each w m is a C-linear combination of m-cycles in Q. By definition, W = 0 is also a potential. The definition of a non-degenerate potential can be found in [DWZ1,Section 7].
The category mod(C Q ) of finite-dimensional left C Q -modules can be identified with the category nil(Q) of nilpotent representations of Q.
By m we denote the arrow ideal in C Q , which is generated by the arrows of Q. Thus for p ≥ 0 we have We call an algebra Λ basic if Λ = C Q /I for some quiver Q and some admissible ideal I of C Q .
A representation of a basic algebra Λ = C Q /I is a nilpotent representation of Q, which is annihilated by the ideal I. We identify the category rep(Λ) of representations of Λ with the category mod(Λ) of finite-dimensional left Λ-modules. Up to isomorphism the simple representations of Λ are the 1-dimensional representations S 1 , . . . , S n .
The category of all (possibly infinite dimensional) Λ-modules is denoted by Mod(Λ), we consider rep(Λ) as a subcategory of Mod(Λ).
see for example [G].  is the m-adic closure of I in C Q . We obtain the following commutative diagram with exact rows, where all morphisms, whose label contains the symbol ι (resp. π) are the obvious canonical monomorphisms (resp. epimorphisms).
Lemma 2.1. For any basic algebra Λ we have For 1 ≤ i ≤ n and p ≥ 2 let I i,p ∈ rep(Λ p ), I i ∈ Mod(Λ) and I i ∈ Mod(Λ) be the injective envelopes of the simple module S i . The above embedding functors yield a chain I i,2 ⊆ I i,3 ⊆ · · · ⊆ I i,p ⊆ · · · I i of submodules of I i , and we have Lemma 2.2. Let Λ = C Q /I be a basic algebra. Then the following hold: Proof. Let a m · · · a 1 be a path of length m in Q, and let M be a representation of Λ. We can see M as a nilpotent representation of Q. For any non-zero vector v 0 ∈ M set v i := a i · · · a 1 v 0 for 1 ≤ i ≤ m. Assume that each of the vectors v 1 , . . . , v m is non-zero. We claim that v 0 , v 1 , . . . , v m are pairwise different and linearly independent. Let b be a path of maximal length such that bv 0 = 0. Such a path b exists, because M is nilpotent. By induction v 1 , . . . , v m are linearly independent. Assume now that Since ba i · · · a 1 is either zero or a path of length i + length(b), we have ba i · · · a 1 v 0 = 0 for all 1 ≤ i ≤ m. Since bv 0 = 0, this is a contradiction. Therefore v 0 , v 1 , . . . , v m are linearly independent. It follows that for any M ∈ decrep(Λ) with dim(M) = (d, v) and any path b with length(b) ≥ |d| we have bM = 0. This implies (i). Parts (ii) and (iv) are easy consequences of (i). Any extension of representations M and N of Λ is a representation of Λ of dimension dim(M ) + dim(N ). This implies (iii).
3. E-invariants and g-vectors of decorated representations 3.1. Definition of E-invariants and g-vectors. Let Q be a quiver, and let W be a potential of Q. Let Λ = P(Q, W ) be the associated Jacobian algebra [DWZ1,Section 3]. For decorated representations M and N of Λ the g-vector g(M) and the invariants E inj (M) and E inj (M, N ) were defined in [DWZ2], where E inj (M) is called the E-invariant of M. We define invariants g Λ (M), E Λ (M) and E Λ (M, N ) of decorated representations M and N of an arbitrary basic algebra Λ = C Q /I as follows.
Proof. Part (i) follows from [P1,Lemma 4.7,Proposition 4.8]. It can also be shown in a more elementary way by using the exact sequence displayed in [DWZ2,Equation (10.4)]. Part (ii) is a direct consequence of (i) and the definition of E Λ (M, N ) and E inj (M, N ).

Homological interpretation of the
be the injective envelope of the simple representation S i of Λ. One easily checks that the socle soc(I i ) of I i is isomorphic to S i , and that We will need the following theorem due to Auslander and Reiten.
