Deformed Cartan matrices and generalized preprojective algebras II: general type

We propose a definition of deformed symmetrizable generalized Cartan matrices with several deformation parameters, which admit a categorical interpretation by graded modules over the generalized preprojective algebras in the sense of Geiß–Leclerc–Schröer. Using the categorical interpretation, we deduce a combinatorial formula for the inverses of our deformed Cartan matrices in terms of braid group actions. Under a certain condition, which is satisfied in all the symmetric cases or in all the finite and affine cases, our definition coincides with that of the mass-deformed Cartan matrices introduced by Kimura–Pestun in their study of quiver W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {W}$$\end{document}-algebras.


Introduction
In their study of the deformed W-algebras, Frenkel-Reshetikhin [FR98] introduced a certain 2-parameter deformation C(q, t) of the Cartan matrix of finite type.In the previous work [FM23], the present authors gave a categorical interpretation of this deformed Cartan matrix C(q, t) in terms of bigraded modules over the generalized preprojective algebras in the sense of Geiß-Leclerc-Schröer [GLS17].More precisely, we have shown that the entries of the matrix C(q, t) and its inverse C(q, t) can be expressed by the Euler-Poincaré pairings of certain bigraded modules.
The definition of the generalized preprojective algebra is given in a generality of arbitrary symmetrizable Kac-Moody type by [GLS17], and it admits a Weyl group symmetry [GLS17, AHI + 22] and a geometric realization of crystal bases [GLS18].As a sequel of [FM23], the main purpose of the present paper is to propose a categorification of a several parameter deformation of arbitrary symmetrizable generalized Cartan matrix (GCM for short) by considering multi-graded modules over the generalized preprojective algebra.In the context of theoretical physics, Kimura-Pestun [KP18b,KP18a] introduced the mass-deformed Cartan matrix, a deformation of GCM with several deformation parameters, in their study of (fractional) quiver W-algebras, which is a generalization of Frenkel-Reshetikhin's deformed W-algebras.Our deformation essentially coincides with Kimura-Pestun's mass-deformed Cartan matrix under a certain condition which is satisfied in all the symmetric cases or in all the finite and affine cases (see §3.1).
To explain our results more precisely, let us prepare some kinds of terminology.Let C = (c ij ) i,j∈I be a GCM with a symmetrizer D = diag(d i | i ∈ I).We put g ij := gcd(|c ij |, |c ji |) and f ij := |c ij |/g ij for i, j ∈ I with c ij < 0. Associated with these data, we have the generalized preprojective algebra Π defined over an arbitrary field (see [GLS17] for the precise definition or §2.3 for our convention).We introduce the (multiplicative) abelian group Γ generated by the elements {q, t} ∪ {µ (g) ji = 1 for all i, j ∈ I with c ij < 0 and 1 ≤ g ≤ g ij .These elements play the role of deformation parameters.Here, we introduced the parameters µ (g) ij in addition to q and t inspired by [KP18b,KP18a], where the counterparts are called mass-parameters.We endow a certain Γ-grading on the algebra Π as in (2.1) below.We can show that this grading on Π is universal under a reasonable condition, see §3.2.With the terminology, we give the following definition of (q, t, µ)-deformation C(q, t, µ) of GCM C, and propose a categorical framework which organizes some relevant combinatorics in terms of the Γ-graded Π-modules: Definition & Claim.We define the Z[Γ]-valued I × I-matrix C(q, t, µ) by the formula where [k] q = (q k − q −k )/(q − q −1 ) is the standard q-integer.We establish the following statements: (1) Each entry of C(q, t, µ) and its inverse C(q, t, µ) can be expressed as the Euler-Poincaré paring of certain Γ-graded Π-modules.( §2.5) (2) Moreover, when C is of infinite type, the formal expansion at t = 0 of each entry of C(q, t, µ) coincides with the Γ-graded dimension of a certain Π-module, and hence its coefficients are non-negative.(Corollary 2.15) (3) For general C, the formal expansion at t = 0 of C(q, t, µ) admits a combinatorial expression in terms of a braid group symmetry.( § §1.5 & 2.6) Note that if we consider the above (3) for each finite type and some specific reduced words, then it recovers the combinatorial formula obtained by [HL15] and [KO23a] after some specialization.We might see our generalization as a kind of aspects of the Weyl/braid group symmetry of Π about general reduced expressions (e.g.[FG19,Mur22b]).When C is of finite type, these results are essentially same as the results in our previous work [FM23].
When C is of infinite type, the algebra Π is no longer finite-dimensional.In this case, we find it suitable to work with the category of Γ-graded modules which are bounded from below with respect to the t-grading, and its completed Grothendieck group.Then, the discussion is almost parallel to the case of finite type.Indeed, we give a uniform treatment which deals with the cases of finite type and of infinite type at the same time.
In the case of finite type, the above combinatorial aspects of the deformed Cartan matrices play an important role in the representation theory of quantum loop algebras, see our previous work [FM23] and references therein.We may expect that our results here on the deformed GCM are also useful in the study of quiver W-algebras and the representation theory of quantum affinizations of Kac-Moody algebras in the future.
This paper is organized as follows.In §1, after fixing our notation, we discuss combinatorial aspects (i.e., a braid group action in §1.3 and the formula for C(q, t, µ) using it in §1.5) of our deformed Cartan matrices.The proofs of several assertions require the categorical interpretation and hence are postponed to the next section.In §2, we discuss the categorical interpretation of our deformed GCM in terms of the graded modules over the generalized preprojective algebras.The final §3 consists of three remarks, which are logically independent from the other parts of the paper.In §3.1, we compare our deformed GCM with the mass-deformed Cartan matrix in the sense of Kimura-Pestun [KP18a].In §3.2, we show that our Γ-grading on Π is universal among the gradings valued at free abelian groups.In §3.3, we briefly discuss the t-deformed GCM, which is obtained from our C(q, t, µ) by evaluating all the deformation parameters except for t at 1, and its categorical interpretation by the classical representation theory of modulated graphs in the sense of Dlab-Ringel [DR80].
