Deformed Cartan matrices and generalized preprojective algebras II: general type

Abstract

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 $\mathcal {W}$-algebras.

1 Introduction

In their study of the deformed $\mathcal {W}$-algebras, Frenkel–Reshetikhin [12] introduced a certain 2-parameter deformation C(q, t) of the Cartan matrix of finite type. In the previous work [15], 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 [20]. More precisely, we have shown that the entries of the matrix C(q, t) and its inverse $\widetilde{C}(q,t)$ 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 Geiss et al. [20], and it admits a Weyl group symmetry [2, 20] and a geometric realization of crystal bases [21]. As a sequel of Fujita and Murakami [15], 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 [29, 30] introduced the mass-deformed Cartan matrix, a deformation of GCM with several deformation parameters, in their study of (fractional) quiver $\mathcal {W}$-algebras, which is a generalization of Frenkel-Reshetikhin’s deformed $\mathcal {W}$-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 Sect. 4.1).

To explain our results more precisely, let us prepare some kinds of terminology. Let $C = (c_{ij})_{i,j \in I}$ be a GCM with a symmetrizer $D = \mathop {\text {diag}}\nolimits (d_i \mid i \in I)$. We put $g_{ij} \,{:=}\,\gcd (\vert c_{ij}\vert , \vert c_{ji}\vert )$ and $f_{ij} \,{:=}\,\vert c_{ij}\vert /g_{ij}$ for $i,j \in I$ with $c_{ij} < 0$. Associated with these data, we have the generalized preprojective algebra $\Pi$ defined over an arbitrary field (see Geiss et al. [20] for the precise definition or Sect. 3.3 for our convention). We introduce the (multiplicative) abelian group $\Gamma$ generated by the elements

$$\begin{aligned} \{q,t\} \cup \left\{ \mu _{ij}^{(g)} \mid (i,j, g) \in I\times I \times \mathbb {Z}, c_{ij} < 0, 1 \le g \le g_{ij} \right\} \end{aligned}$$

which subject to the relations

$$\begin{aligned} \mu ^{(g)}_{ij} \mu ^{(g)}_{ji} = 1 \quad \text {for all } i, j \in I \text { with } c_{ij} < 0 \text { and } 1 \le g \le g_{ij}. \end{aligned}$$

These elements play the role of deformation parameters. Here, we introduced the parameters $\mu _{ij}^{(g)}$ in addition to q and t inspired by Kimura and Pestun [29, 30], where the counterparts are called mass-parameters. We endow a certain $\Gamma$-grading on the algebra $\Pi$ as in (3.1) below. We can show that this grading on $\Pi$ is universal under a reasonable condition, see Sect. 4.2. With the terminology, we give the following definition of $(q, t, \underline{\mu })$-deformation $C(q,t, \underline{\mu })$ of GCM C, and propose a categorical framework which organizes some relevant combinatorics in terms of the $\Gamma$-graded $\Pi$-modules:

Definition and Claim

We define the $\mathbb {Z}[\Gamma ]$-valued $I \times I$-matrix $C(q,t, \underline{\mu })$ by the formula

$$\begin{aligned} C_{ij}(q,t, \underline{\mu }) = {\left\{ \begin{array}{ll} q^{d_i}t^{-1} + q^{-d_i}t &{}\quad \text {if } i=j; \\ - [f_{ij}]_{q^{d_i}} \sum _{g = 1}^{g_{ij}}\mu _{ij}^{(g)} &{}\quad \text {if } c_{ij} < 0; \\ 0 &{}\quad \text {otherwise}, \end{array}\right. } \end{aligned}$$

(1.1)

where $[k]_q = (q^{k}-q^{-k})/(q-q^{-1})$ is the standard q-integer. We establish the following statements:

• (1) Each entry of $C(q,t, \underline{\mu })$ and its inverse $\widetilde{C}(q,t,\underline{\mu })$ can be expressed as the Euler–Poincaré paring of certain $\Gamma$-graded $\Pi$-modules (Sect. 3.5).

• (2) Moreover, when C is of infinite type, the formal expansion at $t=0$ of each entry of $\widetilde{C}(q,t,\underline{\mu })$ coincides with the $\Gamma$-graded dimension of a certain $\Pi$-module, and hence its coefficients are non-negative (Corollary 3.15).

• (3) For general C, the formal expansion at $t=0$ of $\widetilde{C}(q,t,\underline{\mu })$ admits a combinatorial expression in terms of a braid group symmetry (Sects. 2.5 and 3.6).

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 Hernandez and Leclerc [24] and Kashiwara and Oh [27] after some specialization. We might see our generalization as a kind of aspects of the Weyl/braid group symmetry of $\Pi$ about general reduced expressions (e.g. [13, 34]). When C is of finite type, these results are essentially same as the results in our previous work [15].

When C is of infinite type, the algebra $\Pi$ is no longer finite-dimensional. In this case, we find it suitable to work with the category of $\Gamma$-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 [15] and references therein. We may expect that our results here on the deformed GCM are also useful in the study of quiver $\mathcal {W}$-algebras and the representation theory of quantum affinizations of Kac–Moody algebras in the future.

This paper is organized as follows. In Sect. 2, after fixing our notation, we discuss combinatorial aspects (i.e., a braid group action in Sect. 2.3 and the formula for $\widetilde{C}(q,t,\underline{\mu })$ using it in Sect. 2.5) of our deformed Cartan matrices. The proofs of several assertions require the categorical interpretation and hence are postponed to the next section. In Sect. 3, we discuss the categorical interpretation of our deformed GCM in terms of the graded modules over the generalized preprojective algebras. The final Sect. 4 consists of three remarks, which are logically independent from the other parts of the paper. In Sect. 4.1, we compare our deformed GCM with the mass-deformed Cartan matrix in the sense of Kimura–Pestun [29]. In Sect. 4.2, we show that our $\Gamma$-grading on $\Pi$ is universal among the gradings valued at free abelian groups. In Sect. 4.3, we briefly discuss the t-deformed GCM, which is obtained from our $C(q,t,\underline{\mu })$ 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 [11].

Conventions

Throughout this paper, we use the following conventions.

• For a statement $\text {P}$, we set $\delta (\text {P})$ to be 1 or 0 according that $\text {P}$ is true or false. We often use the abbreviation $\delta _{x,y} \,{:=}\,\delta (x=y)$ known as Kronecker’s delta.

• For a group G, let $\mathbb {Z}[G]$ denote the group ring and $\mathbb {Z}[\![G]\!]$ the set of formal sums $\{\sum _{g \in G} a_g g \mid a_g \in \mathbb {Z}\}$. Note that $\mathbb {Z}[\![ G ]\!]$ is a $\mathbb {Z}[G]$-module in the natural way. If $\mathbb {Z}[G]$ is a commutative integral domain, we write $\mathbb {Q}(G)$ for its fraction field.

2 Deformed Cartan matrices

In this section, we introduce a novel definition of multiple parameter deformation of the symmetrizable generalized Cartan matrix, inspired by studies of the quiver $\mathcal {W}$-algebra [29, 30]. Additionally, relevant combinatorial materials associated with this deformed Cartan matrix (including the deformation of root lattice, the braid group action, and their combinatorial formulas) are introduced, motivated by several contexts in the representation theory of quantum affine algebra. These various objects are examined in a unified manner from the viewpoint of representation theory of a certain graded algebra in the next Sect. 3.

2.1 Notation

Let I be a finite set. Recall that a $\mathbb {Z}$-valued $I \times I$-matrix $C = (c_{ij})_{i,j \in I}$ is called a symmetrizable generalized Cartan matrix if the following conditions are satisfied:

• (C1) $c_{ii} = 2$, $c_{ij} \in \mathbb {Z}_{\le 0}$ for all $i, j \in I$ with $i\ne j$, and $c_{ij} = 0$ if and only if $c_{ji} = 0$,

• (C2) there is a diagonal matrix $D = \mathop {\text {diag}}\nolimits (d_i \mid i \in I)$ with $d_i \in \mathbb {Z}_{>0}$ for all $i \in I$ 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 $\gcd (d_i \mid i \in I) =1$. For $i, j \in I$, we write $i \sim j$ when $c_{ij} < 0$. We say that a symmetrizable generalized Cartan matrix C is irreducible if, for any $i, j \in I$, there is a sequence $i_1, \ldots , i_l \in I$ satisfying $i \sim i_1 \sim \cdots \sim i_l \sim j$. 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 $C=(c_{ij})_{i,j \in I}$ with its symmetrizer $D= \mathop {\text {diag}}\nolimits (d_i\mid i \in I)$. For any $i, j \in I$ with $i \sim j$, we set

$$\begin{aligned} g_{ij} \,{:=}\,\gcd (\vert c_{ij}\vert , \vert c_{ji}\vert ), \quad f_{ij} \,{:=}\,\vert c_{ij}\vert /g_{ij}, \quad d_{ij} \,{:=}\,\gcd (d_i, d_j). \end{aligned}$$

By definition, we have $g_{ij} = g_{ji}, d_{ij} = d_{ji}$ and $f_{ij} = d_j/d_{ij}$. Let $r \,{:=}\,\mathop {\text {lcm}}\nolimits (d_i \mid i \in I)$. We note that the transpose ${}^{\texttt{t}}{C} = (c_{ji})_{i,j \in I}$ is also a GCM, whose minimal symmetrizer is $rD^{-1} = \mathop {\text {diag}}\nolimits (r/d_i \mid i\in I)$. Following [20], we say that a subset $\Omega \subset I \times I$ is an acyclic orientation of C if the following conditions are satisfied:

• $\{(i,j), (j,i)\} \cap \Omega \ne \varnothing$ if and only if $i \sim j$,

• for any sequence $(i_1, i_2, \ldots , i_l)$ in I with $l > 1$ and $(i_k, i_{k+1}) \in \Omega$ for all $1 \le k < l$, we have $i_1 \ne i_l$.

Let $\textsf{Q}= \bigoplus _{i \in I}\mathbb {Z} \alpha _i$ be the root lattice of the Kac–Moody algebra associated with C, where $\alpha _i$ is the i-th simple root for each $i \in I$. We write $s_i$ for the i-th simple reflection, which is an automorphism of $\textsf{Q}$ given by $s_i \alpha _j = \alpha _j - c_{ij}\alpha _i$ for $j \in I$. The Weyl group W is defined to be the subgroup of $\mathop {\text {Aut}}\nolimits (\textsf{Q})$ generated by all the simple reflections $\{ s_i\}_{i \in I}$. The pair $(W,\{s_i\}_{i \in I})$ forms a Coxeter system.

2.2 Deformed Cartan matrices

Let $\Gamma$ be the (multiplicative) abelian group defined in Introduction. As an abelian group, $\Gamma$ is free of finite rank. Let $\underline{\mu }^\mathbb {Z}$ denote the subgroup of $\Gamma$ generated by all the elements in $\{\mu _{ij}^{(g)} \mid i,j \in I, i \sim j, 1 \le g \le g_{ij}\}$. Then we have $\Gamma = q^\mathbb {Z} \times t^\mathbb {Z} \times \underline{\mu }^\mathbb {Z}$, where $x^{\mathbb {Z}} \,{:=}\,\{ x^k \mid k \in \mathbb {Z}\}$. If we choose an acyclic orientation $\Omega$ of C, we have $\underline{\mu }^\mathbb {Z} = \prod _{(i,j)\in \Omega } \prod _{g=1}^{g_{ij}} (\mu _{ij}^{(g)})^\mathbb {Z}$. In particular, the rank of $\Gamma$ is $2 + \sum _{(i,j) \in \Omega } g_{ij}$. Consider the group ring $\mathbb {Z}[\Gamma ]$ of $\Gamma$. Given an acyclic orientation $\Omega$ of C, it is identical to the ring of Laurent polynomials in the variables q, t and $\mu _{ij}^{(g)}$ with $(i,j) \in \Omega$.

We define the deformed generalized Cartan matrix (deformed GCM for short) $C(q,t, \underline{\mu })$ to be the $\mathbb {Z}[\Gamma ]$-valued $I \times I$-matrix whose (i, j)-entry $C_{ij}(q,t, \underline{\mu })$ is given by the formula (1.1) in Introduction. We often evaluate all the parameters $\mu _{ij}^{(g)}$ at 1 and write C(q, t) for the resulting $\mathbb {Z}[q^{\pm 1}, t^{\pm 1}]$-valued matrix. More explicitly, its (i, j)-entry is given by

$$\begin{aligned} C_{ij}(q,t) \,{:=}\,\delta _{i,j} (q^{d_i}t^{-1} + q^{-d_i}t) - \delta (i \sim j) g_{ij}[f_{ij}]_{q^{d_i}}. \end{aligned}$$

We refer to the matrix C(q, t) as the (q, t)-deformed GCM. Note that we have $[d_i]_q C_{ij}(q,t) = g_{ij}[d_i f_{ij}]_q$ whenever $i\ne j$, and hence the matrix $([d_i]_q C_{ij}(q,t))_{i,j \in I}$ is symmetric.

Remark 2.1

When the GCM C is of finite type, the matrix C(q, t) coincides with the (q, t)-deformed Cartan matrix considered in Frenkel and Reshetikhin [12]. A deformed GCM of general type is also considered in Kimura and Pestun [29, 30], called the mass deformed Cartan matrix. We discuss the difference between our definition and the definition in Kimura and Pestun [29] in Sect. 4.1.