. This implies (i). By the construction of minimal injective presentations, I Λ 1 (M ) is the injective envelope of Coker(f ). It follows that soc(Coker(f )) ∼ = soc(I Λ 1 (M )). We apply the functor Here we used that I Λ 0 (M ) is injective, which implies Ext 1 Λ (S i , I Λ 0 (M )) = 0. By (i) we know that F is an isomorphism. Thus G is also an isomorphism. This implies (ii).

Combinining Lemma 2.2 and Lemma 3.3 yields the following result.
Lemma 3.4. Let M = (M, V ) be a decorated representation of a basic algebra Λ, and let g Λ (M) = (g 1 , . . . , g n ) be the g-vector of M. If p > dim(M ), then The following result is a homological interpretation of the E-invariant in terms of Auslander-Reiten translations. This can be seen as a generalization of [DWZ2,Corollary 10.9].
In particular, we have Proof. Since p > dim(M ), dim(N ) we can apply Lemma 2.2 and get be a minimal injective presentation of N , where we regard N now as a representation of Λ p . It follows from Lemma 3.3 and Equation (2) that This implies The first equality follows from Lemmas 2.2, 3.3 and 3.4. The second equality says that This finishes the proof.
where B Q and g Λ (M) are defined as in Sections 2.1 and 3.1, respectively. Note that The definition of C Λ (M) is motivated by the (different versions of) Caldero-Chapoton functions appearing in the theory of cluster algebras, see for example [Pa,Section 1]. Such functions first appeared in work of Caldero and Chapoton [CC,Section 3], where they show that the cluster variables of a cluster algebra of a Dynkin quiver are Caldero-Chapoton functions.
Proof. Part (i) follows directly from the definitions and from the additivity of the functors Hom Λ (−, ?) and Ext 1 Λ (−, ?). To prove (ii), let M = (M, V ) be a decorated representation of Λ. For the decorated representation (0, V ) we have . Thus (ii) holds. Now (iii) follows from (i), (ii) and the well known formula where the sum runs over all pairs (e ′ , e ′′ ) ∈ N n × N n such that e ′ + e ′′ = e, see for example [DWZ2, Proof of Proposition 3.2].

4.2.
Definition of a Caldero-Chapoton algebra. In the previous section, we associated to a basic algebra Λ the set . , x ± n ] generated by the variables x 1 , . . . , x n . By definition the Caldero-Chapoton algebra A Λ associated to Λ is the C-subalgebra of C[x ± 1 , . . . , x ± n ] generated by C Λ . The following is a direct consequence of Lemma 4.1(iii).
Lemma 4.2. The set C Λ generates A Λ as a C-vector space.
In this generality, Caldero-Chapoton algebras might not be so useful. (One could generalize even more by replacing the matrix B Q in the definition of the functions C Λ (M) by any other matrix in M n (Z).) But the case where Λ is the Jacobian algebra P(Q, W ) of a quiver Q with non-degenerate potential W (see [DWZ1] for missing definitions) should certainly be of interest. In this case, based on work of Palu [Pa], Plamondon [P1] considered a version of Caldero-Chapoton functions using the Amiot cluster category [A]. In contrast, we follow Derksen, Weyman and Zelevinsky's [DWZ1,DWZ2] approach and define and study Caldero-Chapoton functions purely in terms of the representation theory of the Jacobian algebra without passing to the cluster category. 4.3. Linear independence of Caldero-Chapoton functions. Let Λ = C Q /I be a basic algebra. Except in some trivial cases, the set C Λ of Caldero-Chapoton functions associated to decorated representations of Λ is linearly dependent. Often the Caldero-Chapoton functions satisfy beautiful relations, which should be studied more intensively. On the other hand, by Lemma 4.2, there are C-bases of A Λ consisting only of Caldero-Chapoton functions. Our aim is to provide a candidate B Λ for such a basis. Before constructing B Λ in Section 5, we prove the following criterion for linear independence of certain sets of Caldero-Chapoton functions. Let Q n ≥0 := {(a 1 , . . . , a n ) ∈ Q n | a i ≥ 0 for all i}, Q n >0 := {(a 1 , . . . , a n ) ∈ Q n | a i > 0 for all i}. Proposition 4.3. Let Λ = C Q /I be a basic algebra. Let M j , j ∈ J be decorated representations of Λ. Assume the following: We claim that this defines a partial order on Z n . Clearly, ≤ is reflexive and transitive.