Conventions.Throughout this paper, we use the following conventions.
• For a statement P, we set δ(P) to be 1 or 0 according that P is true or false.We often use the abbreviation δ x,y := δ(x = y) known as Kronecker's delta.
is a commutative integral domain, we write Q(G) for its fraction field.

Deformed Cartan matrices
1.1.Notation.Let I be a finite set.Recall that a Z-valued I × I-matrix C = (c ij ) i,j∈I is called a symmetrizable generalized Cartan matrix if the following conditions are satisfied: (C1) c ii = 2, c ij ∈ Z ≤0 for all i, j ∈ I with i = j, and c ij = 0 if and only if c ji = 0, (C2) there is a diagonal matrix D = diag(d i | i ∈ I) with d i ∈ Z >0 for all i ∈ I such that the product DC is symmetric.We call the diagonal matrix D in (C2) a symmetrizer of C. It is said to be minimal when gcd(d i | i ∈ I) = 1.For i, j ∈ I, we write i ∼ j when c ij < 0. We say that a symmetrizable generalized Cartan matrix C is irreducible if, for any i, j ∈ I, there is a sequence i 1 , . . ., i l ∈ I satisfying i ∼ i 1 ∼ • • • ∼ i l ∼ j.In this case, a minimal symmetrizer of C is unique, and any symmetrizer of C is a scalar multiple of it.From now on, by a GCM, we always mean an irreducible symmetrizable generalized Cartan matrix.We say that C is of finite type if it is positive definite, and it is of infinite type otherwise.
Throughout this section, we fix a GCM C = (c ij ) i,j∈I with its symmetrizer D = diag(d i | i ∈ I).For any i, j ∈ I with i ∼ j, we set By definition, we have We note that the transpose t C = (c ji ) i,j∈I is also a GCM, whose minimal symmetrizer is rD −1 = diag(r/d i | i ∈ I).Following [GLS17], we say that a subset Ω ⊂ I × I is an acyclic orientation of C if the following conditions are satisfied: i∈I Zα i be the root lattice of the Kac-Moody algebra associated with C, where α i is the i-th simple root for each i ∈ I.We write s i for the i-th simple reflection, which is an automorphism of Q given by s i α j = α j − c ij α i for j ∈ I.The Weyl group W is defined to be the subgroup of Aut(Q) generated by all the simple reflections {s i } i∈I .The pair (W, {s i } i∈I ) forms a Coxeter system.1.2.Deformed Cartan matrices.Let Γ be the (multiplicative) abelian group defined in Introduction.As an abelian group, Γ is free of finite rank.Let µ Z denote the subgroup of Γ generated by all the elements in {µ In particular, the rank of Γ is 2 + (i,j)∈Ω g ij .Consider the group ring Z[Γ] of Γ.Given an acyclic orientation Ω of C, it is identical to the ring of Laurent polynomials in the variables q, t and µ (g) ij with (i, j) ∈ Ω.We define the deformed generalized Cartan matrix (deformed GCM for short) C(q, t, µ) to be the Z[Γ]-valued I ×I-matrix whose (i, j)-entry C ij (q, t, µ) is given by the formula (0.1) in Introduction.We often evaluate all the parameters µ (g) ij at 1 and write C(q, t) for the resulting Z[q ±1 , t ±1 ]-valued matrix.More explicitly, its (i, j)-entry is given by We refer to the matrix C(q, t) as the (q, t)-deformed GCM.Note that we have ] q whenever i = j, and hence the matrix ([d i ] q C ij (q, t)) i,j∈I is symmetric.
Remark 1.1.When the GCM C is of finite type, the matrix C(q, t) coincides with the (q, t)-deformed Cartan matrix considered in [FR98].A deformed GCM of general type is also considered in [KP18b,KP18a], called the mass deformed Cartan matrix.We discuss the difference between our definition and the definition in [KP18a] in §3.1.
Theorem 1.3.When C is of infinite type, the matrix C(q, t, µ) has non-negative coefficients, namely we have A proof will be given in the next section (see Corollary 2.15 (2) below).
Remark 1.4.If we evaluate all the deformation parameters except for q at 1 in (1.2), we get a q-deformed Cartan matrix C(q), which is different from the naive q-deformation C ′ (q), where Note that C(q) is invertible, while C ′ (q) is not invertible.See also Remark 3.4 below for a related discussion on q-deformed Cartan matrices.In the context of the representation theory of quantum affinizations, the choice of q-deformation of GCM affects the definition of the algebra.For the quantum affinization of sl 2 , the matrix C(q) was used by Nakajima [Nak11, Remark 3.13] and also adopted by Hernandez in [Her11].See [Her11, Remark 4.1].
1.3.Braid group actions.Let Q(Γ) denote the fraction field of Z[Γ].Let φ be the automorphism of the group Γ given by φ(q) = q, φ(t) = t, and φ(µ ji for all possible i, j ∈ I and g.It induces the automorphisms of Z[[Γ]] and Q(Γ), for which we again write φ.We often write a φ instead of φ(a).
Consider the Q(Γ)-vector space Q Γ given by We endow Q Γ with a non-degenerate φ-sesquilinear hermitian form (−, −) Γ by for each i, j ∈ I.Here the term "φ-sesquilinear hermitian" means that it satisfies It is thought of a deformation of simple coroots.We have (α ∨ i , α j ) Γ = q −d i tC ij (q, t, µ) for any i, j ∈ I. Let {̟ ∨ i } i∈I denote the dual basis of {α i } i∈I with respect to (−, −) Γ .We also consider the element With these conventions, we have For each i ∈ I, we define a Q(Γ)-linear automorphism T i of Q Γ by In terms of the basis {α i } i∈I , we have (1.5) Thus, the action (1.4) can be thought of a deformation of the i-th simple reflection s i .Note that our Q(Γ)-linear automorphisms T i (i ∈ I) of Q Γ recover the braid group actions that were introduced in [Cha02] and [BP98] for finite type cases after certain specializations (see [FM23, Section 1.3]).