Let $\Gamma _0 \,{:=}\,q^\mathbb {Z} \times \underline{\mu }^\mathbb {Z} \subset \Gamma$. Since $\Gamma = t^\mathbb {Z} \times \Gamma _0$, we have $\mathbb {Z}[\Gamma ] = \mathbb {Z}[\Gamma _0][t^{\pm 1}]$. Letting $q^{\pm D} \,{:=}\,\mathop {\text {diag}}\nolimits (q^{\pm d_i} \mid i \in I)$, we can write

$$\begin{aligned} C(q,t,\underline{\mu }) = (\text {id}-tX)t^{-1}q^{D}, \end{aligned}$$

(2.1)

for some $\mathbb {Z}[\Gamma _0][t]$-valued matrix X. In particular, the matrix $C(q,t,\underline{\mu })$ is invertible as a $\mathbb {Z}[\Gamma _0](\!(t)\!)$-valued matrix and its inverse $\widetilde{C}(q,t,\underline{\mu }) = (\widetilde{C}_{ij}(q,t,\underline{\mu }))$ is given by

$$\begin{aligned} \widetilde{C}(q,t,\underline{\mu }) = q^{-D}t\left( \text {id}+ \sum _{k > 0} X^k t^{k} \right) . \end{aligned}$$

Example 2.2

Even if we begin with a non-invertible GCM C, we obtain $C(q, t, \underline{\mu })$ as an invertible matrix. For example, if we take

$$\begin{aligned}C=\begin{pmatrix} 2 &{}\quad -2 \\ -2 &{}\quad 2 \end{pmatrix}\quad \text {and}\quad D= \mathop {\text {diag}}\nolimits (1, 1),\end{aligned}$$

then we obtain

$$\begin{aligned} C(q,t,\underline{\mu })= \begin{pmatrix} qt^{-1}+q^{-1}t &{}\quad -(\mu _{12}^{(1)}+\mu _{12}^{(2)})\\ -(\mu _{21}^{(1)}+\mu _{21}^{(2)}) &{}\quad qt^{-1}+q^{-1}t \end{pmatrix}. \end{aligned}$$

(2.2)

Since $\det C(q,t,\underline{\mu }) = q^2 t^{-2}- (\mu _{12}^{(1)}\mu _{21}^{(2)}+\mu _{21}^{(1)}\mu _{12}^{(2)})+q^{-2}t^2 \in \mathbb {Z}[\Gamma _0](\!(t)\!)^{\times }$, our $C(q,t,\underline{\mu })$ is invertible.

Theorem 2.3

When C is of infinite type, the matrix $\widetilde{C}(q,t,\underline{\mu })$ has non-negative coefficients, namely we have $\widetilde{C}_{ij}(q,t,\underline{\mu }) \in \mathbb {Z}_{\ge 0}[\Gamma _0][\![t]\!]$ for any $i,j \in I$.

A proof will be given in the next section (see Corollary 3.15 (2) below).

Remark 2.4

If we evaluate all the deformation parameters except for q at 1 in (2.2), we get a q-deformed Cartan matrix C(q), which is different from the naive q-deformation $C'(q)$, where

$$\begin{aligned} C(q) = \begin{pmatrix} [2]_q&{}\quad -2 \\ -2 &{}\quad [2]_q\end{pmatrix}, \qquad C'(q) = \begin{pmatrix} [2]_q&{}\quad [-2]_q \\ [-2]_q &{}\quad [2]_q \end{pmatrix}. \end{aligned}$$

Note that C(q) is invertible, while $C'(q)$ is not invertible. See also Remark 4.4 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 $\widehat{\mathfrak {sl}}_2$, the matrix C(q) was used by Nakajima [35, Remark 3.13] and also adopted by Hernandez [23]. See [23, Remark 4.1].

2.3 Braid group actions

Let $\mathbb {Q}(\Gamma )$ denote the fraction field of $\mathbb {Z}[\Gamma ]$. Let $\phi$ be the automorphism of the group $\Gamma$ given by $\phi (q) = q$, $\phi (t) = t$, and $\phi (\mu _{ij}^{(g)}) = \mu _{ji}^{(g)}$ for all possible $i,j \in I$ and g. It induces the automorphisms of $\mathbb {Z}[\![\Gamma ]\!]$ and $\mathbb {Q}(\Gamma )$, for which we again write $\phi$. We often write $a^{\phi }$ instead of $\phi (a)$.

Consider the $\mathbb {Q}(\Gamma )$-vector space $\textsf{Q}_\Gamma$ given by

$$\begin{aligned} \textsf{Q}_\Gamma \,{:=}\,\mathbb {Q}(\Gamma )\otimes _{\mathbb {Z}}\textsf{Q}= \bigoplus _{i \in I} \mathbb {Q}(\Gamma )\alpha _i. \end{aligned}$$

We endow $\textsf{Q}_\Gamma$ with a non-degenerate $\phi$-sesquilinear hermitian form $(-,-)_\Gamma$ by

$$\begin{aligned} (\alpha _i, \alpha _j)_\Gamma \,{:=}\,[d_i]_q C_{ij}(q,t, \underline{\mu }) \end{aligned}$$

for each $i,j \in I$. Here the term “$\phi$-sesquilinear hermitian" means that it satisfies

$$\begin{aligned} (ax,by)_{\Gamma } = a^\phi b (x,y)_\Gamma , \quad (x,y)_\Gamma = (y,x)_\Gamma ^\phi \end{aligned}$$

for any $x,y \in \textsf{Q}_\Gamma$ and $a,b \in \mathbb {Q}(\Gamma )$. Let $\{\alpha _i^\vee \}_{i \in I}$ be another basis of $\textsf{Q}_\Gamma$ defined by

$$\begin{aligned} \alpha _i^\vee \,{:=}\,q^{-d_i}t[d_i]_q^{-1} \alpha _i. \end{aligned}$$

It is thought of a deformation of simple coroots. We have

$$\begin{aligned} (\alpha _i^\vee , \alpha _j)_\Gamma = q^{-d_i}tC_{ij}(q,t,\underline{\mu }) \end{aligned}$$

for any $i,j \in I$. Let $\{ \varpi _i^\vee \}_{i\in I}$ denote the dual basis of $\{\alpha _i\}_{i \in I}$ with respect to $(-,-)_\Gamma$. We also consider the element $\varpi _i \,{:=}\,[d_i]_q \varpi _i^\vee$ for each $i \in I$. With these conventions, we have

$$\begin{aligned} \alpha _i = \sum _{j \in I}C_{ji}(q,t,\underline{\mu })\varpi _j, \qquad \alpha _i^\vee = q^{-d_i}t\sum _{j \in I} C_{ij}(q,t,\underline{\mu })^\phi \varpi _j^\vee . \end{aligned}$$

(2.3)

For each $i \in I$, we define a $\mathbb {Q}(\Gamma )$-linear automorphism $T_i$ of $\textsf{Q}_\Gamma$ by

$$\begin{aligned} T_i x \,{:=}\,x - (\alpha _i^\vee , x)_\Gamma \alpha _i \end{aligned}$$

(2.4)

for $x \in \textsf{Q}_\Gamma$. In terms of the basis $\{ \alpha _i \}_{i \in I}$, we have

$$\begin{aligned} T_i \alpha _j = \alpha _j - q^{-d_i}tC_{ij}(q,t,\underline{\mu }) \alpha _i. \end{aligned}$$

(2.5)

Thus, the action (2.4) can be thought of a deformation of the i-th simple reflection $s_i$. Note that our $\mathbb {Q}(\Gamma )$-linear automorphisms $T_i\,(i\in I)$ of $\textsf{Q}_\Gamma$ recover the braid group actions that were introduced in Chari [8] and Bouwknegt and Pilch [5] for finite type cases after certain specializations (see [15, Section 1.3]).

Proposition 2.5

The operators $\{ T_i \}_{i \in I}$ define an action of the braid group associated to the Coxeter system $(W, \{s_i\}_{i \in I})$, i.e., they satisfy the braid relations:

$$\begin{aligned} T_i T_j&= T_j T_i{} & {} \text {if } c_{ij}=0, \\ T_i T_j T_i&= T_j T_i T_j{} & {} \text {if } c_{ij}c_{ji}=1, \\ (T_i T_j)^k&= (T_j T_i)^k{} & {} \text {if } c_{ij}c_{ji}=k \text { with } k \in \{2,3\}. \end{aligned}$$

A proof will be given in Sect. 3.6 below (after Lemma 3.18).

Given $w \in W$, we choose a reduced expression $w = s_{i_1}s_{i_2} \cdots s_{i_l}$ and set $T_w \,{:=}\,T_{i_1} T_{i_2} \cdots T_{i_l}$. By Proposition 2.5, $T_w$ does not depend on the choice of reduced expression.

2.4 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 $\mu _{ij}$ instead of $\mu _{ij}^{(1)}$. For any $(i,j) \in I$, we define $\mu _{ij} \,{:=}\,\mu _{i,i_1} \mu _{i_1,i_2} \cdots \mu _{i_k,j}$, where $(i_1, \ldots , i_k)$ is any finite sequence in I such that $i \sim i_1 \sim i_2 \sim \cdots \sim i_k\sim j$. Note that the element $\mu _{ij} \in \Gamma$ does not depend on the choice of such a sequence. Let $[-]_{\underline{\mu }=1} :\mathbb {Z}[\Gamma ] \rightarrow \mathbb {Z}[q^{\pm 1}, t^{\pm 1}]$ denote the map induced from the specialization $\underline{\mu }^{\mathbb {Z}} \rightarrow \{1\}$. Recall $C_{ij}(q,t) = [C_{ij}(q,t,\underline{\mu })]_{\underline{\mu }= 1}$ by definition.

Lemma 2.6

When C is of finite type, for any $i,j \in I$ and a sequence $(i_1,\ldots ,i_k)$, we have

$$\begin{aligned} (\varpi _i^\vee , T_{i_1} \cdots T_{i_k} \alpha _{j})_\Gamma = \mu _{ij} [(\varpi _i^\vee , T_{i_1} \cdots T_{i_k} \alpha _{j})_\Gamma ]_{\underline{\mu }=1}. \end{aligned}$$

Proof

By definition, we have $C_{ij}(q,t,\underline{\mu }) = \mu _{ij}C_{ij}(q,t)$ for any $i,j \in I$. Then the assertion follows from (2.5). $\square$

Let $w_0 \in W$ be the longest element. It induces an involution $i \mapsto i^*$ of I by $w_0 \alpha _i = - \alpha _{i^*}$. We consider the $\mathbb {Q}(\Gamma )$-linear automorphism $\nu$ of $\textsf{Q}_\Gamma$ given by $\nu (\alpha _i) = \mu _{i^*i} \alpha _{i^*}$ for each $i \in I$. It is easy to see that $\nu$ is involutive and the pairing $(-,-)_\Gamma$ is invariant under $\nu$. In particular, we have $\nu (\varpi _i^\vee ) = \mu _{ii^*}\varpi _{i^*}^\vee$ for each $i \in I$. Denote the Coxeter and dual Coxeter numbers associated with C by h and $h^\vee$ respectively.

Proposition 2.7

Assume that C is of finite type. We have $T_{w_0} = - q^{-rh^\vee }t^h \nu .$

Proof

We know that the assertion holds when $\underline{\mu }= 1$ [15, Theorem 1.6]. It lifts to the desired formula thanks to Lemma 2.6. $\square$

2.5 Combinatorial inversion formulas

Let C be a GCM of general type.

Let $(i_k)_{k \in \mathbb {Z}_{>0}}$ and $(j_k)_{k \in \mathbb {Z}_{>0}}$ be two sequences in I. We say that $(i_k)_{k \in \mathbb {Z}_{>0}}$ is commutation-equivalent to $(j_k)_{k \in \mathbb {Z}_{>0}}$ if there is a bijection $\sigma :\mathbb {Z}_{> 0} \rightarrow \mathbb {Z}_{>0}$ such that $i_{\sigma (k)} = j_k$ for all $k \in \mathbb {Z}_{>0}$ and we have $c_{i_k, i_l} =0$ whenever $k < l$ and $\sigma (k)> \sigma (l)$.

Theorem 2.8

Let $(i_k)_{k \in \mathbb {Z}_{>0}}$ be a sequence in I satisfying the following condition:

• (1) if C is of finite type, $(i_k)_{k \in \mathbb {Z}_{>0}}$ is commutation-equivalent to another sequence $(j_k)_{k \in \mathbb {Z}_{>0}}$ such that the subsequence $(j_1, \ldots , j_l)$ is a reduced word with l being the length of the longest element $w_0 \in W$ and we have $j_{k+l} = j_{k}^*$ for all $k \in \mathbb {Z}_{>0}$;

• (2) if C is of infinite type, the subsequence $(i_1, i_2, \ldots , i_k)$ is a reduced word for all $k \in \mathbb {Z}_{>0}$, and we have $\vert \{ k \in \mathbb {Z}_{>0} \mid i_k = i\}\vert = \infty$ for each $i \in I$.

Then, for any $i, j \in I$, we have

$$\begin{aligned} \widetilde{C}_{ij}(q,t, \underline{\mu }) = q^{-d_j}t\sum _{k \in \mathbb {Z}_{>0}; i_k = j}(\varpi _i^\vee , T_{i_1} \cdots T _{i_{k-1}}\alpha _j)_\Gamma . \end{aligned}$$

(2.6)

Proof of Theorem 2.8 for finite type

Note that the RHS of (2.6) is unchanged if we replace the sequence $(i_1,i_2,\ldots )$ with another commutation-equivalent sequence thanks to Proposition 2.5. When C is of finite type, we know that the equality (2.6) holds at $\underline{\mu }=1$ by Fujita and Murakami [15, Proposition 3.16]. Since we have $\widetilde{C}_{ij}(q,t,\underline{\mu }) = \mu _{ij}\widetilde{C}_{ij}(q,t)$ for any $i,j \in I$, we can deduce (2.6) for general $\underline{\mu }$ thanks to Lemma 2.6. $\square$

A proof when C is of infinite type will be given in Sect. 3.6 below (after Corollary 3.19).