The partial order ≤ on Z n induces obviously a partial order on the set of Laurent monomials in the variables x 1 , . . . , x n . Namely, set Among the Laurent monomials x g Λ (M)+B Q e occuring in the expression Without loss of generality we assume that λ j = 0 for all j. There is a (not necessarily unique) index s such that It follow that the Laurent monomial x g Λ (Ms) does not occur as a summand of any of the Laurent polynomials C Λ (M j ) with j = s. (Here we use that the g-vectors of the decorated representations M j are pairwise different.) This implies λ s = 0, a contradiction. Thus For a = (a 1 , . . . , a n ) and b = (b 1 , . . . , b n ) in Z n set a · b := a 1 b 1 + · · · + a n b n .
Note that condition (d) in the following lemma coincides with condition (i) in Proposition 4.3.
is positive. Clearly, there is some e ∈ Q n such that r i j · e = 1 for all 1 ≤ j ≤ m. (The (k × n)-matrix with rows r i 1 , . . . , r im has rank m. Thus, we can see it as a surjective homomorphism Q n → Q m .) Now observe that the kth entry of B Q e is λ (k) 1 + · · · + λ (k) m for all 1 ≤ k ≤ n with k / ∈ {i 1 , . . . , i m } and that this entry is positive. It follows that . We get B Q e = 0, and therefore −aB Q e = b · e = 0. Since b has only positive entries and e has only non-negative entries, we get e = 0. This finishes the proof.
For the example, where Λ is the path algebra of an affine quiver of type A 2 , the main argument used in the proof of Proposition 4.3 can already be found in [C,Section 6.1]. If we replace condition (i) by condition (a), Proposition 4.3 was first proved by Fu and Keller [FK,Corollary 4.4]. Essentially the same argument was later also used by Plamondon [P1]. That the Fu-Keller argument can be applied under condition (b) was observed by Geiß and Labardini. To any triangulation T of a punctured Riemann surface with non-empty boundary, one can associate a 2-acyclic quiver Q T . It is shown in [GLaS] that there is always a triangulation T such that the matrix B Q T satisfies condition (b). Note that any irreducible component Z ∈ decIrr(Λ) can be seen as an irreducible component in Irr(Λ dec ), where Λ dec := Λ× C × · · ·× C is defined as the product of Λ with n copies of C. In fact, we can identify decrep(Λ) and rep(Λ dec ). Thus statements on varieties of representations can be carried over to varieties of decorated representations.

Strongly reduced components of representation varieties
By definition we have We have an isomorphism decrep d,v (Λ) → rep d (Λ) of affine varieties mapping (M, C v ) to M . Thus the irreducible components of decrep d,v (Λ) can be interpreted as irreducible components of rep d (Λ). For Z ∈ decIrr d,v (Λ) let πZ be the corresponding component in Irr d (Λ).

It is easy to construct examples of components
. Namely, let Λ = CQ be the path algebra of the Kronecker quiver, and let Z ∈ decIrr d,v (Λ) with d = (1, 1) and v = (0, 0). (Since Λ is a path algebra of an acyclic quiver, decrep d,v (Λ) is irreducible for all d, v.) An easy calculation shows that end Λ (Z) = e Λ (Z) = E Λ (Z) = 1 and hom Λ (Z, Z) = ext 1 Λ (Z, Z) = E Λ (Z, Z) = 0. A further discussion of this example can be found in Section 9.4.3.
Proof. Let d = dim(πZ) and d i = dim(πZ i ) for i = 1, 2. Choose some p ≥ 2|d|, |d 1 |+ |d 2 |. By Lemma 2.2 we can regard all the representations in Z, Z 1 and Z 2 as representations of Λ p . Thus we can interpret Z, Z 1 and Z 2 as irreducible components in decIrr(Λ p ). Now Proposition 3.5 allows us to assume without loss of generality that Λ = Λ p . Voigt's Lemma [G,Proposition 1.1] implies that c Λ (Z) ≤ e Λ (Z). The Auslander-Reiten formula . (Here we used again Proposition 3.5.) Following [GLS] we call an irreducible component Z ∈ decIrr(Λ) strongly reduced provided

Given irreducible components
It is quite easy to show that Z 1 ⊕ · · · ⊕ Z t is an irreducible closed subset of decrep d,v (Λ), but in general it is not an irreducible component.