Proposition 1.5.The operators {T i } i∈I define an action of the braid group associated to the Coxeter system (W, {s i } i∈I ), i.e., they satisfy the braid relations : A proof will be given in §2.6 below (after Lemma 2.18).Given w ∈ W , we choose a reduced expression By Proposition 1.5, T w does not depend on the choice of reduced expression.
1.4.Remark on finite type.In this subsection, we assume that C is of finite type.Since we always have g ij = 1 in this case, we write µ ij instead of µ (1) ij .For any (i, j) ∈ I, we define Note that the element µ ij ∈ Γ does not depend on the choice of such a sequence.Let [−] µ=1 : Z[Γ] → Z[q ±1 , t ±1 ] denote the map induced from the specialization µ Z → {1}.Recall C ij (q, t) = [C ij (q, t, µ)] µ=1 by definition.
Lemma 1.6.When C is of finite type, for any i, j ∈ I and a sequence (i 1 , . . ., i k ), we have By definition, we have C ij (q, t, µ) = µ ij C ij (q, t) for any i, j ∈ I. Then the assertion follows from (1.5).
Let w 0 ∈ W be the longest element.It induces an involution i → i * of I by w It is easy to see that ν is involutive and the pairing (−, −) Γ is invariant under ν.In particular, we have ν( Denote the Coxeter and dual Coxeter numbers associated with C by h and h ∨ respectively.
Proposition 1.7.Assume that C is of finite type.We have Proof.We know that the assertion holds when µ = 1 [FM23, Theorem 1.6].It lifts to the desired formula thanks to Lemma 1.6.1.5.Combinatorial inversion formulas.Let C be a GCM of general type.
Let (i k ) k∈Z >0 and (j k ) k∈Z >0 be two sequences in I.We say that (i k ) k∈Z >0 is commutationequivalent to (j k ) k∈Z >0 if there is a bijection σ : Z >0 → Z >0 such that i σ(k) = j k for all k ∈ Z >0 and we have c i k ,i l = 0 whenever k < l and σ(k) > σ(l).
Theorem 1.8.Let (i k ) k∈Z >0 be a sequence in I satisfying the following condition: (1) if C is of finite type, (i k ) k∈Z >0 is commutation-equivalent to another sequence (j k ) k∈Z >0 such that the subsequence (j 1 , . . ., j l ) is a reduced word with l being the length of the longest element w 0 ∈ W and we have Then, for any i, j ∈ I, we have Proof of Theorem 1.8 for finite type.Note that the RHS of (1.6) is unchanged if we replace the sequence (i 1 , i 2 , . ..) with another commutation-equivalent sequence thanks to Proposition 1.5.When C is of finite type, we know that the equality (1.6) holds at µ = 1 by [FM23,Proposition 3.16].Since we have C ij (q, t, µ) = µ ij C ij (q, t) for any i, j ∈ I, we can deduce (1.6) for general µ thanks to Lemma 1.6.
A proof when C is of infinite type will be given in §2.6 below (after Corollary 2.19).
In the remaining part of this section, we discuss the special case of the above inversion formula (1.6) when the sequence comes from a Coxeter element and deduce a recursive algorithm to compute C(q, t, µ).Fix an acyclic orientation Ω of C. We say that a total ordering Taking a compatible total ordering, we define the Coxeter element τ Ω := s i 1 • • • s in .The assignment Ω → τ Ω gives a well-defined bijection between the set of acyclic orientations of C and the set of Coxeter elements of W .In what follows, we abbreviate T Ω := T τ Ω .Letting I = {i 1 , . . ., i n } be a total ordering compatible with Ω, for each i ∈ I, we set Note that the resulting element β Ω i is independent of the choice of the compatible ordering.Proposition 1.9.Let Ω be an acyclic orientation of C. For any i, j ∈ I, we have Proof.Choose a total ordering I = {i 1 , . . ., i n } compatible with Ω.Then we have When C is of infinite type, this sequence satisfies the condition in Theorem 1.8 by [Spe09], and hence we obtain (1.8).When C is of finite type, we know that the subsequence (i 1 , . . ., i 2l ) = (i 1 , . . ., i n ) h is commutation-equivalent to a sequence (j 1 , . . ., j 2l ) such that (j 1 , . . ., j l ) is a reduced word (adapted to Ω) for the longest element w 0 and j k+l = j * k for all 1 ≤ k ≤ l.Indeed, when C is of simply-laced type, it follows from [Béd99].When C is of non-simply-laced type, we simply have τ h/2 Ω = w 0 and (i 1 , . . ., i n ) h/2 is a reduced word for w 0 .Therefore Theorem 1.8 again yields (1.8).
Lemma 1.10.For each i ∈ I and k ∈ N, we have (1.9) Proof.For any i, j ∈ I, we have by definition.Using this identity, we obtain Applying T k Ω (1 − T Ω ) yields (1.9).Once we fix a total ordering I = {i 1 , . . ., i n } compatible with Ω, the equalities (1.7) and (1.9) compute the elements T k Ω β Ω i for all (k, i) ∈ Z ≥0 × I recursively along the lexicographic total ordering of Z ≥0 ×I.Thus, together with (1.8), we have obtained a recursive algorithm to compute C ij (q, t, µ).
We say that a GCM C is bipartite if there is a function ǫ : I → Z/2Z such that ǫ(i) = ǫ(j) implies i ∼ j.When C is bipartite, we can simplify the above recursive formula by separating the parameter t as explained below.
For each i ∈ I, we consider a Q(Γ)-linear automorphism Ti of Q Γ obtained from T i by evaluating the parameter t at 1.More precisely, it is given by Ti for all j ∈ I.The operators { Ti } i∈I define another action of the braid group, under which the A height function ξ gives an acyclic orientation Ω ξ of C such that we have (i, j) ∈ Ω ξ if i ∼ j and ξ(j) = ξ(i) + 1.When i ∈ I is a sink of Ω ξ , in other words, when ξ(i) < ξ(j) holds for all j ∈ I with j ∼ i, we define another height function s i ξ by Remark 1.12.There exists a height function for C if and only if C is bipartite.