In the remaining part of this section, we discuss the special case of the above inversion formula (2.6) when the sequence comes from a Coxeter element and deduce a recursive algorithm to compute $\widetilde{C}(q,t,\underline{\mu })$. Fix an acyclic orientation $\Omega$ of C. We say that a total ordering $I =\{i_1, \ldots , i_n\}$ is compatible with $\Omega$ if $(i_k, i_l) \in \Omega$ implies $k < l$. Taking a compatible total ordering, we define the Coxeter element $\tau _\Omega \,{:=}\,s_{i_1}\cdots s_{i_n}$. The assignment $\Omega \mapsto \tau _\Omega$ 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 $T_\Omega \,{:=}\,T_{\tau _\Omega }$. Letting $I = \{i_1,\ldots ,i_n \}$ be a total ordering compatible with $\Omega$, for each $i \in I$, we set

$$\begin{aligned} \beta ^\Omega _i \,{:=}\,(1-T_\Omega )\varpi _i = q^{-d_i}tT_{i_1} T_{i_2} \cdots T_{i_{k-1}}\alpha _{i_k} \qquad \text {if } i = i_k. \end{aligned}$$

(2.7)

Note that the resulting element $\beta ^\Omega _i$ is independent of the choice of the compatible ordering.

Proposition 2.9

Let $\Omega$ be an acyclic orientation of C. For any $i,j \in I$, we have

$$\begin{aligned} \widetilde{C}_{ij}(q,t,\underline{\mu }) = \sum _{k=0}^\infty (\varpi _i^\vee , T^{k}_\Omega \beta ^\Omega _j)_\Gamma . \end{aligned}$$

(2.8)

Proof

Choose a total ordering $I = \{i_1, \ldots , i_n\}$ compatible with $\Omega$. Then we have $T_\Omega = T_{i_1} \cdots T_{i_n}$. We extend the sequence $(i_1, \ldots , i_n)$ to an infinite sequence $(i_k)_{k \in \mathbb {Z}_{>0}}$ so that $i_{k+n} = i_k$ for all $k \in \mathbb {N}$. When C is of infinite type, this sequence satisfies the condition in Theorem 2.8 by Speyer [39], and hence we obtain (2.8). When C is of finite type, we know that the subsequence $(i_1,\ldots , i_{2l}) = (i_1, \ldots , i_n)^{h}$ is commutation-equivalent to a sequence $(j_1, \ldots , j_{2l})$ such that $(j_1,\ldots ,j_l)$ is a reduced word (adapted to $\Omega$) for the longest element $w_0$ and $j_{k+l} = j_k^*$ for all $1 \le k \le l$. Indeed, when C is of simply-laced type, it follows from Bédard [4]. When C is of non-simply-laced type, we simply have $\tau _\Omega ^{h/2} = w_0$ and $(i_1, \ldots , i_n)^{h/2}$ is a reduced word for $w_0$. Therefore Theorem 2.8 again yields (2.8). $\square$

Lemma 2.10

For each $i \in I$ and $k \in \mathbb {N}$, we have

$$\begin{aligned} q^{d_i} t^{-1} T^{k+1}_\Omega \beta ^\Omega _i + q^{-d_i}t T^{k}_\Omega \beta ^\Omega _i + \sum _{j \sim i}C_{ji}(q,t,\underline{\mu }) T_\Omega ^{k+\delta ((j,i)\in \Omega )}\beta ^\Omega _j =0. \end{aligned}$$

(2.9)

Proof

For any $i, j \in I$, we have

$$\begin{aligned} T_i \varpi _j = {\left\{ \begin{array}{ll} -q^{-2d_i}t^{2}\varpi _i - q^{-d_i}t\sum _{i' \sim i}C_{i'i}(q,t,\underline{\mu })\varpi _{i'} &{}\quad \text {if } i=j, \\ \varpi _j &{}\quad \text {if } i\ne j \end{array}\right. } \end{aligned}$$

by definition. Using this identity, we obtain

$$\begin{aligned} q^{d_i}t T_\Omega \varpi _i =- q^{-d_i}t\varpi _i -\sum _{j \sim i} C_{ji}(q,t,\underline{\mu }) T_\Omega ^{\delta ((j,i)\in \Omega )}\varpi _j. \end{aligned}$$

Applying $T^{k}_\Omega (1-T_\Omega )$ yields (2.9). $\square$

Once we fix a total ordering $I= \{i_1,\ldots ,i_n\}$ compatible with $\Omega$, the equalities (2.7) and (2.9) compute the elements $T^{k}_\Omega \beta ^\Omega _i$ for all $(k,i) \in \mathbb {Z}_{\ge 0} \times I$ recursively along the lexicographic total ordering of $\mathbb {Z}_{\ge 0} \times I$. Thus, together with (2.8), we have obtained a recursive algorithm to compute $\widetilde{C}_{ij}(q,t, \underline{\mu })$.

We say that a GCM C is bipartite if there is a function $\epsilon :I \rightarrow \mathbb {Z} / 2\mathbb {Z}$ such that $\epsilon (i) = \epsilon (j)$ implies $i \not \sim j$. When C is bipartite, we can simplify the above recursive formula by separating the parameter t as explained below.

For each $i \in I$, we consider a $\mathbb {Q}(\Gamma )$-linear automorphism $\bar{T}_i$ of $\textsf{Q}_\Gamma$ obtained from $T_i$ by evaluating the parameter t at 1. More precisely, it is given by

$$\begin{aligned} \bar{T}_i \alpha _j = \alpha _j - q^{-d_i}C_{ij}(q,1,\underline{\mu }) \alpha _i \end{aligned}$$

for all $j \in I$. The operators $\{\bar{T}_i\}_{i \in I}$ define another action of the braid group, under which the $\mathbb {Q}(\Gamma _0)$-subspace $\textsf{Q}_{\Gamma _0} \,{:=}\,\bigoplus _{i \in I}\mathbb {Q}(\Gamma _0)\alpha _i$ of $\textsf{Q}_\Gamma$ is stable.

Definition 2.11

A function $\xi :I \rightarrow \mathbb {Z}$ is called a height function (for C) if

$$\begin{aligned} \vert \xi (i) - \xi (j)\vert = 1 \quad \text {for all } i, j \in I \text { with } i \sim j. \end{aligned}$$

A height function $\xi$ gives an acyclic orientation $\Omega _\xi$ of C such that we have $(i, j) \in \Omega _\xi$ if $i \sim j$ and $\xi (j) = \xi (i) + 1$. When $i \in I$ is a sink of $\Omega _\xi$, in other words, when $\xi (i) < \xi (j)$ holds for all $j \in I$ with $j \sim i$, we define another height function $s_i \xi$ by

$$\begin{aligned} (s_i \xi )(j) \,{:=}\,\xi (j)+2\delta _{i,j}. \end{aligned}$$

Remark 2.12

There exists a height function for C if and only if C is bipartite.

Given a function $\xi :I \rightarrow \mathbb {Z}$, we define a linear automorphism $t^\xi$ of $\textsf{Q}_{\Gamma }$ by

$$\begin{aligned} t^\xi \alpha _i \,{:=}\,t^{\xi (i)}\alpha _i \end{aligned}$$

for each $i \in I$. When $\xi :I \rightarrow \mathbb {Z}$ is a height function and $i \in I$ a sink of $\Omega _\xi$, a straightforward computation yields $t^{\xi } T_i = \bar{T}_i t^{s_i \xi }$, from which we deduce

$$\begin{aligned} t^{\xi } T_{\Omega _\xi } = \bar{T}_{\Omega _\xi } t^{\xi + 2}. \end{aligned}$$

(2.10)

Definition 2.13

Let $\xi :I \rightarrow \mathbb {Z}$ be a height function. Define a map $\Phi _{\xi } :I \times \mathbb {Z} \rightarrow \textsf{Q}_{\Gamma _0}$ by

$$\begin{aligned} \Phi _\xi (i,u) \,{:=}\,{\left\{ \begin{array}{ll} \bar{T}^{k}_{\Omega _\xi }(1-\bar{T}_{\Omega _\xi })\varpi _i &{}\quad \text {if } u = \xi (i) + 2k \text { for some } k \in \mathbb {Z}_{\ge 0}, \\ 0 &{} \quad \text {else}. \end{array}\right. } \end{aligned}$$

The next proposition is a consequence of Proposition 2.9 and (2.10).

Proposition 2.14

Let $\xi :I \rightarrow \mathbb {Z}$ be a height function. For any $i,j \in I$, we have

$$\begin{aligned} \widetilde{C}_{ij}(q,t,\underline{\mu }) = \sum _{u = \xi (j)}^\infty \left( \varpi _i^\vee , \Phi _\xi (j,u)\right) _{\Gamma }t^{u-\xi (i)+1}. \end{aligned}$$

(2.11)

Now, Lemma 2.10 specializes to the following.

Lemma 2.15

Let $\xi :I \rightarrow \mathbb {Z}$ be a height function. For any $(i,u) \in I \times \mathbb {Z}$ with $u > \xi (i)$, we have

$$\begin{aligned} q^{-d_i} \Phi _\xi (i,u-1)+q^{d_i}\Phi _\xi (i,u+1) + \sum _{j \sim i} C_{ji}(q,1,\underline{\mu }) \Phi _\xi (j,u) =0. \end{aligned}$$

(2.12)

In particular, (2.12) enables us to compute recursively all the $\Phi _\xi (i,u)$ starting from

$$\begin{aligned} \Phi _{\xi }(i, \xi (i)) = (1-\bar{T}_{\Omega _\xi })\varpi _i= q^{-d_i}\bar{T}_{i_1}\cdots \bar{T}_{i_{k-1}}\alpha _i \quad \text {for all } i \in I, \end{aligned}$$

where $I = \{i_1,\ldots ,i_n\}$ is a total ordering compatible with $\Omega _\xi$ and $i_k=i$.

Thus, Proposition 2.14 combined with Lemma 2.15 gives a simpler recursive algorithm to compute $\widetilde{C}_{ij}(q,t,\underline{\mu })$ when C is bipartite.

Remark 2.16

When C is of finite type, the formula (2.11) recovers the formulas in [24, Proposition 2.1] (type ADE) and [27, Theorem 4.7] (type BCFG) after the specialization $\Gamma _0 \rightarrow \{1\}$.

Remark 2.17

When C is of finite type, the above algorithm can be used to compute $\widetilde{C}(q,t,\underline{\mu })$ (or $\widetilde{C}(q,t)$) completely. For example, let us give an explicit formula of $\widetilde{C}(q,t)$ for type $\text {F}_4$. We use the convention $I = \{1,2,3,4\}$ with $1 \sim 2 \sim 3 \sim 4$ and $(d_1,d_2,d_3,d_4) = (2,2,1,1)$. Then, for any $i \le j$, we have

$$\begin{aligned} \widetilde{C}_{ij}(q,t) = \frac{f_{ij}(q,t)+f_{ij}(q^{-1},t^{-1})}{q^9t^{-6}+q^{-9}t^6} \end{aligned}$$

where $f_{ij} = f_{ij}(q,t)$ is given by

$$\begin{aligned} f_{11}&= q^7t^{-5}+qt^{-1},&f_{12}&= q^5t^{-4}+q^3t^{-2}+q, \\ f_{13}&= q^4t^{-3}+q^2t^{-1},&f_{14}&= q^3t^{-2}, \\ f_{22}&= q^7t^{-5}+(q^5+q^3)t^{-3}+(q^3+2q)t^{-1},&f_{23}&= q^6t^{-4}+(q^4+q^2)t^{-2}+1, \\ f_{24}&= q^5t^{-3}+qt^{-1},&f_{33}&= q^8t^{-5}+(q^6+q^4)t^{-3}+(2q+1)t^{-1}, \\ f_{34}&= q^7t^{-4}+q^3t^{-2}+q,&f_{44}&= q^8t^{-5}+q^2t^{-1}. \end{aligned}$$

For the other case $i > j$, we can use the relation $[d_i]_q \widetilde{C}_{ij}(q,t) = [d_j]_q \widetilde{C}_{ji}(q,t)$.

When C is of type $\text {ABCD}$, an explicit formula of $\widetilde{C}(q,t)$ is given in [12, Appendix C]. When C is of type $\text {ADE}$, we have $\widetilde{C}(q,t) = \widetilde{C}(qt^{-1},1)$ and an explicit formula of $\widetilde{C}(q) = \widetilde{C}(q, 1)$ is given in [18, Appendix A] (see also [27, Sections 4.4.1, 4.4.2]).

3 Generalized preprojective algebras

In this section, we present a categorical characterization of the combinatorial objects that we introduced in Sect. 2. Our interpretation permits the translation of several unproven combinatorial properties of these objects (including Theorem 2.3, Proposition 2.5, and Theorem 2.8) into well-established categorical properties from the vantage point of the representation theory of generalized preprojective algebras [20].

Throughout this section, we fix an arbitrary field $\Bbbk$. Unless specified otherwise, vector spaces and algebras are defined over $\Bbbk$, and modules are left modules.

3.1 Conventions

Let Q be a finite quiver. We understand it as a quadruple $Q=(Q_0, Q_1, \text {s}, \text {t})$, where $Q_0$ is the set of vertices, $Q_1$ is the set of arrows and $\text {s}$ (resp. $\text {t}$) is the map $Q_1 \rightarrow Q_0$ which assigns each arrow with its source (resp. target). For a quiver Q, we set $\Bbbk Q_0 \,{:=}\,\bigoplus _{i \in Q_0} \Bbbk e_i$ and $\Bbbk Q_1 \,{:=}\,\bigoplus _{\alpha \in Q_1} \Bbbk \alpha$. We endow $\Bbbk Q_0$ with a $\Bbbk$-algebra structure by $e_i \cdot e_j = \delta _{ij} e_i$ for any $i,j \in Q_0$, and $\Bbbk Q_1$ with a $(\Bbbk Q_0, \Bbbk Q_0)$-bimodule structure by $e_i \cdot \alpha = \delta _{i, \text {t}(\alpha )} \alpha$ and $\alpha \cdot e_i = \delta _{i, \text {s}(\alpha )} \alpha$ for any $i \in Q_0$ and $\alpha \in Q_1$. Then the path algebra of Q is defined to be the tensor algebra $\Bbbk Q \,{:=}\,T_{\Bbbk Q_0}(\Bbbk Q_1)$.

Let G be a multiplicative abelian group with unit 1. By a G-graded quiver, we mean a quiver Q equipped with a map $\deg :Q_1 \rightarrow G$. We regard its path algebra $\Bbbk Q$ as a G-graded algebra in the natural way.