The next lemma is an easy exercise.
Lemma 5.4. For 1 ≤ i ≤ n and any decorated representation Proof. Let Z ∈ decIrr s.r. d,v (Λ) for some d, v, and let Z ′ be the corresponding irreducible component of decIrr d,0 (Λ). We clearly have c Lemma 5.6. Let Z ∈ decIrr d,v (Λ), and assume that p > |d|. Then the following are equivalent: Proof. Since p > |d|, we can apply Lemma 2.2 and Proposition 3.5 and get c Λp (Z) = c Λ (Z) and E Λp (Z) = E Λ (Z). This yields the result.
The additivity of the functor Hom Λ (−, ?) and upper semicontinuity imply the following lemma.
Proof. For i = 1, 2 let (d i , v i ) := dim(Z i ), and let (d, v) := dim(Z). We have dim(Z) = dim(Z 1 ) + dim(Z 2 ) and dim(πZ i ) = d i . The map N 1 , N 2 ). The fibres of f M 1 ,M 2 are of dimension Using Equation (1), an easy calculation yields Let M be in the image of f . We want to compute the dimension of the fibre f −1 (M). Let It follows from the Krull-Remak-Schmidt Theorem that T is a finite set. Thus the fibre

Thus by upper semicontinuity there is a dense open subset
This finishes the proof.
Lemma 5.10. For Z, Z 1 , Z 2 ∈ decIrr(Λ) with Z = Z 1 ⊕ Z 2 we have Proof. For i = 1, 2 let (d i , v i ) := dim(Z i ), and let (d, v) := dim(Z). We get The first equality follows directly from the definition of c Λ (Z). The second equality uses Lemma 5.7(i) and Lemma 5.9.
The following result is a version of Theorem 5.3 for strongly reduced components.
Assume that Z := Z 1 ⊕ Z 2 is a strongly reduced component. Thus we have c Λ (Z) = E Λ (Z). Applying Lemma 5.10 and Lemma 5.7(ii) this implies To show the converse, assume that Z 1 and Z 2 are strongly reduced with E Λ (Z 1 , Z 2 ) = E Λ (Z 2 , Z 1 ) = 0. We claim that For the first equality we use Lemma 5.10, the second equality is just our assumption that Z 1 and Z 2 are strongly reduced. Finally, the third equality follows from Lemma 5.7 together with our assumption that E Λ (Z 1 , Z 2 ) and E Λ (Z 2 , Z 1 ) are both zero. Thus Z is strongly reduced.
Note that Theorems 5.3 and 5.11 imply that each Z ∈ decIrr s.r. (Λ) is of the form Z = Z 1 ⊕ · · · ⊕ Z t with Z i ∈ decIrr s.r. (Λ) and Z i indecomposable for all i. The next lemma follows directly from upper semicontinuity and Lemma 4.1(i).
Proof. It follows from the definitions that . Now the claim follows, since c Λ (Z) = E Λ (Z).
For a representation M let add(M ) be the category of all finite direct sums of direct summands of M . Plamondon [P2] obtains the following striking result.
Theorem 5.15 (Plamondon). For any finite-dimensional basic algebra Λ the following hold: Note that Plamondon works with irreducible components, and not with decorated irreducible components. But his results translate easily from one concept to the other.
We now generalize Theorem 5.15(i) to arbitrary basic algebras Λ. It turns out that decIrr s.r. (Λ) is in general no longer parametrized by Z n but by a subset of Z n . Our proof is based on Plamondon's result and uses additionally truncations of basic algebras.
For a basic algebra Λ let G Λ : decIrr(Λ) → Z n be the map, which sends Z ∈ decIrr(Λ) to the generic g-vector g Λ (Z) of Z.