Given a function ξ : I → Z, we define a linear automorphism t ξ of Q Γ by t ξ α i := t ξ(i) α i for each i ∈ I.When ξ : I → Z is a height function and i ∈ I a sink of Ω ξ , a straightforward computation yields t ξ T i = Ti t s i ξ , from which we deduce (1.10) Definition 1.13.Let ξ : I → Z be a height function.Define a map Φ ξ : The next proposition is a consequence of Proposition 1.9 and (1.10).
Proposition 1.14.Let ξ : I → Z be a height function.For any i, j ∈ I, we have Now, Lemma 1.10 specializes to the following.
Lemma 1.15.Let ξ : I → Z be a height function.For any (i, u) ∈ I × Z with u > ξ(i), we have (1.12) In particular, (1.12) enables us to compute recursively all the Φ ξ (i, u) starting from where I = {i 1 , . . ., i n } is a total ordering compatible with Ω ξ and i k = i.
Thus, Proposition 1.14 combined with Lemma 1.15 gives a simpler recursive algorithm to compute C ij (q, t, µ) when C is bipartite.
For the other case i > j, we can use the relation When C is of type ABCD, an explicit formula of C(q, t) is given in [FR98, Appendix C].When C is of type ADE, we have C(q, t) = C(qt −1 , 1) and an explicit formula of C(q) = C(q, 1) is given in [GTL17, Appendix A] (see also [KO23a, § §4.4.1,4.4.2]).

Generalized preprojective algebras
Throughout this section, we fix an arbitrary field k.Unless specified otherwise, vector spaces and algebras are defined over k, and modules are left modules.
2.1.Conventions.Let Q be a finite quiver.We understand it as a quadruple Q = (Q 0 , Q 1 , s, t), where Q 0 is the set of vertices, Q 1 is the set of arrows and s (resp.t) is the map Q 1 → Q 0 which assigns each arrow with its source (resp.target).For a quiver Q, we set kQ 0 := i∈Q 0 ke i and kQ 1 := α∈Q 1 kα.We endow kQ 0 with a k-algebra structure by e i • e j = δ ij e i for any i, j ∈ Q 0 , and kQ 1 with a (kQ 0 , kQ 0 )-bimodule structure by e i • α = δ i,t(α) α and α • e i = δ i,s(α) α for any i ∈ Q 0 and α ∈ Q 1 .Then the path algebra of Q is defined to be the tensor algebra kQ := T kQ 0 (kQ 1 ).
Let G be a multiplicative abelian group with unit 1.By a G-graded quiver, we mean a quiver Q equipped with a map deg : Q 1 → G.We regard its path algebra kQ as a G-graded algebra in the natural way.
We say that a G-graded vector space V = g∈G V g is locally finite if V g is of finite dimension for all g ∈ G.In this case, we define its graded dimension dim G V to be the formal sum For a G-graded vector space V and an element x ∈ G, we define the grading shift xV = g∈G (xV ) g by (xV ) g = V x −1 g .More generally, for a = g∈G a g g ∈ Z ≥0 [[G]], we set V ⊕a := g∈G (gV ) ⊕ag .When V ⊕a happens to be locally finite, we have dim 2.2.Preliminary on positively graded algebras.Let t Z denote a free abelian group generated by a non-trivial element t.In what follows, we consider the case when G is a direct product G = G 0 × t Z , where G 0 is another abelian group.Our principal example is the group Γ = t Z × Γ 0 in §1.2.For G-graded vector space V = g∈G V g and n ∈ Z, we define the G 0 -graded subspace V n ⊂ V by V n := g∈G 0 V t n g .By definition, we have V = n∈Z V n .We use the notation V ≥n := m≥n V m and V >n := m>n V m .
We consider a G-graded algebra Λ satisfying the following condition: (A) Λ = Λ ≥0 and dim k Λ n < ∞ for each n ∈ Z ≥0 .In particular, Λ 0 is a G 0 -graded finite dimensional algebra.Let {S j } j∈J be a complete collection of G 0 -graded simple modules of Λ 0 up to isomorphism and grading shift.It also gives a complete collection of G-graded simple modules of Λ.For a G-graded Λ-module M, the subspace M ≥n ⊂ M is a Λ-submodule for each n ∈ Z.Let Λ-mod ≥n G denote the category of G-graded Λ-modules M satisfying M = M ≥n and dim k M m < ∞ for all m ≥ n, whose morphisms are G-homogeneous Λ-homomorphisms.This is a k-linear abelian category.Let Λ-mod + G := n∈Z Λ-mod ≥n G .Note that Λ-mod + G contains all the finitely generated G-graded Λ-modules, because it contains their projective covers by the condition (A).
Lemma 2.1.Given n ∈ Z and M ∈ Λ-mod ≥n G , there is a surjection P ։ M from a projective Λ-module P belonging to Λ-mod ≥n G .Proof.For each m ≥ n, let P m ։ M m be a projective cover of M m regarded as a G 0 -graded Λ 0 -module.Then consider the G-graded projective Λ-module P := Λ ⊗ Λ 0 m≥n t m P m , which carries a natural surjection P ։ M.This P belongs to Λ-mod ≥n For an abelian category C, we denote by K(C) its Grothendieck group.We regard K(Λ-mod ≥n G ) as a subgroup of K(Λ-mod + G ) via the inclusion for any n ∈ Z.Then, the collection of subgroups {K(Λ-mod ≥n G )} n∈Z gives a filtration of K(Λ-mod + G ).We define the completed Grothendieck group K(Λ-mod + G ) to be the projective limit is free with a basis {[S j ]} j∈J .Proof.For any n ∈ Z and M ∈ Λ-mod ≥n G , we have a unique expression denotes the G 0 -graded Jordan-Hölder multiplicity of S j in the finite length G 0 -graded Λ 0 -module M m .This proves the assertion.