We say that a G-graded vector space $V = \bigoplus _{g \in G}V_g$ is locally finite if $V_g$ is of finite dimension for all $g \in G$. In this case, we define its graded dimension $\dim _G V$ to be the formal sum $\sum _{g \in G}\dim _\Bbbk (V_g) g \in \mathbb {Z}[\![G]\!]$. For a G-graded vector space V and an element $x \in G$, we define the grading shift $xV = \bigoplus _{g \in G}(xV)_{g}$ by $(xV)_g = V_{x^{-1}g}$. More generally, for $a = \sum _{g \in G} a_g g \in \mathbb {Z}_{\ge 0}[\![G]\!]$, we set $V^{\oplus a} \,{:=}\,\bigoplus _{g \in G} (gV)^{\oplus a_g}$. When $V^{\oplus a}$ happens to be locally finite, we have $\dim _G V^{\oplus a} = a \dim _G V$.

3.2 Preliminary on positively graded algebras

Let $t^{\mathbb {Z}}$ 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 $G = G_0 \times t^{\mathbb {Z}}$, where $G_0$ is another abelian group. Our principal example is the group $\Gamma = t^\mathbb {Z} \times \Gamma _0$ in Sect. 2.2. For G-graded vector space $V = \bigoplus _{g \in G} V_g$ and $n \in \mathbb {Z}$, we define the $G_0$-graded subspace $V_n \subset V$ by $V_n \,{:=}\,\bigoplus _{g \in G_0}V_{t^n g}$. By definition, we have $V = \bigoplus _{n \in \mathbb {Z}}V_n$. We use the notation $V_{\ge n} \,{:=}\,\bigoplus _{m \ge n} V_m$ and $V_{>n} \,{:=}\,\bigoplus _{m > n}V_m$.

We consider a G-graded algebra $\Lambda$ satisfying the following condition:

• (A) $\Lambda = \Lambda _{\ge 0}$ and $\dim _\Bbbk \Lambda _n < \infty$ for each $n \in \mathbb {Z}_{\ge 0}$.

In particular, $\Lambda _0$ is a $G_0$-graded finite dimensional algebra. Let $\{ S_j\}_{j \in J}$ be a complete collection of $G_0$-graded simple modules of $\Lambda _0$ up to isomorphism and grading shift. It also gives a complete collection of G-graded simple modules of $\Lambda$. For a G-graded $\Lambda$-module M, the subspace $M_{\ge n} \subset M$ is a $\Lambda$-submodule for each $n \in \mathbb {Z}$. Let $\Lambda \text {-}\text {mod}_G^{\ge n}$ denote the category of G-graded $\Lambda$-modules M satisfying $M = M_{\ge n}$ and $\dim _\Bbbk M_m < \infty$ for all $m \ge n$, whose morphisms are G-homogeneous $\Lambda$-homomorphisms. This is a $\Bbbk$-linear abelian category. Let $\Lambda \text {-}\text {mod}_G^+ \,{:=}\,\bigcup _{n \in \mathbb {Z}} \Lambda \text {-}\text {mod}_G^{\ge n}$. Note that $\Lambda \text {-}\text {mod}_G^{+}$ contains all the finitely generated G-graded $\Lambda$-modules, because it contains their projective covers by the condition (A).

Lemma 3.1

Given $n \in \mathbb {Z}$ and $M \in \Lambda \text {-}\text {mod}_G^{\ge n}$, there is a surjection $P \twoheadrightarrow M$ from a projective $\Lambda$-module P belonging to $\Lambda \text {-}\text {mod}_G^{\ge n}$.

Proof

For each $m \ge n$, let $P_m \twoheadrightarrow M_m$ be a projective cover of $M_m$ regarded as a $G_0$-graded $\Lambda _0$-module. Then consider the G-graded projective $\Lambda$-module $P \,{:=}\,\Lambda \otimes _{\Lambda _0}\bigoplus _{m \ge n} t^mP_m$, which carries a natural surjection $P \twoheadrightarrow M$. This P belongs to $\Lambda \text {-}\text {mod}_G^{\ge n}$ because $\dim _G P$ is not greater than $\dim _{G} \Lambda \cdot \sum _{m \ge n}t^m \dim _{G_0} P_m$ which belongs to $\mathbb {Z}[G_0][\![t]\!] t^n$. $\square$

For an abelian category $\mathcal {C}$, we denote by $K(\mathcal {C})$ its Grothendieck group. We regard $K(\Lambda \text {-}\text {mod}_G^{\ge n})$ as a subgroup of $K(\Lambda \text {-}\text {mod}_G^+)$ via the inclusion for any $n \in \mathbb {Z}$. Then, the collection of subgroups $\{ K(\Lambda \text {-}\text {mod}_G^{\ge n})\}_{n \in \mathbb {Z}}$ gives a filtration of $K(\Lambda \text {-}\text {mod}_G^+)$. We define the completed Grothendieck group $\hat{K}(\Lambda \text {-}\text {mod}_G^+)$ to be the projective limit

$$\begin{aligned} \hat{K}(\Lambda \text {-}\text {mod}_G^+) \,{:=}\,\varprojlim _{n} K(\Lambda \text {-}\text {mod}_G^{+})/K(\Lambda \text {-}\text {mod}_G^{\ge n}). \end{aligned}$$

Note that $\hat{K}(\Lambda \text {-}\text {mod}_G^+)$ carries a natural $\mathbb {Z}[G_0](\!( t )\!)$-module structure given by $a [M] = [M^{\oplus a_+}] - [M^{\oplus a_-}]$, where we choose $a_+, a_- \in \mathbb {Z}_{\ge 0}[G_0](\!(t)\!)$ so that $a = a_+ - a_-$.

Lemma 3.2

The $\mathbb {Z}[G_0](\!(t)\!)$-module $\hat{K}(\Lambda \text {-}\text {mod}_G^+)$ is free with a basis $\{[S_j]\}_{j \in J}$.

Proof

For any $n \in \mathbb {Z}$ and $M \in \Lambda \text {-}\text {mod}^{\ge n}_G$, we have a unique expression

$$\begin{aligned}{}[M] = \sum _{j\in J}\left( \sum _{m \ge n}[M_m:S_j]_{G_0}t^m\right) [S_j] \end{aligned}$$

in $\hat{K}(\Lambda \text {-}\text {mod}_G^+)$, where $[M_m:S_j]_{G_0} \in \mathbb {Z}[G_0]$ denotes the $G_0$-graded Jordan-Hölder multiplicity of $S_j$ in the finite length $G_0$-graded $\Lambda _0$-module $M_m$. This proves the assertion.

$\square$

3.3 Generalized preprojective algebras

We fix a GCM $C = (c_{ij})_{i,j \in I}$ and its symmetrizer $D = \mathop {\text {diag}}\nolimits (d_i \mid i \in I)$ as in Sect. 2.1. Recall the free abelian group $\Gamma$ in Sect. 2.2. We consider the quiver $\widetilde{Q}= (\widetilde{Q}_0, \widetilde{Q}_1, \text {s}, \text {t})$ given as follows:

$$\begin{aligned} \widetilde{Q}_0= & {} I, \quad \widetilde{Q}_1 = \left\{ \alpha _{ij}^{(g)} \mid (i,j, g) \in I \times I \times \mathbb {Z}, i \sim j, 1 \le g \le g_{ij} \right\} \cup \{ \varepsilon _i \mid i \in I \}, \\ \text {s}(\alpha _{ij}^{(g)})= & {} j, \quad \text {t}(\alpha _{ij}^{(g)})= i, \quad \text {s}(\varepsilon _i)=\text {t}(\varepsilon _i)=i. \end{aligned}$$

We equip the quiver $\widetilde{Q}$ with a $\Gamma$-grading by

$$\begin{aligned} \deg (\alpha _{ij}^{(g)}) = q^{-d_if_{ij}}t\mu ^{(g)}_{ij}, \qquad \deg (\varepsilon _i) = q^{2d_i}. \end{aligned}$$

(3.1)

Let $\Omega$ be an acyclic orientation of C. We define the associated potential $W_\Omega \in \Bbbk \widetilde{Q}$ by

$$\begin{aligned} W_\Omega = \sum _{i, j \in I; i \sim j} \sum _{g=1}^{g_{ij}} \text {sgn}_\Omega (i,j)\alpha _{ij}^{(g)} \alpha _{ji}^{(g)}\varepsilon _{i}^{f_{ij}}, \end{aligned}$$

(3.2)

where $\text {sgn}_\Omega (i,j) \,{:=}\,(-1)^{\delta ((j,i) \in \Omega )}$. Note that $W_\Omega$ is homogeneous of degree $t^2$. We define the $\Gamma$-graded $\Bbbk$-algebra $\widetilde{\Pi }$ to be the quotient of $\Bbbk \widetilde{Q}$ by the ideal generated by all the cyclic derivations of $W_\Omega$. In other words, the algebra $\widetilde{\Pi }$ is the quotient of $\Bbbk \widetilde{Q}$ by the following two kinds of relations:

• (R1) $\varepsilon _i^{f_{ij}} \alpha _{ij}^{(g)} = \alpha _{ij}^{(g)} \varepsilon _j^{f_{ji}}$ for any $i,j \in I$ with $i \sim j$ and $1 \le g \le g_{ij}$;

• (R2) $\displaystyle \sum _{j \in I: j\sim i}\sum _{g=1}^{g_{ij}}\sum _{k = 0}^{f_{ij}-1}\text {sgn}_\Omega (i,j) \varepsilon _i^k \alpha _{ij}^{(g)} \alpha _{ji}^{(g)} \varepsilon _i^{f_{ij}-1-k} =0$ for each $i \in I$.

Remark 3.3

Although the definition of the algebra $\widetilde{\Pi }$ depends on the choice of acyclic orientation $\Omega$, it is irrelevant. In fact, a different choice of $\Omega$ gives rise to an isomorphic $\Gamma$-graded algebra. Moreover, one may define $\widetilde{\Pi }$ with more general orientation (i.e., without acyclic condition, as in Sect. 4.1 below). Even if we do so, the resulting $\Gamma$-graded algebra is isomorphic to our $\widetilde{\Pi }$.

For a positive integer $\ell \in \mathbb {Z}_{>0}$, we define the $\Gamma$-graded algebra $\Pi (\ell )$ to be the quotient

$$\begin{aligned} \Pi (\ell ) = \widetilde{\Pi }/ (\varepsilon ^\ell ), \end{aligned}$$

where $\varepsilon \,{:=}\,\sum _{i \in I} \varepsilon _i^{r/d_i}$. Note that $\varepsilon$ is homogeneous and central in $\widetilde{\Pi }$. In other words, it is the quotient of $\Bbbk \widetilde{Q}$ by the three kinds of relations: (R1), (R2), and

• (R3) $\varepsilon _i^{r\ell / d_i} = 0$ for each $i \in I$.

Remark 3.4

The algebra $\Pi (\ell )$ is identical to the generalized preprojective algebra $\Pi ({}^\texttt{t}C, \ell rD^{-1}, \Omega )$ in the sense of Geiss et al. [20].

Lemma 3.5

For any $\ell \in \mathbb {Z}_{>0}$, the algebra $\Pi (\ell )$ satisfies the condition (A) in Sect. 3.2.

Proof

The fact $\Pi (\ell )_{\ge 0} =\Pi (\ell )$ is clear from the definition (3.1). For any $n \in \mathbb {Z}_{\ge 0}$, thanks to the relation (R3), the vector space $\Pi (\ell )_n$ is spanned by a finite number of vectors in

$$\begin{aligned} \{ \varepsilon _{i_0}^{m_0} \alpha ^{(g_1)}_{i_0,i_1} \varepsilon _{i_1}^{m_1} \alpha ^{(g_2)}_{i_1,i_2} \cdots \varepsilon _{i_{n-1}}^{m_{n-1}}\alpha ^{(g_{n})}_{i_{n-1},i_n}\varepsilon _{i_n}^{m_n} \mid i_k \in I, 0 \le m_k < r \ell /d_{i_k}, 1 \le g_k \le g_{i_{k-1},i_k} \}. \end{aligned}$$

Therefore, we have $\dim _\Bbbk \Pi (\ell )_n < \infty$. $\square$

In what follows, we fix $\ell \in \mathbb {Z}_{>0}$ and write $\Pi$ for $\Pi (\ell )$ for the sake of brevity.

By the definition, we have

$$\begin{aligned} \Pi _0 \cong \prod _{i \in I} H_i, \qquad \text {where } H_i \,{:=}\,\Bbbk [\varepsilon _i]/(\varepsilon _i^{r\ell /d_i}). \end{aligned}$$

In particular, for each $M \in \Pi \text {-}\text {mod}_\Gamma ^+$ and $n \in \mathbb {Z}$, the subspace $e_i M_n$ is a finite-dimensional $H_i$-module for each $i \in I$. We say that M is locally free if $e_i M_n$ is a free $H_i$-module for any $n \in \mathbb {Z}$ and $i \in I$, or equivalently $M_n$ is a projective $\Pi _0$-module for any $n \in \mathbb {Z}$. In this case, we set ${\text {rank}}_{i} M \,{:=}\,\dim _\Gamma e_i (M/\varepsilon _i M) \in \mathbb {Z}[\Gamma _0](\!(t)\!)$.

Theorem 3.6

(Geiss et al. [20, Section 11]) As a (left) $\Pi$-module, $\Pi$ is locally free in itself.