Theorem 5.17. For a basic algebra Λ the following hold: Proof. Since Λ p is finite-dimensional for all p, we know from Plamondon's Theorem 5.15 Λ . Note also that for every Z ∈ decIrr(Λ) we have G Λ (Z) = G Λp (Z) for some large enough p.) To show the converse, assume that Λ is infinite dimensional. Recall that Λ = lim ← − (Λ p ) and where I i is the injective envelope of the simple Λ-module S i , and I i,p is the injective envelope of S i considered as a Λ p -module, and Λ p = C Q /(I + m p ). We have It follows that there exists some 1 ≤ i ≤ n such that I i is infinite dimensional.
As a vector space, e i Λ p is generated by the residue classes a + (I + m p ) of all paths a in Q with t(a) = i. We have This implies dim(I i,2 ) ≥ 2. (Otherwise, there is no arrow a with t(a) = i, which implies I i,p = I i,2 for all p ≥ 2, a contradiction since I i is infinite dimensional.) Now suppose that I i,p−1 = I i,p for some p ≥ 3. This implies e i Λ p−1 = e i Λ p . Thus we have e i (I + m p−1 ) = e i (I + m p ). It follows that e i (I + m p+1 ) = e i I + e i (I + m p )m = e i I + e i (I + m p−1 )m = e i (I + m p ).
In other words, we have I i,p+1 = I i,p . By induction we get I i,q = I i,p−1 for all q ≥ p, a contradiction since I i is infinite dimensional. Thus we proved that dim(I i,p ) ≥ p for all p ≥ 2. Now assume that −e i is in the image of G s.r. Λ . (Here e i denotes the ith standard basis vector of Z n .) In other words, there is some Z ∈ decIrr s.r. (Λ) such that G s.r. Λ (Z) = −e i . By Lemma 5.16(ii) we know that Z ∈ decIrr s.r. <p (Λ p ) for some p ≥ 1. Since g Λ (Z) = −e i , we have I Λp 0 (Z) = I i,p and I Λp 1 (Z) = 0. (Here we use Theorem 5.15(ii).) This implies that Z is the closure of the orbit of the decorated representation (I i,p , 0). But dim(I i,p ) ≥ p and the dimension of all representations in Z is strictly smaller than p, a contradiction.
6. Component graphs and CC-clusters 6.1. The graph of strongly reduced components. Let Λ be a basic algebra. In [CBS] the component graph Γ(Irr(Λ)) of Λ is defined as follows: The vertices of Γ(Irr(Λ)) are the indecomposable irreducible components in Irr(Λ). There is an edge between (possibly equal) vertices Z 1 and Z 2 if ext 1 Λ (Z 1 , Z 2 ) = ext 1 Λ (Z 2 , Z 1 ) = 0. We want to define an analogue of Γ(Irr(Λ)) for strongly reduced components. The graph Γ(decIrr s.r. (Λ)) of strongly reduced components has as vertices the indecomposable components in decIrr s.r. (Λ), and there is an edge between (possibly equal) vertices Z 1 and 6.2. Component clusters. Let Γ be a graph, and let Γ 0 be the set of vertices of Γ. We allow only single edges, and we allow loops, i.e. edges from a vertex to itself. For a subset U ⊆ Γ 0 let Γ U be the full subgraph, whose set of vertices is U . The subgraph Γ U is complete if for each i, j ∈ U with i = j there is an edge between i and j. A complete subgraph Γ U is maximal if for each complete subgraph Γ U ′ with U ⊆ U ′ we have U = U ′ . We call a subgraph Γ U loop-complete if Γ U is complete and there is a loop at each vertex in U . Proof. Assume that Z 1 , . . . , Z n+1 are pairwise different vertices of a loop-complete subgraph Γ J of Γ(decIrr s.r. (Λ)). For 1 ≤ i ≤ n + 1 let g Λ (Z i ) be the generic g-vector of Z i . Since Γ J is loop-complete we know by Theorem 5.11 that is again a strongly reduced component for each a = (a 1 , . . . , a n+1 ) ∈ N n+1 . By the additivity of g-vectors we get g Λ (Z a ) = a 1 g Λ (Z 1 ) + · · · + a n+1 g Λ (Z n+1 ).