2.3.Generalized preprojective algebras.We fix a GCM C = (c ij ) i,j∈I and its symmetrizer D = diag(d i | i ∈ I) as in §1.1.Recall the free abelian group Γ in §1.2.We consider the quiver Q = ( Q 0 , Q 1 , s, t) given as follows: We equip the quiver Q with a Γ-grading by deg(α Let Ω be an acyclic orientation of C. We define the associated potential where sgn Ω (i, j) := (−1) δ((j,i)∈Ω) .Note that W Ω is homogeneous of degree t 2 .We define the Γ-graded k-algebra Π to be the quotient of k Q by the ideal generated by all the cyclic derivations of W Ω .In other words, the algebra Π is the quotient of k Q by the following two kinds of relations: (R1) ε Remark 2.3.Although the definition of the algebra Π depends on the choice of acyclic orientation Ω, it is irrelevant.In fact, a different choice of Ω gives rise to an isomorphic Γ-graded algebra.Moreover, one may define Π with more general orientation (i.e., without acyclic condition, as in §3.1 below).Even if we do so, the resulting Γ-graded algebra is isomorphic to our Π.
For a positive integer ℓ ∈ Z >0 , we define the Γ-graded algebra Π(ℓ) to be the quotient where . Note that ε is homogeneous and central in Π.In other words, it is the quotient of k Q by the three kinds of relations: (R1), (R2), and Remark 2.4.The algebra Π(ℓ) is identical to the generalized preprojective algebra Π( t C, ℓrD −1 , Ω) in the sense of [GLS17].
Proof.The fact Π(ℓ) ≥0 = Π(ℓ) is clear from the definition (2.1).For any n ∈ Z ≥0 , thanks to the relation (R3), the vector space Π(ℓ) n is spanned by a finite number of vectors in In what follows, we fix ℓ ∈ Z >0 and write Π for Π(ℓ) for the sake of brevity.By the definition, we have where ).
In particular, for each M ∈ Π-mod + Γ and n ∈ Z, the subspace e i M n is a finite-dimensional H i -module for each i ∈ I.We say that M is locally free if e i M n is a free H i -module for any n ∈ Z and i ∈ I, or equivalently M n is a projective Π 0 -module for any n ∈ Z.In this case, we set rank Theorem 2.6 ([GLS17, §11]).As a (left) Π-module, Π is locally free in itself.
For each i ∈ I, let P i := Πe i be the indecomposable projective Π-module associated to the vertex i and S i its simple quotient.Consider the two-sided ideal J i := Π(1 − e i )Π.We have Π/J i ∼ = H i as Γ-graded algebras.We write E i for Π/J i when we regard it as a Γ-graded left Π-module.This is a locally free Π-module characterized by rank j E i = δ i,j .In K(Π-mod + Γ ), we have (

2.3)
There is the anti-involution φ : Π → Π op given by the assignment φ(e i ) := e i , φ(α Recall the automorphism of the group Γ also denoted by φ in §1.3.By definition, if x ∈ Π is homogeneous of degree γ ∈ Γ, then φ(x) is homogeneous of degree φ(γ).For a left Π-module M, let M φ be the right Π-module obtained by twisting the original left Π-module structure by the opposition φ.If M is Γ-graded, M φ is again Γ-graded by setting (M φ ) γ := M φ(γ) .In particular, for M ∈ Π-mod + Γ , we have dim Γ (M φ ) = (dim Γ M) φ .2.4.Projective resolutions.Following [GLS17, §5.1], for each i, j ∈ I with i ∼ j, we define the bigraded (H i , H j )-bimodule i H j by i H j := It is free as a left H i -module and free as a right H j -module.Moreover, the relation (R1) gives the following: In particular, we get the following lemma.
Lemma 2.7.For i, j ∈ I with i ∼ j, we have two isomorphisms as Γ-graded left H i -modules and as Γ-graded right H j -modules respectively.
Consider the following sequence of Γ-graded (Π, Π)-bimodules: where ⊗ i := ⊗ H i and the morphisms ψ and ϕ are given by The other arrows i∈I Πe i ⊗ i e i Π → Π → 0 are canonical.The relation (R2) ensures that the sequence (2.4) forms a complex.For each i ∈ I, applying (−) ⊗ Π E i to (2.4) yields the following complex of Γ-graded (left) Π-modules: Here we used Lemma 2.7.
(1) When C is of infinite type, we have Ker ψ = 0 and Ker ψ (i) = 0 for all i ∈ I.
(2) When C is of finite type, we have Ker 2.5.Euler-Poincaré pairing.For a Γ-graded right Π-module M and a Γ-graded left Πmodule N, the vector space ]t m+n under the assumption.This proves the assertion for k = 0.The other case when k > 0 follows from this case and Lemma 2.1.
We consider the following finiteness condition for a pair (M, N) of objects in Π-mod + Γ : (B) For each γ ∈ Γ, the space tor Π k (M φ , N) γ vanishes for k ≫ 0. If (M, N) satisfies the condition (B), their Euler-Poincaré (EP) pairing and we have M ⊕a , N ⊕b Γ = a φ b M, N Γ .
Proposition 2.11.For any i, j ∈ I, the pair (S i , S j ) satisfy the condition (B) and we have (2.7) Here we understand (1 Proof.The former formula (2.6) directly follows from Theorem 2.8.The latter (2.7) follows from Theorem 2.8 and the fact that S i has an E i -resolution of the form: See the proof of [FM23, Proposition 3.11] for some more details.