For each $i \in I$, let $P_i \,{:=}\,\Pi e_i$ be the indecomposable projective $\Pi$-module associated to the vertex i and $S_i$ its simple quotient. Consider the two-sided ideal $J_i \,{:=}\,\Pi (1-e_i) \Pi$. We have $\Pi / J_i \cong H_i$ as $\Gamma$-graded algebras. We write $E_i$ for $\Pi / J_i$ when we regard it as a $\Gamma$-graded left $\Pi$-module. This is a locally free $\Pi$-module characterized by ${\text {rank}}_j E_i = \delta _{i,j}$. In $\hat{K}(\Pi \text {-}\text {mod}_\Gamma ^+)$, we have

$$\begin{aligned}{}[E_i] = \frac{1-q^{2r\ell }}{1-q^{2d_i}}[S_i]. \end{aligned}$$

(3.3)

There is the anti-involution $\phi :\Pi \rightarrow \Pi ^{\text {op}}$ given by the assignment

$$\begin{aligned} \phi (e_i) \,{:=}\,e_i, \qquad \phi (\alpha _{ij}^{(g)}) \,{:=}\,\alpha _{ji}^{(g)}, \qquad \phi (\varepsilon _i) \,{:=}\,\varepsilon _i. \end{aligned}$$

Recall the automorphism of the group $\Gamma$ also denoted by $\phi$ in Sect. 2.3. By definition, if $x \in \Pi$ is homogeneous of degree $\gamma \in \Gamma$, then $\phi (x)$ is homogeneous of degree $\phi (\gamma )$. For a left $\Pi$-module M, let $M^\phi$ be the right $\Pi$-module obtained by twisting the original left $\Pi$-module structure by the opposition $\phi$. If M is $\Gamma$-graded, $M^\phi$ is again $\Gamma$-graded by setting $(M^\phi )_\gamma \,{:=}\,M_{\phi (\gamma )}$. In particular, for $M \in \Pi \text {-}\text {mod}_\Gamma ^+$, we have $\dim _\Gamma (M^\phi ) = (\dim _\Gamma M)^\phi$.

3.4 Projective resolutions

Following [20, Section 5.1], for each $i, j \in I$ with $i \sim j$, we define the bigraded $(H_i, H_j)$-bimodule ${}_i H_j$ by

$$\begin{aligned} {}_i H_j \,{:=}\,\sum _{g=1}^{g_{ij}} H_i \alpha _{ij}^{(g)} H_j \subset \Pi . \end{aligned}$$

It is free as a left $H_i$-module and free as a right $H_j$-module. Moreover, the relation (R1) gives the following:

$$\begin{aligned} {}_i H_j = \bigoplus _{k=0}^{f_{ji}-1}\bigoplus _{g=1}^{g_{ij}}H_i \left( \alpha _{ij}^{(g)}\varepsilon _j^k\right) = \bigoplus _{k=0}^{f_{ij}-1}\bigoplus _{g=1}^{g_{ij}}\left( \varepsilon _i^k \alpha _{ij}^{(g)}\right) H_j. \end{aligned}$$

In particular, we get the following lemma.

Lemma 3.7

For $i, j\in I$ with $i\sim j$, we have two isomorphisms

$$\begin{aligned} {}_{H_i}({}_i H_j) \cong H_i^{\oplus (-q^{-d_j}tC_{ji}(q,t,\underline{\mu })^\phi )}, \qquad ({}_i H_j) {}_{H_j} \cong H_j^{\oplus (-q^{-d_i}tC_{ij}(q,t, \underline{\mu }))} \end{aligned}$$

as $\Gamma$-graded left $H_i$-modules and as $\Gamma$-graded right $H_j$-modules respectively.

Consider the following sequence of $\Gamma$-graded $(\Pi , \Pi )$-bimodules:

$$\begin{aligned} \bigoplus _{i\in I} q^{-2d_i} t^2 \Pi e_i \otimes _i e_i \Pi \xrightarrow {\psi } \bigoplus _{i, j \in I: i \sim j} \Pi e_j \otimes _j {}_jH_i \otimes _i e_i \Pi \xrightarrow {\varphi } \bigoplus _{i\in I} \Pi e_i \otimes _i e_i \Pi \rightarrow \Pi \rightarrow 0,\nonumber \\ \end{aligned}$$

(3.4)

where $\otimes _i \,{:=}\,\otimes _{H_i}$ and the morphisms $\psi$ and $\varphi$ are given by

$$\begin{aligned} \psi (e_i \otimes e_i)&\,{:=}\,\sum _{j \sim i}\sum _{g=1}^{g_{ij}} \sum _{\begin{array}{c} k,l \ge 0, \\ k+l = f_{ij}-1 \end{array}}\text {sgn}_\Omega (i,j) \left( \varepsilon _i^{k} \alpha _{ij}^{(g)} \otimes \alpha _{ji}^{(g)} \varepsilon _i^{l}\otimes e_i + e_i \otimes \varepsilon _i^{k} \alpha _{ij}^{(g)} \otimes \alpha _{ji}^{(g)} \varepsilon _i^{l}\right) , \\&\quad \varphi (e_j \otimes x\otimes e_i) \,{:=}\,x \otimes e_i + e_j \otimes x. \end{aligned}$$

The other arrows $\bigoplus _{i\in I} \Pi e_i \otimes _i e_i \Pi \rightarrow \Pi \rightarrow 0$ are canonical. The relation (R2) ensures that the sequence (3.4) forms a complex. For each $i \in I$, applying $(-)\otimes _{\Pi } E_i$ to (3.4) yields the following complex of $\Gamma$-graded (left) $\Pi$-modules:

$$\begin{aligned} q^{-2d_i}t^2 P_i \xrightarrow {\psi ^{(i)}} \bigoplus _{j\sim i} P_j^{\oplus (-q^{-d_i}tC_{ij}(q,t,\underline{\mu })^\phi )} \xrightarrow {\varphi ^{(i)}} P_i \rightarrow E_i \rightarrow 0. \end{aligned}$$

(3.5)

Here we used Lemma 3.7.

Theorem 3.8

(Geiss et al. [20, Proposition 12.1 and Corollary 12.2], Fujita and Murakami [15, Theorem 3.3]) The complexes (3.4) and (3.5) are exact. Moreover, the followings hold.

• (1) When C is of infinite type, we have $\mathop {\text {Ker}}\nolimits \psi = 0$ and $\mathop {\text {Ker}}\nolimits \psi ^{(i)} =0$ for all $i \in I$.

• (2) When C is of finite type, we have $\mathop {\text {Ker}}\nolimits \psi ^{(i)} \cong q^{-rh^\vee }t^h\mu _{i^*i} E_{i^*}$ for each $i \in I$.

3.5 Euler–Poincaré pairing

For a $\Gamma$-graded right $\Pi$-module M and a $\Gamma$-graded left $\Pi$-module N, the vector space $M \otimes _\Pi N$ is naturally $\Gamma$-graded. Let $\mathop {\text {tor}}\nolimits _k^\Pi (M, N)$ denote the k-th left derived functor of $M \mapsto M \otimes _\Pi N$ (or equivalently, that of $N \mapsto M \otimes _\Pi N$).

Lemma 3.9

If $M \in \Pi ^\text {op}\text {-}\text {mod}_\Gamma ^{\ge m}$ and $N \in \Pi \text {-}\text {mod}_\Gamma ^{\ge n}$, we have

$$\begin{aligned} \mathop {\text {tor}}\nolimits _k^\Pi (M, N) \in \Bbbk \text {-}\text {mod}_\Gamma ^{\ge m+n} \quad \text {for any } k \in \mathbb {Z}_{\ge 0}. \end{aligned}$$

Proof

We see that $\dim _\Gamma (M\otimes _\Pi N)$ is not grater than $\dim _\Gamma M \cdot \dim _\Gamma N$ which belongs to $\mathbb {Z}[\Gamma _0][\![t]\!]t^{m+n}$ under the assumption. This proves the assertion for $k =0$. The other case when $k > 0$ follows from this case and Lemma 3.1. $\square$

We consider the following finiteness condition for a pair (M, N) of objects in $\Pi \text {-}\text {mod}_\Gamma ^+$:

• (B) For each $\gamma \in \Gamma$, the space $\mathop {\text {tor}}\nolimits ^\Pi _k(M^\phi , N)_\gamma$ vanishes for $k \gg 0$.

If (M, N) satisfies the condition (B), their Euler–Poincaré (EP) pairing

$$\begin{aligned} \langle M,N \rangle _\Gamma \,{:=}\,\sum _{k =0}^{\infty } (-1)^k \dim _\Gamma \mathop {\text {tor}}\nolimits ^\Pi _k(M^\phi , N). \end{aligned}$$

is well-defined as an element of $\mathbb {Z}[\![\Gamma ]\!]$. The next lemma is immediate from the definition.

Lemma 3.10

Let $M, N \in \Pi \text {-}\text {mod}_\Gamma ^+$.

• (1) If (M, N) satisfies (B), the opposite pair (N, M) also satisfies (B) and we have $\langle N,M \rangle _\Gamma = \langle M, N \rangle _\Gamma ^\phi$.

• (2) If (M, N) satisfies (B), the pair $(M^{\oplus a}, N^{\oplus b})$ also satisfies (B) for any $a, b \in \mathbb {Z}_{\ge 0}[\Gamma ]$ and we have $\langle M^{\oplus a}, N^{\oplus b} \rangle _\Gamma = a^\phi b \langle M, N \rangle _\Gamma$.

• (3) Suppose that there is an exact sequence $0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ in $\Pi \text {-}\text {mod}_\Gamma ^+$, and the pairs $(M', N)$ and $(M'', N)$ both satisfy (B). Then the pair (M, N) also satisfies (B) and we have $\langle M, N\rangle _\Gamma = \langle M', N \rangle _\Gamma + \langle M'', N \rangle _\Gamma .$

Proposition 3.11

For any $i,j \in I$, the pair $(S_i, S_j)$ satisfy the condition (B) and we have

$$\begin{aligned} \langle E_i, S_j \rangle _\Gamma&= {\left\{ \begin{array}{ll} \displaystyle \frac{q^{-d_i}t \left( C_{ij}(q,t,\underline{\mu }) - q^{-rh^\vee }t^{h}\mu _{ii^*} C_{i^*j}(q,t,\underline{\mu }) \right) }{1-q^{-2rh^\vee }t^{2h}}&{}\quad \text {if } C \text { is of finite type}, \\ q^{-d_i}tC_{ij}(q,t,\underline{\mu }) &{}\quad \text {otherwise}, \end{array}\right. } \end{aligned}$$

(3.6)

$$\begin{aligned} \langle S_i, S_j\rangle _\Gamma&= \frac{1-q^{2d_i}}{1-q^{2r\ell }} \langle E_i, S_j \rangle _\Gamma . \end{aligned}$$

(3.7)

Here we understand $(1-\gamma )^{-1} = \sum _{k \ge 0}\gamma ^{k} \in \mathbb {Z}[\![\Gamma ]\!]$ for $\gamma \in \Gamma {\setminus } \{1\}$.

Proof

The former formula (3.6) directly follows from Theorem 3.8. The latter (3.7) follows from Theorem 3.8 and the fact that $S_i$ has an $E_i$-resolution of the form:

$$\begin{aligned} \cdots \rightarrow q^{2r\ell + 2d_i}E_i \rightarrow q^{2r\ell } E_i \rightarrow q^{2d_i}E_i \rightarrow E_i \rightarrow S_i \rightarrow 0. \end{aligned}$$

See the proof of Fujita and Murakami [15, Proposition 3.11] for some more details. $\square$

Corollary 3.12

For any $M,N \in \Pi \text {-}\text {mod}_\Gamma ^+$, the pair (M, N) satisfies the condition (B). Moreover, the EP pairing induces a $\phi$-sesquilinear hermitian form on the $\mathbb {Z}[\Gamma _0](\!(t)\!)$-module $\hat{K}(\Pi \text {-}\text {mod}_\Gamma ^+)$ valued at $\mathbb {Z}[\Gamma _0][(1-q^{2r\ell })^{-1}](\!(t)\!)$.

Proof

Given $M,N \in \Pi \text {-}\text {mod}_\Gamma ^+$, we shall show that (M, N) satisfies the condition (B). Without loss of generality, we may assume that $M, N \in \Pi \text {-}\text {mod}_\Gamma ^{\ge 0}$. For $\gamma \in \Gamma$ fixed, take $n \in \mathbb {Z}$ such that $\gamma \not \in \Gamma _0 t^{\mathbb {Z}_{> n}}$. By Lemma 3.9, we have $\mathop {\text {tor}}\nolimits _k^\Pi (M^\phi _{>n},N)_{\gamma } =0$ and therefore $\mathop {\text {tor}}\nolimits _k^\Pi (M^\phi ,N)_\gamma \simeq \mathop {\text {tor}}\nolimits _k^\Pi (M^\phi /M^\phi _{>n},N)_\gamma$ for any $k \in \mathbb {Z}_{\ge 0}$. Similarly, we have $\mathop {\text {tor}}\nolimits _k^\Pi (M^\phi /M^\phi _{>n},N)_\gamma \simeq \mathop {\text {tor}}\nolimits _k^\Pi (M^\phi /M^\phi _{>n},N/N_{>n})_\gamma$ and hence $\mathop {\text {tor}}\nolimits _k^\Pi (M^\phi ,N)_\gamma \simeq \mathop {\text {tor}}\nolimits _k^\Pi (M^\phi /M^\phi _{>n},N/N_{>n})_\gamma$ for any $k \in \mathbb {Z}_{\ge 0}$. By Lemma 3.10 and Proposition 3.11, we know that the condition (B) is satisfied for any finite-dimensional modules. Therefore, for k large enough, we have $\mathop {\text {tor}}\nolimits _k^\Pi (M^\phi /M^\phi _{>n},N/N_{>n})_\gamma = 0$. Thus, the pair (M, N) satisfies the condition (B). Now, by Lemma 3.10 (3) and Proposition 3.11, the EP pairing induces a pairing on the Grothendieck group $K(\Pi \text {-}\text {mod}_G^+)$ valued at $\mathbb {Z}[\Gamma _0][(1-q^{2r\ell })^{-1}](\!(t)\!)$. Lemma 3.9 tells us that this is continuous with respect to the topology given by the filtration $\{ K(\Pi \text {-}\text {mod}_G^{\ge n})\}_{n \in \mathbb {Z}}$. Therefore, it descends to a pairing on the completion $\hat{K}(\Pi \text {-}\text {mod}_\Gamma ^+)$ satisfying the desired properties. $\square$

Let $\mathbb {F}$ be an algebraic closure of the field $\mathbb {Q}(\Gamma _0)(\!( t )\!)$. We understand that $\mathbb {Q}(\Gamma )$ is a subfield of $\mathbb {F}$ by considering the Laurent expansions at $t = 0$. By Corollary 3.12 above, the EP pairing linearly extends to a $\phi$-sesquilinear hermitian form, again written by $\langle -, - \rangle _\Gamma$, on

$$\begin{aligned} \hat{K}(\Pi \text {-}\text {mod}_\Gamma ^+)_\mathbb {F}\,{:=}\,\hat{K}(\Pi \text {-}\text {mod}_\Gamma ^+)\otimes _{\mathbb {Z}[\Gamma _0](\!(t)\!)} \mathbb {F}\end{aligned}$$

valued at $\mathbb {F}$. Note that the set $\{ [E_i] \}_{i \in I}$ forms an $\mathbb {F}$-basis of $\hat{K}(\Pi \text {-}\text {mod}_\Gamma ^+)_\mathbb {F}$ by Lemma 3.2 and (3.3), and that, if $M \in \Pi \text {-}\text {mod}_\Gamma ^+$ is locally free, we have $[M] = \sum _{i \in I}({\text {rank}}_i M) [E_i]$.