Furthermore, we know from Theorem 5.3 that Z a = Z b if and only if a = b. Now one can essentially copy the proof of [GS,Theorem 1.1] to show that there are a, b ∈ N n+1 with g Λ (Z a ) = g Λ (Z b ) but a = b. By Theorem 5.17 different strongly reduced components have different g-vectors. Thus we have a contradiction.
Corollary 6.2. Let Λ be a finite-dimensional basic algebra. Let M be a representation of Λ with Hom Λ (τ − Λ (M ), M ) = 0. Then M has at most n isomorphism classes of indecomposable direct summands.
The following conjecture might be a bit too optimistic. But it is true for Λ = C Q the path algebra of an acyclic quiver Q, see [DW,Corollary 21] and Section 9.1. Conjecture 6.3. For any basic algebra Λ the following hold: (i) The component clusters of Λ have cardinality at most n.
(ii) The E-rigid component clusters of Λ are exactly the component clusters of cardinality n.
6.3. E-rigid representations. After most of this work was done, we learned that Iyama and Reiten [IR] obtained some beautiful results on socalled τ -rigid modules over finitedimensional algebras, which fit perfectly into the framework of Caldero-Chapoton algebras.
Adapting their terminology to decorated representations of basic algebras, a decorated representation M of a basic algebra Λ is called E-rigid provided E Λ (M) = 0. The following theorem is just a reformulation of Iyama and Reiten's results on τ -rigid modules. Part (i) follows also directly from the more general statement in Theorem 6.1.
For M ∈ decrep(Λ) let Σ(M) be the number of isomorphism classes of indecomposable direct summands of M.
Theorem 6.4 ( [IR]). Let Λ = C Q /I be a finite-dimensional basic algebra. For M ∈ decrep(Λ) the following hold: It is easy to find examples of infinite dimensional basic algebras Λ such that Theorem 6.4(iii) does not hold, see Section 9.3.1.
A basic algebra Λ is representation-finite if there are only finitely many isomorphism classes of indecomposable representations in rep(Λ). One easily checks that Λ is finitedimensional in this case.
Corollary 6.5. Assume that Λ is a representation-finite basic algebra. Then the following hold: Proof. Since Λ is representation-finite, every irreducible component Z ∈ decIrr(Λ) is a union of finitely many orbits, and exactly one of these orbits has do be dense in Z.
6.4. Generic Caldero-Chapoton functions. For each (d, v) ∈ N n × N n let If B Λ is a basis of A Λ , then we call B Λ the generic basis of A Λ . 6.5. CC-clusters. For a component cluster U of a basic algebra Λ let (In each of the products above we assume that a Z = 0 for all but finitely many Z ∈ U .) The set C U is called a CC-cluster of Λ, and the elements in M U are CC-cluster monomials.
(The letters CC just indicate that we deal with sets of Caldero-Chapoton functions.) A CC- The following result is a direct consequence of the definition of B Λ and Theorem 5.11. Proposition 6.7. Let Λ be a basic algebra. Then where the union is over all component clusters U of Λ. 6.6. A change of perspective. The CC-clusters are a generalization of the clusters of a cluster algebra defined by Fomin and Zelevinsky. In general, the Fomin-Zelevinsky cluster monomials form just a small subset of the set of CC-cluster monomials. Recall that the Fomin-Zelevinsky cluster monomials are obtained by the inductive procedure of cluster mutation [FZ1,FZ2], and the relation between neighbouring clusters is described by the exchange relations. One can see the exchange relations as part of the definition of a cluster algebra. On the other hand, the definition of a Caldero-Chapoton algebra does not involve any mutations of CC-clusters. The CC-clusters are given by collections of irreducible components, and they do not have to be constructed inductively. One can find a meaningful notion of neighbouring CC-clusters, and it remains quite a challenge to actually determine an analogue of the exchange relations. 6.7. Open problems. In this section let Λ be any basic algebra. The following conjecture is again quite optimistic in this generality.
Conjecture 6.8 is true for every Λ = C Q with Q an acyclic quiver and also for numerous other examples, see [GLS].