Corollary 2.12.For any M, N ∈ Π-mod + Γ , the pair (M, N) satisfies the condition (B).Moreover, the EP pairing induces a φ-sesquilinear hermitian form on the Γ , we shall show that (M, N) satisfies the condition (B).Without loss of generality, we may assume that M, N ∈ Π-mod ≥0 Γ .For γ ∈ Γ fixed, take n ∈ Z such that γ ∈ Γ 0 t Z>n .By Lemma 2.9, we have tor Π k (M φ >n , N) γ = 0 and therefore tor . By Lemma 2.10 and Proposition 2.11, we know that the condition (B) is satisfied for any finite-dimensional modules.Therefore, for k large enough, we have tor Π k (M φ /M φ >n , N/N >n ) γ = 0. Thus, the pair (M, N) satisfies the condition (B).Now, by Lemma 2.10 (3) and Proposition 2.11, the EP pairing induces a pairing on the Grothendieck group K(Π-mod + G ) valued at Z[Γ 0 ][(1 − q 2rℓ ) −1 ]((t)).Lemma 2.9 tells us that this is continuous with respect to the topology given by the filtration {K(Π-mod ≥n G )} n∈Z .Therefore, it descends to a pairing on the completion K(Π-mod + Γ ) satisfying the desired properties.Let F be an algebraic closure of the field Q(Γ 0 )((t)).We understand that Q(Γ) is a subfield of F by considering the Laurent expansions at t = 0.By Corollary 2.12 above, the EP pairing linearly extends to a φ-sesquilinear hermitian form, again written by

and (2.3), and that, if
It is useful to introduce the module Pi := P i /P i ε i = (Π/Πε i )e i for each i ∈ I.We can easily prove the following (see [FM23, Lemma 2.5]).
Lemma 2.13.If M ∈ Π-mod + Γ is locally free, we have Pi , M Γ = rank i M. In particular, we have Pi , E j Γ = δ i,j and rank i P j = (dim Γ e j Pi ) φ for any i, j ∈ I.

Since the basis {[P
by Lemma 2.13), the property (3) follows from the property (2).
(1) When C is of finite type, we have (2) When C is of infinite type, we have Proof.It follows from Theorem 2.14 and the inversion of (1.3).
Remark 2.16.Since P i , Pj Γ = dim Γ (e i Pj ), Corollary 2.15 interprets the matrix C(q, t, µ) in terms of the EP pairing between the bases {[P i ]} i∈I and {[ Pi ]} i∈I .In this sense, Corollary 2.15 is dual to (2.6) in Proposition 2.11.
Remark 2.17.In the previous paper [FM23], we dealt with GCMs of finite type and finite dimensional (q, t)-graded Π-modules.Therein, we used the modules Īi := D( P φ i ) and the graded extension groups ext k Π , where D is the graded k-dual functor, instead of the modules Pi and the graded torsion groups tor Π k .Note that the two discussions are mutually equivalent thanks to the usual adjunction (cf.[ASS06, §A.4 Proposition 4.11]), i.e., we have In this sense, our discussion here is a slight generalization of that in [FM23] with the additional µ-grading.
2.6.Braid group action.Recall the two-sided ideal J i = Π(1 − e i )Π.For any M ∈ Π-mod + Γ and k ∈ Z ≥0 , we see that tor Π k (J i , M) also belongs to Π-mod + Γ .When C is of infinite type, J φ i = J i has projective dimension at most 1.In particular, the derived tensor product ) and the canonical map K(Π-mod + Γ ) → K(Π-mod + Γ ), it gives the element , where the second equality follows since Sending this equality by the isomorphism χ ℓ in Theorem 2.14, we get (2.9) In particular, we obtain the following analogue of [AIRT12, Proposition 2.10].
Lemma 2.18.When C is of infinite type, we have for any M ∈ Π-mod + Γ and i ∈ I. Proof.When C is of infinite type, we have Ψ = id by definition.Thus, the equation (2.9) coincides with the defining equation (1.5) of T i in this case.
Proof of Proposition 1.5.When C is of finite type, we can reduce the proof to the case of affine type since the collection {T i } i∈I can be extended to the collection {T i } i∈I∪{0} of the corresponding untwisted affine type.Hence, it suffices to consider the case when C is of infinite type.In this case, the braid relations for {T i } i∈I follow from Lemma 2.18 and the fact that the ideals {J i } i∈I satisfy the braid relations with respect to multiplication, which is due to [FG19, Theorem 4.7].For example, when c ij c ji = 1, we have Moreover, if we assume that M is locally free, so is Proof.The first assertion is a direct consequence of Lemma 2.18.For C of infinite type, since projective dimension of J i is at most 1 by Theorem 2.8, our involution φ yields When C is of finite type, our assertion follows easily from the exact embedding to the corresponding untwisted affine type Π. Namely, we have an isomorphism Proof of Theorem 1.8 when C is of infinite type.Assume that C is of infinite type.Let (i k ) k∈Z >0 be a sequence in I satisfying the condition (2) in Theorem 1.8.We have a filtration This filtration {F k } k≥0 is exhaustive, i.e., k≥0 F k = 0. Indeed, since the algebra Π satisfies the condition (A), its radical filtration {R k } k≥0 as a right Π-module is exhaustive.Note that, for any right Π-module M and i ∈ I, the right module M/MJ i is the largest quotient of M such that (M/MJ i )e j = 0 for j = i.Thanks to this fact and our assumption on the sequence (i k ) k∈Z >0 , we can find for each k > 0 a large integer K such that The filtration {F k } k≥0 induces an exhaustive filtration {F k e i } k≥0 of the projective module P i such that for each k ≥ 1.Therefore, in K(Π-mod + Γ ), we have Applying the isomorphism χ ℓ in Theorem 2.14 to this equality, we obtain

This is rewritten as
3), we obtain the desired equality (1.6) from this.Remark 2.20.In [IR08], they proved that the ideal semigroup J i | i ∈ I gives the set of isoclasses of classical tilting Π-modules for any symmetric affine type C with D = id.In our situation, our two-sided ideals are Γ-graded tilting objects whose Γ-graded endomorphism algebras are isomorphic to Π when C is of infinite type by arguments in [BIRS11,FG19].In particular, our braid group symmetry on K(Π-mod + Γ ) is induced from auto-equivalences on the derived category (cf.[MP19, §2]).

Remarks
3.1.Comparison with Kimura-Pestun's deformation.In their study of (fractional) quiver W-algebras, Kimura-Pestun [KP18a] introduced a deformation of GCM called the mass-deformed Cartan matrix.In this subsection, we compare their mass-deformed Cartan matrix with our deformed GCM C(q, t, µ).