It is useful to introduce the module $\bar{P}_i \,{:=}\,P_i/P_i \varepsilon _i = (\Pi /\Pi \varepsilon _i)e_i$ for each $i \in I$. We can easily prove the following (see [15, Lemma 2.5]).

Lemma 3.13

If $M \in \Pi \text {-}\text {mod}_\Gamma ^+$ is locally free, we have $\langle \bar{P}_i, M \rangle _\Gamma = {\text {rank}}_i M$. In particular, we have $\langle \bar{P}_i, E_j \rangle _\Gamma = \delta _{i,j}$ and ${\text {rank}}_i P_j = (\dim _\Gamma e_j \bar{P}_i)^\phi$ for any $i,j \in I$.

On the other hand, we consider the $\mathbb {F}$-vector space $\textsf{Q}_\Gamma \otimes _{\mathbb {Q}(\Gamma )}\mathbb {F}$, on which the pairing $(-,-)_\Gamma$ extends linearly. Let $\Psi$ be the $\mathbb {F}$-linear automorphism of $\textsf{Q}_\Gamma \otimes _{\mathbb {Q}(\Gamma )}\mathbb {F}$ given by

$$\begin{aligned} \Psi \,{:=}\,{\left\{ \begin{array}{ll} \displaystyle (1+ q^{-rh^\vee }t^h \nu )^{-1} = \frac{\text {id}-q^{-rh^\vee }t^h\nu }{1-q^{-2rh^\vee }t^{2h}} &{}\quad \text {if } C \text { is of finite type}, \\ \text {id}&{}\quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

Here $\nu$ is the linear operator on $\textsf{Q}_\Gamma$ given by $\nu (\alpha _i) = \mu _{i^*i}\alpha _{i^*}$, which we have already defined in Sect. 2.3. Let us choose an element $\kappa _\ell \in \mathbb {F}$ satisfying $\kappa _\ell ^2 = q^{r\ell }[r\ell ]_qt^{-1}$.

Theorem 3.14

The assignment $[E_i] \mapsto \kappa _\ell \alpha _i^\vee \, (i \in I)$ gives an $\mathbb {F}$-linear isomorphism

$$\begin{aligned} \chi _\ell :\hat{K}(\Pi \text {-}\text {mod}_\Gamma ^+)_\mathbb {F}\rightarrow \textsf{Q}_\Gamma \otimes _{\mathbb {Q}(\Gamma )} \mathbb {F}\end{aligned}$$

satisfying the following properties:

• (1) For any $i \in I$, we have $\chi _\ell [S_i] = \kappa _\ell ^{-1} \alpha _i$.

• (2) For any $x,y \in \hat{K}(\Pi \text {-}\text {mod}_\Gamma ^+)_\mathbb {F}$, we have $\langle x,y \rangle _\Gamma = ( \Psi \chi _\ell (x), \chi _\ell (y))_\Gamma .$

• (3) For any $i \in I$, we have $\varpi _i^\vee = \kappa _\ell ^{-1}\Psi \chi _\ell [P_i]$ and $\varpi _i = q^{-d_i}t\kappa _\ell \Psi \chi _\ell [\bar{P}_i]$.

Proof

As the set $\{ [E_i]\}_{i \in I}$ forms an $\mathbb {F}$-basis of $\hat{K}(\Pi \text {-}\text {mod}_\Gamma ^+)_\mathbb {F}$, the linear map $\chi _\ell$ is an isomorphism. The properties (1) and (2) follow from the identities (3.3), (3.6) and (3.7). Since the basis $\{[P_i]\}_{i \in I}$ (resp. $\{[\bar{P}_i]\}_{i \in I}$) is dual to the basis $\{[S_i]\}_{i \in I}$ (resp. $\{[E_i]\}_{i \in I}$ by Lemma 3.13), the property (3) follows from the property (2). $\square$

Corollary 3.15

Let $i,j \in I$.

• (1) When C is of finite type, we have

  $$\begin{aligned} \widetilde{C}_{ij}(q,t,\underline{\mu }) = \frac{q^{-d_i}t}{1-q^{-2rh^\vee }t^{2h}} \left( \dim _{\Gamma }(e_i \bar{P}_j) - q^{-rh^\vee }t^h \mu _{ii^*}\dim _{\Gamma }(e_{i^*} \bar{P}_j) \right) . \end{aligned}$$

• (2) When C is of infinite type, we have

  $$\begin{aligned} \widetilde{C}_{ij}(q,t,\underline{\mu }) = q^{-d_j}t\dim _{\Gamma }(e_i \bar{P}_j). \end{aligned}$$

Proof

It follows from Theorem 3.14 and the inversion of (2.3). $\square$

In particular, Corollary 3.15 (2) proves Theorem 2.3.

Remark 3.16

Since $\langle P_i, \bar{P}_j \rangle _{\Gamma } = \dim _\Gamma (e_i \bar{P}_j)$, Corollary 3.15 interprets the matrix $\widetilde{C}(q,t,\underline{\mu })$ in terms of the EP pairing between the bases $\{[P_i]\}_{i \in I}$ and $\{[\bar{P}_i]\}_{i \in I}$. In this sense, Corollary 3.15 is dual to (3.6) in Proposition 3.11.

Remark 3.17

In the previous paper [15], we dealt with GCMs of finite type and finite dimensional (q, t)-graded $\Pi$-modules. Therein, we used the modules $\bar{I}_i \,{:=}\,\mathbb {D}(\bar{P}_i^\phi )$ and the graded extension groups $\mathop {\text {ext}}\nolimits _\Pi ^k$, where $\mathbb {D}$ is the graded $\Bbbk$-dual functor, instead of the modules $\bar{P}_i$ and the graded torsion groups $\mathop {\text {tor}}\nolimits ^\Pi _k$. Note that the two discussions are mutually equivalent thanks to the usual adjunction (cf. [3, Section A.4 Proposition 4.11]), i.e., we have $\mathbb {D}(\mathop {\text {tor}}\nolimits _k^\Pi (\mathbb {D}(M),N)) \simeq \mathop {\text {ext}}\nolimits ^k_\Pi (N,M)$ for $M,N \in \Pi \text {-}\text {mod}_\Gamma ^+$. In this sense, our discussion here is a slight generalization of that in Fujita and Murakami [15] with the additional $\underline{\mu }$-grading.

3.6 Braid group action

In this subsection, we interpret our braid group symmetry on the deformed root lattice $\textsf{Q}_\Gamma$ (see Sect. 2) as a $\Gamma$-graded counterpart of the categorical braid group symmetry on the module category over $\Pi$ and its Grothendieck group [1, 7, 13, 25], etc. via our argument in Sect. 3.5. In particular, we establish Proposition 2.5 and Theorem 2.8 as corollaries of these categorical symmetries.

Recall the two-sided ideal $J_i = \Pi (1-e_i) \Pi$. For any $M \in \Pi \text {-}\text {mod}_\Gamma ^+$ and $k \in \mathbb {Z}_{\ge 0}$, we see that $\mathop {\text {tor}}\nolimits _k^\Pi (J_i, M )$ also belongs to $\Pi \text {-}\text {mod}_\Gamma ^+$. When C is of infinite type, $J_i^{\phi } = J_i$ has projective dimension at most 1. In particular, the derived tensor product $J_i \overset{{\textbf {L}}}{\otimes }_\Pi M$ is an object in the bounded derived category $\mathcal {D}^b (\Pi \text {-}\text {mod}_\Gamma ^+)$ for each $M \in \Pi \text {-}\text {mod}_\Gamma ^+$. By the natural identification $K(\mathcal {D}^b (\Pi \text {-}\text {mod}_\Gamma ^+))\cong K(\Pi \text {-}\text {mod}_\Gamma ^+)$ and the canonical map $K(\Pi \text {-}\text {mod}_\Gamma ^+)\rightarrow \hat{K}(\Pi \text {-}\text {mod}_\Gamma ^+)$, it gives the element

$$\begin{aligned}{}[J_i \overset{{\textbf {L}}}{\otimes }_\Pi M] = \sum _{k = 0}^{\infty }(-1)^k[\mathop {\text {tor}}\nolimits ^\Pi _k(J_i, M)] = \sum _{j \in I}\langle J_i e_j, M \rangle _\Gamma [S_j], \end{aligned}$$

(3.8)

of $\hat{K}(\Pi \text {-}\text {mod}_\Gamma ^+)$, where the second equality follows since $J_i^\phi = J_i$. When C is of finite type, we define the element $[J_i \overset{{\textbf {L}}}{\otimes }_\Pi M]$ of $\hat{K}(\Pi \text {-}\text {mod}_\Gamma ^+)$ by (3.8). Recalling the relation $E_i = \Pi /J_i$, we have $[J_ie_j] = [P_j] -\delta _{i,j}[E_i]$ for each $j \in I$, and hence

$$\begin{aligned}{}[J_i \overset{{\textbf {L}}}{\otimes }_\Pi M] = [M] - \langle E_i, M \rangle _\Gamma [S_i]. \end{aligned}$$

Sending this equality by the isomorphism $\chi _\ell$ in Theorem 3.14, we get

$$\begin{aligned} \chi _\ell [J_i \overset{{\textbf {L}}}{\otimes }_\Pi M] = \chi _\ell [M]-(\Psi \alpha _i^\vee , \chi _\ell [M])_\Gamma \alpha _i. \end{aligned}$$

(3.9)

In particular, we obtain the following analogue of Amiot et al. [1, Proposition 2.10].

Lemma 3.18

When C is of infinite type, we have

$$\begin{aligned} \chi _\ell [J_i \overset{{\textbf {L}}}{\otimes }_\Pi M] = T_i \chi _\ell [M] \qquad \text {for any } M \in \Pi \text {-}\text {mod}_\Gamma ^+ \text { and } i \in I. \end{aligned}$$

Proof

When C is of infinite type, we have $\Psi = \text {id}$ by definition. Thus, the Eq. (3.9) coincides with the defining Eq. (2.5) of $T_i$ in this case. $\square$

Proof of Proposition 2.5

When C is of finite type, we can reduce the proof to the case of affine type since the collection $\{ T_i \}_{i \in I}$ can be extended to the collection $\{T_i\}_{i \in I\cup \{0\}}$ 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 $\{ T_i \}_{i \in I}$ follow from Lemma 3.18 and the fact that the ideals $\{J_i\}_{i \in I}$ satisfy the braid relations with respect to multiplication, which is due to Fu and Geng [13, Theorem 4.7]. For example, when $c_{ij}c_{ji}=1$, we have

$$\begin{aligned}{} & {} J_i \overset{{{\textbf {L}}}}{\otimes }_\Pi J_j \overset{{{\textbf {L}}}}{\otimes }_\Pi J_i \simeq J_i \otimes _\Pi J_j \otimes _\Pi J_i \simeq J_iJ_jJ_i \\{} & {} \quad = J_jJ_iJ_j \simeq J_j \otimes _\Pi J_i \otimes _\Pi J_j \simeq J_j \overset{{{\textbf {L}}}}{\otimes }_\Pi J_i \overset{{{\textbf {L}}}}{\otimes }_\Pi J_j, \end{aligned}$$

which implies the desired braid relation $T_iT_jT_i = T_jT_iT_j$. $\square$

Corollary 3.19

Let $M \in \Pi \text {-}\text {mod}_\Gamma ^+$ with $\mathop {\text {tor}}\nolimits _1^\Pi (J_i, M)=0$ for $i \in I$. We have

$$\begin{aligned} \chi _\ell [J_i \otimes _{\Pi } M] = T_i\chi _\ell [M]. \end{aligned}$$

Moreover, if we assume that M is locally free, so is $J_i \otimes _{\Pi } M$.