Problem 6.9. Does the set We say that A Λ has the Laurent phenomenon property, if for any E-rigid component cluster {Z 1 , . . . , Z n } of Λ, we have The first part of the following proposition is a consequence of [DWZ2,Lemma 5.2], compare also the calculation at the end of [GLS,Section 6.3]. The rest follows from [DWZ2,Corollary 7.2].
Proposition 7.1. We have In general, the sets M Q , M Q,W and B Q,W are pairwise different. 7.2. Example. Let Q be the quiver It is not difficult to check that P(Q, W 1 ) is infinite dimensional and P(Q, W 2 ) is finitedimensional. By [BFZ,Proposition 1.26] the algebras A Q and A up Q do not coincide. The potentials W 1 and W 2 are both non-degenerate, see [DWZ1,Example 8.6] and [La2,Example 8.2], respectively. Furthermore, by [P2,Example 4.3] the set B Q,W 2 of generic functions is not contained in A Q . In particular, A Q and A Q,W 2 do not coincide. We conjecture that

Sign-coherence of generic g-vectors
The following result implies Theorem 1.4. The special case, where Λ = P(Q, W ) is a Jacobian algebra with non-degenerate potential W and U is an E-rigid component cluster, is proved in [P2,Theorem 3.7(1)].
Theorem 8.1. Let Λ be a basic algebra, and let U be a component cluster of Λ. Then the set {g Λ (Z) | Z ∈ U } is sign-coherent.
Proof. Assume that {g Λ (Z) | Z ∈ U } is not sign-coherent. Thus there are Z 1 , Z 2 ∈ U such that the set {g Λ (Z 1 ), g Λ (Z 2 )} is not sign-coherent. Since U is a component cluster, we know from Theorem 5.11 that Z := Z 1 ⊕ Z 2 is a strongly reduced component. By Lemma 5.12 we have g Λ (Z) = g Λ (Z 1 ) + g Λ (Z 2 ). By Lemma 5.16(ii) there is some p such that Z, Z 1 , Z 2 ∈ decIrr s.r.
Next, assume that v 1 and v 2 are both non-zero. The components Z 1 and Z 2 are indecomposable. It follows that Z 1 and Z 2 are just the orbits of some negative simple representations. But then {g Λ (Z 1 ), g Λ (Z 2 )} has to be sign-coherent, a contradiction.
Finally, let v 1 = 0 and v 2 = 0. Thus we get Z 2 = O(S − i ) for some 1 ≤ i ≤ n. This implies g Λ (Z 2 ) = e i . Since {g Λ (Z 1 ), g Λ (Z 2 )} is not sign-coherent, the ith entry of g Λ (Z 1 ) has to be negative. It follows that the socle of each representation in Z 1 has S i as a composition factor. In particular, the ith entry d i of d 1 is non-zero. But we also have E Λ (Z 1 , Z 2 ) = 0. Now Lemma 5.4 implies that d i = 0, a contradiction. 9. Examples 9.1. Strongly reduced components for hereditary algebras.
9.1.1. Assume that Λ = C Q with Q an acyclic quiver. Thus Λ is equal to the ordinary path algebra CQ. Clearly, for each (d, v) ∈ N n × N n the variety decrep d,v (Λ) is an affine space. In particular, it has just one irreducible component, namely Z d,v := decrep d,v (Λ).
The following result is a direct consequence of Lemma 9.1 and Schofield's [Scho] ground breaking work on general representations of acyclic quivers. For all unexplained terminology we refer to [Scho].
Proposition 9.2. Let Λ = C Q with Q an acyclic quiver. Then the indecomposable strongly reduced components are the components Z d,0 , where d is a Schur root, and the components Z 0,e 1 , . . . , Z 0,en , where e i is the ith standard basis vector of Z n .