Let Q be a quiver without loops and d : Q 0 → Z >0 be a function.Following [KP18a, §2.1], we call such a pair (Q, d) a fractional quiver.We set d i := d(i) and d ij := gcd(d i , d j ) for i, j ∈ Q 0 .Let C = (c ij ) i,j∈I be a GCM.We say that a fractional quiver (Q, d) is of type C if Q 0 = I and the following condition is satisfied: {s(e), t(e)} = {i, j}}| for any i, j ∈ I. (3.1) In this case, D = diag(d i | i ∈ I) is a symmetrizer of C and we have See §1.1 for the definitions.For a given fractional quiver (Q, d) of type C, Kimura-Pestun introduced a matrix C KP = (C KP ij ) i,j∈I , whose (i, j)-entry C KP ij is a Laurent polynomial in the formal parameters q 1 , q 2 and µ e for each e ∈ Q 1 given by The parameters µ e are called mass-parameters.If we evaluate all the parameters to 1, the matrix C KP coincides with the GCM C by (3.1).
3.2.Universality of the grading.In this subsection, we briefly explain how one can think that our grading (2.1) on the algebra Π is universal.It is stated as follows.
We keep the notation in §2.3.Let G be the (multiplicative) abelian group generated by the finite number of formal symbols Proposition 3.5.The degree map (2.1) gives an isomorphism deg : Proof.Choose integers {a i } i∈I satisfying i∈I a i d i = d.Let e and w be the elements of G f given by e := i∈I [ε i ] a i and w = [α respectively.Note that w does not depend on the choice of i, j ∈ I with i ∼ j and 1 ≤ g ≤ g ij by (3.6).We define a group homomorphism ι : Γ ′ → G f by ι(q 2d ) := e, ι(t 2 ) := w and ι(q for any i, j ∈ I with i ∼ j.Since C is assumed to be irreducible, it follows that [ε i ] r/d i = [ε j ] r/d j for any i, j ∈ I. Furthermore, since G f is torsion-free, we get for any i, j ∈ I. (3.7) Using (3.7), for each i ∈ I, we find The equality ι(deg[α is obvious.Thus we conclude that ι • deg = id holds.
In particular, we have the isomorphism of group rings Using the notation in the above proof, we consider the formal roots e 1/d and w 1/2 .Then we obtain the isomorphism . This means that our deformed GCM C(q, t, µ) can be specialized to any other deformation of C which arises from a grading of the quiver Q respecting the potential W Ω (formally adding roots of deformation parameters if necessary).
3.3.t-Cartan matrices and representations of modulated graphs.In this subsection, we discuss the t-Cartan matrix C(1, t), which is obtained from our (q, t)-deformed GCM C(q, t) by evaluating the parameter q at 1.Note that this kind of specialization is also studied by  in the case of finite type very recently.Here we give an interpretation of the t-Cartan matrix from the viewpoint of certain graded algebras arising from an F -species.
First, we briefly recall the notion of acyclic F -species over a base field F [Gab73,Rin76].Let I = {1, . . ., n}.By definition, an F -species (F i , i F j ) over F consists of • a finite dimensional skew-field F i over F for each i ∈ I; • an (F i , F j )-bimodule i F j for each i, j ∈ I such that F acts centrally on i F j and dim F i F j is finite; • There does not exist any sequence i 1 , . . ., i l , i l+1 = i 1 such that i k F i k+1 = 0 for each k = 1, . . ., l.
For i F j = 0, we write . If we put c ii = 2 and c ij = 0 for i F j = 0 = j F i , the matrix C := (c ij ) i,j∈I is clearly a GCM with left symmetrizer D = diag(dim F F i | i ∈ I).We have an acyclic orientation Ω of this GCM determined by the conditions i F j = 0. Following our convention in §1.1, we write dim F F i = d i .For our F -species (F i , i F j ), we set S := i∈I F i and B := (i,j)∈Ω i F j .Note that B is an (S, S)-bimodule.We define a finite dimensional hereditary algebra T = T (C, D, Ω) to be the tensor algebra T := T S (B).Note that we use the same convention for T (C, D, Ω) as that in Geiß-Leclerc-Schröer [GLS16], unlike our dual convention for the algebra Π(ℓ).
We can also define the preprojective algebra (see [DR80] for details).For (i, j) ∈ Ω, there exists a F j -basis {x 1 , . . ., x |c ji | } of i F j and a F j -basis {y 1 , . . ., y |c ji | } of Hom F j ( i F j , F j ) such that for every x ∈ i F j we have x = which does not depend on our choice of basis {x i } and {y j }.Letting j F i := Hom F j ( i F j , F j ) for (i, j) ∈ Ω, we can also define the similar canonical element c ji ∈ j F i ⊗ F i i F j .We put B := (i,j)∈Ω ( i F j ⊕ j F i ), and define the preprojective algebra Π T = Π T (ℓ) of the algebra T as Although there is obviously no nontrivial Z-grading on S by the fact F i is a finite dimensional skew-field, we can nevertheless endow T and Π T with a t Z -grading induced from their tensor algebra descriptions.Each element of i F j has degree t.We remark that if we specifically choose a decomposition of each i F j like F ((ε))-species H in [GLS20, §4.1] and define its preprojective algebra, then we can also endow these algebras with natural µ Z -gradings and homogeneous relations by using [DR80, Lemma 1.1].But we only consider the t Z -grading here since our aim is to interpret the t-Cartan matrix.By our t Z -grading, our algebra Π T satisfies the condition (A) in §2.2 (with k = F ).
We have the following complex of t-graded modules for each simple module F i : Lemma 3.6.The complex is exact.Moreover, the followings hold.
(1) When C is of infinite type, Ker ψ (i) = 0 for all i ∈ I.In particular, each object in Proof.The statement (1) is deduced from the Auslander-Reiten theory for T (e.g.[AHI + 22, Proposition 7.8]).The statement (2) follows from [Söd21,§6].Note that C is of finite type if and only if Π T is a self-injective finite dimensional algebra and its Nakayama permutation can be similarly computed as Theorem 2.8 by an analogue of [Miz14, §3] (see Remark 3.8).
Corollary 3.7.For any i, j ∈ I, the followings hold.
(1) When C is of finite type, we have (2) When C is of infinite type, we have Here dim t Z denotes the graded dimension of t Z -graded F -vector spaces.