Proof

The first assertion is a direct consequence of Lemma 3.18. For C of infinite type, since projective dimension of $J_i$ is at most 1 by Theorem 3.8, our involution $\phi$ yields that $\mathop {\text {tor}}\nolimits _k^\Pi (J_i, M)=0$ also for $k \ge 2$. This shows $[J_i \otimes _{\Pi } M]= [J_i \overset{{\textbf {L}}}{\otimes }_{\Pi } M]$ when C is of infinite type. When C is of finite type, our assertion follows easily from the exact embedding to the corresponding untwisted affine type $\hat{\Pi }$. Namely, we have an isomorphism $J_i\otimes _{\Pi } M \simeq \hat{J}_i \otimes _{\hat{\Pi }} M$, where $\hat{J}_i \,{:=}\,\hat{\Pi }(1-e_i)\hat{\Pi }$. The last assertion is just an analogue of Geiss et al. [20, Proof of Proposition 9.4]. $\square$

Proof of Theorem 2.8 when C is of infinite type

Assume that C is of infinite type. Let $(i_k)_{k \in \mathbb {Z}_{>0}}$ be a sequence in I satisfying the condition (2) in Theorem 2.8. We have a filtration $\Pi = F_0 \supset F_1 \supset F_2 \supset \cdots$ of $(\Pi ,\Pi )$-bimodules given by $F_k \,{:=}\,J_{i_1}J_{i_2} \cdots J_{i_k}$. This filtration $\{F_k\}_{k \ge 0}$ is exhaustive, i.e., $\bigcap _{k \ge 0}F_k = 0$. Indeed, since the algebra $\Pi$ satisfies the condition $\text {(A)}$, its radical filtration $\{ R_k\}_{k \ge 0}$ as a right $\Pi$-module is exhaustive. Note that, for any right $\Pi$-module M and $i \in I$, the right module $M/MJ_i$ is the largest quotient of M such that $(M/MJ_i)e_j \ne 0$ for $j \ne i$. Thanks to this fact and our assumption on the sequence $(i_k)_{k \in \mathbb {Z}_{>0}}$, we can find for each $k > 0$ a large integer K such that $F_K \subset R_k$. Thus, we have $\bigcap _{k} F_k = \bigcap _k R_k = 0$. Moreover, by Murakami [33, Proposition 3.8], we have

$$\begin{aligned} F_{k-1}/F_{k} \simeq J_{i_1} J_{i_2}\cdots J_{i_{k-1}}\otimes _\Pi E_{i_k} \qquad \text {as } \Gamma \text {-graded left } \Pi \text {-modules} \end{aligned}$$

for each $k \ge 1$. Note that we have an equality $\mathop {\text {tor}}\nolimits _1^\Pi (J_{i_1}, J_{i_2}\cdots J_{i_{k-1}}\otimes _\Pi E_{i_k})=0$ by Murakami [33, Proof of Proposition 3.8]. This yields $\chi _\ell [J_{i_1}J_{i_2}\cdots J_{i_{k-1}}\otimes _\Pi E_{i_k}] = \kappa _\ell T_{i_1}, T_{i_2}\cdots T_{i_{k-1}} \alpha ^{\vee }_{i_k}$ inductively by Corollary 3.19.

The filtration $\{F_k\}_{k \ge 0}$ induces an exhaustive filtration $\{ F_k e_i \}_{k \ge 0}$ of the projective module $P_i$ such that

$$\begin{aligned} F_{k-1}e_i / F_{k}e_i \simeq {\left\{ \begin{array}{ll} J_{i_1} \cdots J_{i_{k-1}}\otimes _\Pi E_{i} &{}\quad \text {if } i_k = i, \\ 0 &{}\quad \text {otherwise} \end{array}\right. } \end{aligned}$$

for each $k \ge 1$. Therefore, in $\hat{K}(\Pi \text {-}\text {mod}_\Gamma ^+)$, we have

$$\begin{aligned}{}[P_i] = \sum _{k=1}^{\infty }[F_{k-1}e_i/F_{k}e_i] = \sum _{k :i_k = i}[J_{i_1} \cdots J_{i_{k-1}}\otimes _\Pi E_{i}]. \end{aligned}$$

Applying the isomorphism $\chi _\ell$ in Theorem 3.14 to this equality, we obtain

$$\begin{aligned} \varpi _i^\vee = \sum _{k :i_k = i}T_{i_1}\cdots T_{i_{k-1}} \alpha _i^\vee . \end{aligned}$$

This is rewritten as

$$\begin{aligned} \varpi _i = q^{-d_i}t \sum _{k :i_k = i}T_{i_1}\cdots T_{i_{k-1}} \alpha _i. \end{aligned}$$

Since $\widetilde{C}_{ij}(q,t,\underline{\mu }) = (\varpi _i^\vee , \varpi _j)_\Gamma$ by (2.3), we obtain the desired equality (2.6) from this. $\square$

Remark 3.20

In Iyama and Reiten [25], they proved that the ideal semigroup $\langle J_i \mid i \in I \rangle$ gives the set of isoclasses of classical tilting $\Pi$-modules for any symmetric affine type C with $D = \text {id}$. In our situation, our two-sided ideals are $\Gamma$-graded tilting objects whose $\Gamma$-graded endomorphism algebras are isomorphic to $\Pi$ when C is of infinite type by arguments in [7, 13]. In particular, our braid group symmetry on $\hat{K}(\Pi \text {-}\text {mod}_\Gamma ^+)$ is induced from auto-equivalences on the derived category (cf. [31, Section 2]).

4 Remarks

4.1 Comparison with Kimura–Pestun’s deformation

In their study of (fractional) quiver $\mathcal {W}$-algebras, Kimura–Pestun Kimura and Pestun [29] 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 $C(q,t,\underline{\mu })$.

Let Q be a quiver without loops and $d :Q_0 \rightarrow \mathbb {Z}_{>0}$ be a function. Following [29, Section 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 $i,j \in Q_0$. Let $C = (c_{ij})_{i,j \in I}$ 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:

$$\begin{aligned} c_{ij} = 2\delta _{i,j} - (d_j/d_{ij})\vert \{ e \in Q_1 \mid \{\text {s}(e), \text {t}(e)\} = \{ i,j \}\}\vert \quad \text {for any } i,j \in I. \end{aligned}$$

(4.1)

In this case, $D = \mathop {\text {diag}}\nolimits (d_i \mid i \in I)$ is a symmetrizer of C and we have

$$\begin{aligned} g_{ij} = \vert \{ e \in Q_1 \mid \{\text {s}(e), \text {t}(e)\} = \{ i,j \}\}\vert , \quad f_{ij} = d_j/d_{ij} \quad \text {when } i \sim j. \end{aligned}$$

See Sect. 2.1 for the definitions. For a given fractional quiver (Q, d) of type C, Kimura–Pestun introduced a matrix $C^{\text {KP}}= (C^{\text {KP}}_{ij})_{i,j \in I}$, whose (i, j)-entry $C^{\text {KP}}_{ij}$ is a Laurent polynomial in the formal parameters $q_1, q_2$ and $\mu _e$ for each $e \in Q_1$ given by

$$\begin{aligned} C^{\text {KP}}_{ij} \,{:=}\,\delta _{i,j} (1+q_1^{-d_i}q_2^{-1}) - \frac{1-q_1^{-d_j}}{1-q_1^{-d_{ij}}}\left( \sum _{e :i \rightarrow j}\mu _e^{-1} + \sum _{e :j \rightarrow i}\mu _e q_1^{-d_{ij}}q_2^{-1}\right) . \end{aligned}$$

(4.2)

The parameters $\mu _e$ are called mass-parameters. If we evaluate all the parameters to 1, the matrix $C^{\text {KP}}$ coincides with the GCM C by (4.1).

Now we fix a function $g :Q_1 \rightarrow \mathbb {Z}_{>0}$ whose restriction induces a bijection between $\{ e \in Q_1 \mid \{\text {s}(e), \text {t}(e)\} = \{ i,j \}\}$ and $\{ g \in \mathbb {Z} \mid 1 \le g \le g_{ij} \}$ for each $i,j \in I$ with $i \sim j$. Then consider the monomial transformation $\mathbb {Z}[q_1^{\pm 1},q_2^{\pm 1},\mu _e^{\pm 1} \mid e \in Q_1] \rightarrow \mathbb {Z}[\Gamma ]$ given by

$$\begin{aligned} q_1 \mapsto q^2, \qquad q_2 \mapsto t^{-2}, \qquad \mu _e \mapsto q^{d_{ij}}t^{-1}\mu _{ij}^{(g(e))}, \end{aligned}$$

(4.3)

where $i = \text {t}(e)$ and $j = \text {s}(e)$. Note that it induces an isomorphism if we formally add the square roots of $q_1$ and $q_2$. Under this monomial transformation, for any $i,j \in I$, we have

$$\begin{aligned} C^{\text {KP}}_{ij} \mapsto q^{-d_j}t\left( \delta _{i,j} (q^{d_i}t^{-1} + q^{-d_i}t) - \delta (i \sim j) [f_{ij}]_{q^{d_{ij}}} \sum _{g = 1}^{g_{ij}}\mu _{ij}^{(g)} \right) . \end{aligned}$$

(4.4)

Proposition 4.1

Under the monomial transformation (4.3), the matrix $C^{\text {KP}}$ corresponds to the matrix $C(q,t,\underline{\mu })q^{-D}t$ if and only if the following condition is satisfied:

$$\begin{aligned} \text {For any } i,j \in I \text { with } i \sim j, \text { we have } f_{ij} = 1 \text { or } f_{ji}=1. \end{aligned}$$

(4.5)

Proof

Compare (4.4) with (1.1) and note that we have $[f_{ij}]_{q^{d_{ij}}} = [f_{ij}]_{q^{d_i}}$ for any $i,j \in I$ with $i \sim j$ if and only if the condition (4.5) is satisfied. $\square$

Example 4.2

If we take our GCM C and its symmetrizer D as

$$\begin{aligned} C = \begin{pmatrix}2&{}\quad -6\\ -9&{}\quad 2\end{pmatrix} \quad \text {and} \quad D=\mathop {\text {diag}}\nolimits (3, 2), \end{aligned}$$

not satisfying (4.5), then the image of $C^{\text {KP}}$ under (4.3) is

$$\begin{aligned} \begin{pmatrix} 1+q^{-6}t^{2} &{}\quad -(q^{-1}+q^{-3})t(\mu _{12}^{(1)}+\mu _{12}^{(2)}+\mu _{12}^{(3)}) \\ -(q^{-1}+q^{-3}+q^{-5})t(\mu _{21}^{(1)}+\mu _{21}^{(2)}+\mu _{21}^{(3)}) &{}\quad 1+q^{-4}t^2 \end{pmatrix}, \end{aligned}$$

which is different from

$$\begin{aligned}{} & {} C(q, t, \underline{\mu }) q^{-D}t \\{} & {} \quad = \begin{pmatrix} 1+q^{-6}t^2 &{}\quad -(q + q^{-5})t(\mu _{12}^{(1)}+\mu _{12}^{(2)}+\mu _{12}^{(3)})\\ -(q + q^{-3} + q^{-7})t(\mu _{21}^{(1)}+\mu _{21}^{(2)}+\mu _{21}^{(3)}) &{}\quad 1 + q^{-4}t^2 \end{pmatrix}. \end{aligned}$$

Remark 4.3

The condition (4.5) is satisfied for all finite and affine types. It is also satisfied when C is symmetric (i.e., ${}^\texttt{t}C = C$). This (4.5) also appears in Nakajima and Weekes [36, Section C(iv)] as a condition for two possible mathematical definitions of Coulomb branches of quiver gauge theories with symmetrizers to coincide with each other as schemes.

Remark 4.4

When (4.5) is satisfied, we can assure that the evaluation at $t = 1$ makes sense in the inversion formulas. More precisely, assuming (4.5), we see that the matrix X in (2.1) is written in the form $X = q^{-1}X'$ with $X'$ being a $\mathbb {Z}[\underline{\mu }^\mathbb {Z}][q^{-1},t]$-valued matrix (see the proof of Fujita and Oh [16, Lemma 4.3]), and hence we have $\widetilde{C}_{ij}(q,t,\underline{\mu }) \in \mathbb {Z}[\underline{\mu }^\mathbb {Z}][q^{-1},t][\![(q^{-1}t)]\!]$ for any $i,j \in I$. Thus, under (4.5), the evaluation at $t=1$ gives a well-defined element $\widetilde{C}_{ij}(q,1,\underline{\mu })$ of $\mathbb {Z}[\underline{\mu }^{\mathbb {Z}}][\![q^{-1}]\!]$.

4.2 Universality of the grading

In this subsection, we briefly explain how one can think that our grading (3.1) on the algebra $\widetilde{\Pi }$ is universal. It is stated as follows.

We keep the notation in Sect. 3.3. Let $\widetilde{G}$ be the (multiplicative) abelian group generated by the finite number of formal symbols $\{[a] \mid a \in \widetilde{Q}_1 \}$ subject to the relations

$$\begin{aligned}{}[\alpha _{i_1j_1}^{(g_1)}] [\alpha _{j_1i_1}^{(g_1)}][\varepsilon _{i_1}]^{f_{i_1j_1}} = [\alpha _{i_2j_2}^{(g_2)}] [\alpha _{j_2i_2}^{(g_2)}][\varepsilon _{i_2}]^{f_{i_2j_2}} \end{aligned}$$

(4.6)

for any $i_k,j_k \in I$ with $i_k \sim j_k$ and $1 \le g_k \le g_{i_kj_k}$ ($k = 1,2$). Let $\widetilde{G}\twoheadrightarrow \widetilde{G}_f$ be the quotient by the torsion subgroup. By construction, for any free abelian group G, giving a homomorphism $\deg :\widetilde{G}_f \rightarrow G$ is equivalent to giving $\widetilde{Q}$ a structure of G-graded quiver $\deg :\widetilde{Q}_1 \rightarrow G$ such that the potential $W_\Omega$ is homogeneous. In this sense, we can say that the tautological map $\widetilde{Q}_1 \rightarrow \widetilde{G}_f$ gives a universal grading on the algebra $\widetilde{\Pi }$.

Now recall our fixed symmetrizer $D = \mathop {\text {diag}}\nolimits (d_i \mid i \in I)$ and set $d \,{:=}\,\gcd (d_i \mid i \in I)$. Let $\Gamma ' \subset \Gamma$ be the subgroup generated by $\{ \deg (a) \mid a \in \widetilde{Q}_1\}$. Note that $\Gamma '$ is a free abelian group with a basis $\{q^{2d}, t^{2}\} \cup \{q^{-d_if_{ij}}t\mu ^{(g)}_{ij}\mid (i,j) \in \Omega , 1 \le g \le g_{ij} \}$.

Proposition 4.5

The degree map (3.1) gives an isomorphism $\deg :\widetilde{G}_f \simeq \Gamma '$.