For a finite-dimensional path algebra Λ = CQ one can use Schofield's algorithm [Scho] (see also [DW] for a more efficient version of the algorithm) to determine the canonical decomposition of a dimension vector, and one can also use it to decide if ext 1 Λ (Z 1 , Z 2 ) is zero or not. So at least in principle, the graph Γ(decIrr s.r. (Λ)) can be computed. However, even in this case there are numerous interesting open questions on the structure of the graph Γ(decIrr s.r. (Λ)), see [Sche]. 9.2. Strongly reduced components for 1-vertex algebras. Proposition 9.3. Let Λ = C Q /I be a basic algebra with n = 1. Then the following hold: (i) If Λ is finite-dimensional, then the indecomposable strongly reduced components in decIrr(Λ) are O(S − 1 ) and the closure of O((I 1 , 0)), where I 1 is the injective envelope of the simple Λ-module S 1 .
Proof. Assume that Λ is finite-dimensional. Then Theorem 5.17 implies that Im(G s.r. Λ ) = Z. For m ≥ 0, we know that the orbit closures of (S − 1 ) m and (I 1 , 0) m are E-rigid strongly reduced components with generic g-vectors me 1 and −me 1 , respectively. This implies (i). Part (ii) follows from the proof of Theorem 5.17(ii). 9.3. Strongly reduced components for some representation-finite algebras. By Corollary 6.5 each vertex of the component graph of a representation-finite basic algebra has a loop. In the following examples, for each E-rigid indecomposable strongly reduced component, we just display the indecomposable decorated representation whose orbit closure is the component. We describe For p = 2, the component graph Γ(decIrr s.r. (Λ p )) looks as follows: To repair the somewhat non-symmetric graph Γ(decIrr s.r. (Λ)) one could insert a vertex for the infinite-dimensional indecomposable injective Λ-module I 2 . Such aspects will be dealt with elsewhere.
9.3.2. Let Q be the quiver and let Λ := C Q /I, where I is generated by ba. Then Γ(decIrr s.r. (Λ)) looks as follows: For p ≥ 2 the indecomposable representations of the p-truncation Λ p are M 1 , . . . , M p , and we get otherwise.
The following statement says that in our example, there is a positive answer to Problem 6.9. Since Λ is representation-finite, each strongly reduced component contains an E-rigid decorated representation. Each vertex of Γ(decIrr s.r. (Λ)) has a loop. Let Γ(decIrr s.r. (Λ)) • be the graph obtained by deleting these loops. We display Γ(decIrr s.r. (Λ)) • in the following picture. Note that each component cluster is E-rigid and contains exactly three irreducible components. r r r r ▲ ▲ ▲ ▲ ▲ ▲ ▲

· · ·
Thus there is exactly one component cluster {Z} of cardinality one, and there are infinitely many component clusters of cardinality two. One can easily check that E Λ (Z, Z) = 0, hence the loop at Z, but E Λ (Z) = 0. Thus {Z} is not E-rigid. The other component clusters are E-rigid. The CC-cluster monomials are where a, b, i ≥ 0.
The set B Λ of generic Caldero-Chapoton functions is just the set of CC-cluster monomials. Recall from [BFZ] that for any acyclic quiver Q we have A Q = A up Q . In this case, B Λ is a C-basis of A Q , see [GLS].
The set of Schur roots of Q consists of real and imaginary Schur roots. The above picture shows the real Schur roots (and the vectors −e 1 and −e 2 ). The set R + im of imaginary Schur roots consists of all dimension vectors d = (d 1 , d 2 ) ∈ N 2 with d 2 = 0 such that (3 − √ 5)/2 ≤ d 1 /d 2 ≤ (3 + √ 5)/2, see [DW,Section 3] and [K,Section 6]. There is no edge between Z d,0 and any other vertex of Γ(decIrr s.r. (Λ)). In particular, there is no loop at Z d,0 .
The CC-cluster monomials are where a, b, i ≥ 0 and d ∈ R + im . Again it follows from [GLS] that these CC-cluster monomials form a C-basis of A Q . It remains a challenge to compute the exchange relations between all neighbouring CC-clusters. For the E-rigid CC-clusters, the exchange relations are known from the Fomin-Zelevinsky exchange relations arising from mutations of clusters. But for d, d 1 , d 2 ∈ R + im and i ≥ −1 it remains an open problem to express the products C Λ (d)C Λ (p i ), C Λ (d)C Λ (q i ) and C Λ (d 1 )C Λ (d 2 ) as linear combinations of elements from the basis B Λ .