Proof. The equality [P Π
) and an equality dim t Z e i Π T ⊗ Π T F j = δ ij d i immediately yield our assertion by Lemma 3.6 with arguments similar to the case of the generalized preprojective algebras in §2.5.Remark 3.8.In the case of our algebra Π T , the two-sided ideal J i := Π T (1−e i )Π T and the ideal semi-group J 1 , . . ., J n also gives the Weyl group symmetry on its module category analogously to [IR08,BIRS11,Miz14] (see [AHI + 22, §7.1]).Even if we consider the algebra Π T and t Z -homogeneous ideal J i , we can also establish the similar braid group symmetry as §2.6 after the specialization q → 1 and µ → 1 by Lemma 3.6.
Remark 3.9.The algebra Π T is a Koszul algebra for non-finite types and (h − 2, h)almost Koszul algebras for finite types in the sense of [BBK02] with our t Z -gradings.Thus Corollary 3.7 might be interpreted in the context of [BBK02, §3.3].
As a by-product of this description, we have the following generalization of the formula in [HL15, Proposition 2.1] and [Fuj22, Proposition 3.8] for any bipartite symmetrizable Kac-Moody type.For a t-series f (t) = k f k t k ∈ Z[[t, t −1 ]], we write [f (t)] k := f k for k ∈ Z. Proposition 3.10.Assume that C is bipartite and take a height function ξ for C such that Ω ξ = Ω (see §1.5).Let (F i , i F j ) be a modulated graph associated with (C, D, Ω) as above.Let M ≃ τ −k T P T i and N ≃ τ −l T P T j be any two indecomposable preprojective T -modules.When C is of infinite type, we have (ξ(i)+2k)−(ξ(j)+2l)−1 . (3.9) When C is of finite type, the equality (3.9) still holds provided that 1 ≤ (ξ(i) + 2k) − (ξ(j) + 2l) − 1 ≤ h − 1. (3.10) Otherwise, we have Ext 1 T (M, N) = 0. Proof.We may deduce the assertion by a combinatorial thought using the formula (1.11) as in [HL15] or [Fuj22].But, here we shall give another proof using the algebra Π T .
Remark 3.11.In [GLS17], they also introduced the 1-Iwanaga-Gorenstein algebra H over any field k associated with a GCM C, its symmetrizer D, and an orientation Ω.The algebra H has quite similar features to our algebra T , and we can also show a version of Proposition 3.10 for the algebra H with t Z -graded structure of the corresponding generalized preprojective algebra in a similar way.These algebras H and T have the following common dimension property of extension groups due to [GLS16, Proposition 5.5]: We keep the convention in Proposition 3.10.Let X ≃ τ −k H P H i and Y ≃ τ −l H P H j be any two indecomposable preprojective H-modules.Then we have dim k Ext 1 H (X, Y ) = dim F Ext 1 T (M, N).Thanks to this common dimension property between the algebras H and T , Corollary 2.15 specializes to Corollary 3.7 after the specialization q → 1 and µ → 1 with Remark 2.4.Remark 3.12.When the authors almost finished writing this paper, a preprint [KO23b] by Kashiwara-Oh appeared in arXiv, which shows that the t-Cartan matrix of finite type is closely related to the representation theory of quiver Hecke algebra.Combining their main theorem with Proposition 3.10 above, we find a relationship between the representation theory of the modulated graphs and that of quiver Hecke algebras, explained as follows.
Let C be a Cartan matrix of finite type, and let g denote the simple Lie algebra associated with C. Let R be the quiver Hecke algebra associated with C and its minimal symmetrizer D, which categorifies the quantized enveloping algebra U q (g).We are interested in the Z ≥0 -valued invariant d(S, S ′ ) defined by using the R-matrices, which measures how far two "affreal" R-modules S and S ′ are from being mutually commutative with respect to the convolution product (or parabolic induction).Given an (acyclic) orientation Ω of C, we have an affreal R-module S Ω (α) for each positive root α of g, called a cuspidal module.See [KO23b] for details.
On the other hand, we have a generalization of the Gabriel theorem for F -species (see [DR75,DR76,Rin76]).In particular, for each positive root α of g, there exists an indecomposable module M Ω (α) over the algebra T = T (C, D, Ω) satisfying i∈I (dim F i e i M Ω (α))α i = α, uniquely up to isomorphism.Note that every indecomposable T -module is a preprojective module when C is of finite type.
denote the indecomposable projective T -module (resp.Π T -module) associated with i, and τ T the Auslander-Reiten translation for (left) T -modules.Note that this algebra Π T satisfies P Π T i = k≥0 τ −k T P T i by an argument on the preprojective component of the Auslander-Reiten quiver of T similar to [Söd21, Proposition 4.7].Note that our F -species (F i , i F j ) is nothing but a modulated graph associated with (C, D, Ω) in the sense of Dlab-Ringel [DR80], although we will work with these algebras along with a context of a deformation of C.

T
(P Π T i ) [u] ∼ = τ −k T P T i if u = ξ(i) + 2k for k ∈ Z ≥0 , 0 otherwise (3.11)as (ungraded) T -modules.Now, we have for eachM ≃ τ −k T P T i and N ≃ τ −l T P T j dim F Ext 1 T (M, N) = dim F Ext 1 T (τ −k T P T i , τ −l T P T j ) = dim F e j τ (k−l−1) T P T i (cf.[ASS06, §IV 2.13]) the quotient by the torsion subgroup.By construction, for any free abelian group G, giving a homomorphism deg :G f → G is equivalent to giving Q a structure of G-graded quiver deg : Q 1 → G such that the potential W Ω is homogeneous.In this sense, we can say that the tautological map Q 1 → G f gives a universal grading on the algebra Π.Now recall our fixed symmetrizer D = diag(d i | i ∈ I) and set d := gcd(d i | i ∈ I).Let Γ ′ ⊂ Γ be the subgroup generated by {deg(a) | a ∈ Q 1 }.Note that Γ ′ is a free abelian group with a basis {q 2d , t 2