Proof

Choose integers $\{a_{i}\}_{i \in I}$ satisfying $\sum _{i \in I}a_i d_i = d$. Let e and w be the elements of $\widetilde{G}_f$ given by $e \,{:=}\,\prod _{i \in I}[\varepsilon _i]^{a_i}$ and $w = [\alpha _{ij}^{(g)}] [\alpha _{ji}^{(g)}][\varepsilon _{i}]^{f_{ij}}$ respectively. Note that w does not depend on the choice of $i,j \in I$ with $i \sim j$ and $1 \le g \le g_{ij}$ by (4.6). We define a group homomorphism $\iota :\Gamma ' \rightarrow \widetilde{G}_f$ by $\iota (q^{2d}) \,{:=}\,e$, $\iota (t^2) \,{:=}\,w$ and $\iota (q^{-d_if_{ij}}t\mu ^{(g)}_{ij})\,{:=}\,[\alpha ^{(g)}_{ij}]$ for $(i,j) \in \Omega$, $1 \le g \le g_{ij}$. It is easy to see $\deg \circ \iota = \text {id}$. Now we shall prove $\iota \circ \deg = \text {id}$. First, we observe that $[\varepsilon _i]^{f_{ij}} = [\varepsilon _j]^{f_{ji}}$ when $i \sim j$ by (4.6). Since $f_{ij} = d_j / d_{ij}$, we have

$$\begin{aligned}{}[\varepsilon _i]^{r/d_i} = ([\varepsilon _i]^{f_{ij}})^{rd_{ij}/d_id_j} = ([\varepsilon _j]^{f_{ji}})^{rd_{ij}/d_id_j} = [\varepsilon _j]^{r/d_j} \end{aligned}$$

for any $i,j \in I$ with $i \sim j$. Since C is assumed to be irreducible, it follows that $[\varepsilon _i]^{r/d_i} = [\varepsilon _j]^{r/d_j}$ for any $i, j \in I$. Furthermore, since $\widetilde{G}_f$ is torsion-free, we get

$$\begin{aligned}{}[\varepsilon _i]^{d_j/d} = [\varepsilon _j]^{d_i/d} \qquad \text {for any } i, j \in I. \end{aligned}$$

(4.7)

Using (4.7), for each $i \in I$, we find

$$\begin{aligned} \iota ( \deg [\varepsilon _i]) = e^{d_i/d} = \prod _{j \in I}[\varepsilon _j]^{a_jd_i/d} = \prod _{j \in I}[\varepsilon _i]^{a_j d_j /d} = [\varepsilon _i]. \end{aligned}$$

The equality $\iota (\deg [\alpha ^{(g)}_{ij}]) = [\alpha ^{(g)}_{ij}]$ is obvious. Thus we conclude that $\iota \circ \deg = \text {id}$ holds. $\square$

In particular, we have the isomorphism of group rings $\mathbb {Z}[\widetilde{G}_f] \simeq \mathbb {Z}[\Gamma ']$. Using the notation in the above proof, we consider the formal roots $e^{1/d}$ and $w^{1/2}$. Then we obtain the isomorphism $\mathbb {Z}[\widetilde{G}_f][e^{1/d}, w^{1/2}] \simeq \mathbb {Z}[\Gamma ].$ This means that our deformed GCM $C(q,t,\underline{\mu })$ can be specialized to any other deformation of C which arises from a grading of the quiver $\widetilde{Q}$ respecting the potential $W_\Omega$ (formally adding roots of deformation parameters if necessary).

4.3 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 Kashiwara–Oh [27] 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 [17, 37]. Let $I=\{1, \dots , n\}$. By definition, an F-species $(F_i, {}_iF_{j})$ over F consists of

• a finite dimensional skew-field $F_i$ over F for each $i \in I$;

• an $(F_i, F_j)$-bimodule ${}_i F_{j}$ for each $i, j \in I$ such that F acts centrally on ${}_i F_{j}$ and $\dim _{F}{}_i F_{j}$ is finite;

• There does not exist any sequence $i_1, \dots , i_l, i_{l+1}=i_1$ such that ${}_{i_k} F_{i_{k+1}}\ne 0$ for each $k=1, \dots , l$.

For ${}_iF_j\ne 0$, we write ${}_{F_i}({}_iF_j) \simeq F_i^{\oplus {-c_{ij}}}$ and $({}_iF_j)_{F_j} \simeq F_j^{\oplus {-c_{ji}}}$. If we put $c_{ii}=2$ and $c_{ij}=0$ for ${}_iF_j=0={}_jF_i$, the matrix $C\,{:=}\,({c_{ij}})_{i,j \in I}$ is clearly a GCM with left symmetrizer $D = \mathop {\text {diag}}\nolimits (\dim _F F_i \mid i\in I )$. We have an acyclic orientation $\Omega$ of this GCM determined by the conditions ${}_iF_j\ne 0$. Following our convention in Sect. 2.1, we write $\dim _F F_i = {d_i}$. For our F-species $(F_i, {}_iF_j)$, we set $S \,{:=}\,\prod _{i \in I} F_i$ and $B \,{:=}\,\bigoplus _{(i,j) \in \Omega } {}_iF_j$. Note that B is an (S, S)-bimodule. We define a finite dimensional hereditary algebra $T = {T(C, D, \Omega )}$ to be the tensor algebra $T \,{:=}\,T_S(B)$. Note that we use the same convention for $T(C, D, \Omega )$ as that in Geiß–Leclerc–Schröer [19], unlike our dual convention for the algebra $\Pi (\ell )$.

We can also define the preprojective algebra (see [11] for details). For $(i,j)\in \Omega$, there exists a $F_j$-basis $\{x_1, \dots , x_{{\vert c_{ji}\vert }}\}$ of ${}_iF_j$ and a $F_j$-basis $\{y_1, \dots , y_{{\vert c_{ji}\vert }}\}$ of $\mathop {\text {Hom}}\nolimits _{F_j}({}_iF_j, F_j)$ such that for every $x \in {}_iF_j$ we have $x = \sum _{i=1}^{{\vert c_{ji}\vert }} y_i(x)x_i.$ We have the canonical element $\texttt{c}_{ij} = \sum _{i=1}^{{\vert c_{ji}\vert }} x_i \otimes _{F_i} y_i \in {}_iF_j \otimes _{F_j} \mathop {\text{ Hom }}\nolimits _{F_i}({}_iF_j, F_i)$ which does not depend on our choice of basis $\{x_i\}$ and $\{y_j\}$. Letting ${}_jF_i\,{:=}\,\mathop {\text {Hom}}\nolimits _{F_j}({}_iF_j, F_j)$ for $(i,j)\in \Omega$, we can also define the similar canonical element $\texttt{c}_{ji}\in {}_jF_i \otimes _{F_i} {}_iF_j$. We put $\overline{B}\,{:=}\,\bigoplus _{(i,j)\in \Omega }({}_iF_j \oplus {}_jF_i)$, and define the preprojective algebra $\Pi _T=\Pi _T(\ell )$ of the algebra T as

$$\begin{aligned} T_S(\overline{B})/\langle \sum _{(i,j)\in \Omega }\text{ sgn}_{\Omega }(i,j)\texttt{c}_{ij}\rangle . \end{aligned}$$

Let $P^T_i$ (resp. $P^{\Pi _T}_i$) denote the indecomposable projective T-module (resp. $\Pi _T$-module) associated with i, and $\tau _T$ the Auslander–Reiten translation for (left) T-modules. Note that this algebra $\Pi _T$ satisfies $P^{\Pi _T}_i=\bigoplus _{k\ge 0} \tau _T^{-k} P^T_i$ by an argument on the preprojective component of the Auslander–Reiten quiver of T similar to Söderberg [38, Proposition 4.7]. Note that our F-species $(F_i, {}_iF_j)$ is nothing but a modulated graph associated with ${(C, D,\Omega )}$ in the sense of Dlab–Ringel [11], although we will work with these algebras along with a context of a deformation of C.

Although there is obviously no nontrivial $\mathbb {Z}$-grading on S by the fact $F_i$ is a finite dimensional skew-field, we can nevertheless endow T and $\Pi _T$ with a $t^{\mathbb {Z}}$-grading induced from their tensor algebra descriptions. Each element of ${}_iF_j$ has degree t. We remark that if we specifically choose a decomposition of each ${}_iF_j$ like $F(\!(\varepsilon )\!)$-species $\tilde{H}$ in Geiß et al. [22, Section 4.1] and define its preprojective algebra, then we can also endow these algebras with natural $\underline{\mu }^{\mathbb {Z}}$-gradings and homogeneous relations by using [11, Lemma 1.1]. But we only consider the $t^{\mathbb {Z}}$-grading here since our aim is to interpret the t-Cartan matrix. By our $t^{\mathbb {Z}}$-grading, our algebra $\Pi _T$ satisfies the condition (A) in Sect. 3.2 (with $\Bbbk = F$).

We have the following complex of t-graded modules for each simple module $F_i$:

Lemma 4.6

The complex

$$\begin{aligned} t^2 P^{\Pi _T}_i \xrightarrow {\psi ^{(i)}} \bigoplus _{j\sim i} (P_j^{\Pi _T})^{\oplus (-t{C_{ji}(1,t)})} \rightarrow P^{\Pi _T}_i \rightarrow F_i \rightarrow 0. \end{aligned}$$

(4.8)

is exact. Moreover, the followings hold.

• (1) When C is of infinite type, $\mathop {\text {Ker}}\nolimits \psi ^{(i)} =0$ for all $i \in I$. In particular, each object in $\Pi _T\text {-}\text {mod}_{t^\mathbb {Z}}^+$ has projective dimension at most 2.

• (2) When C is of finite type, we have $\mathop {\text {Ker}}\nolimits \psi ^{(i)} \cong t^h F_{i^*}$ for each $i \in I$.

Proof

The statement (1) is deduced from the Auslander–Reiten theory for T (e.g. [2, Proposition 7.8]). The statement (2) follows from Söderberg [38, Section 6]. Note that C is of finite type if and only if $\Pi _T$ is a self-injective finite dimensional algebra and its Nakayama permutation can be similarly computed as Theorem 3.8 by an analogue of Mizuno [32, Section 3] (see Remark 4.8). $\square$

Corollary 4.7

For any $i,j \in I$, the followings hold.

• (1) When C is of finite type, we have

  $$\begin{aligned} { d_i \widetilde{C}_{ij}(1,t) = \frac{t}{1-t^{2h}}\left( \dim _{t^{\mathbb {Z}}}(e_i P_j^{\Pi _T}) - t^h \dim _{t^{\mathbb {Z}}}(e_{i^*} P_j^{\Pi _T}) \right) .} \end{aligned}$$

• (2) When C is of infinite type, we have

  $$\begin{aligned}{ d_i\widetilde{C}_{ij}(1,t) = t\dim _{t^\mathbb {Z}}(e_i P_j^{\Pi _T}).} \end{aligned}$$

Here $\dim _{t^\mathbb {Z}}$ denotes the graded dimension of $t^\mathbb {Z}$-graded F-vector spaces.

Proof

The equality $[P_j^{\Pi _T}]=\sum _{i\in I} (\dim _{t^{\mathbb {Z}}}(e_iP_j^{\Pi _T}) / \dim _F F_i) [F_i]$ in $\hat{K}(\Pi _T\text {-}\text {mod}_{t^\mathbb {Z}})$ and an equality $\dim _{t^{\mathbb {Z}}} e_i \Pi _T \otimes _{\Pi _T} F_j = \delta _{ij} {d_i}$ immediately yield our assertion by Lemma 4.6 with arguments similar to the case of the generalized preprojective algebras in Sect. 3.5. $\square$

Remark 4.8

In the case of our algebra $\Pi _T$, the two-sided ideal $J_i\,{:=}\,\Pi _T(1-e_i)\Pi _T$ and the ideal semi-group $\langle J_1, \dots , J_n \rangle$ also gives the Weyl group symmetry on its module category analogously to Iyama and Reiten [25], Buan et al. [7], Mizuno [32] (see [2, Section 7.1]). Even if we consider the algebra $\Pi _T$ and $t^{\mathbb {Z}}$-homogeneous ideal $J_i$, we can also establish the similar braid group symmetry as Sect. 3.6 after the specialization $q\rightarrow 1$ and $\underline{\mu }\rightarrow 1$ by Lemma 4.6.

Remark 4.9

The algebra $\Pi _T$ is a Koszul algebra for non-finite types and $(h-2, h)$-almost Koszul algebras for finite types in the sense of Brenner et al. [6] with our $t^{\mathbb {Z}}$-gradings. Thus Corollary 4.7 might be interpreted in the context of Brenner et al. [6, Section 3.3].

As a by-product of this description, we have the following generalization of the formula in Hernandez and Leclerc [24, Proposition 2.1] and Fujita [14, Proposition 3.8] for any bipartite symmetrizable Kac–Moody type. For a t-series $f(t) = \sum _k f_k t^k \in \mathbb {Z}[\![t, t^{-1}]\!]$, we write $[f(t)]_k \,{:=}\,f_k$ for $k \in \mathbb {Z}$.

Proposition 4.10

Assume that C is bipartite and take a height function $\xi$ for C such that $\Omega _\xi = \Omega$ (see Sect. 2.5). Let $(F_i, {}_iF_j)$ be a modulated graph associated with ${(C, D,\Omega )}$ as above. Let $M \simeq \tau _T^{-k}P^T_i$ and $N \simeq \tau _T^{-l}P_j^T$ be any two indecomposable preprojective T-modules. When C is of infinite type, we have

$$\begin{aligned} \dim _F \mathop {\text {Ext}}\nolimits _T^1(M,N) = \left[ {d_i\widetilde{C}_{ij}(1,t)}\right] _{(\xi (i)+2k)-(\xi (j)+2l) -1}. \end{aligned}$$

(4.9)

When C is of finite type, the equality (4.9) still holds provided that

$$\begin{aligned} 1 \le (\xi (i)+2k)-(\xi (j)+2l) -1 \le h-1. \end{aligned}$$

(4.10)

Otherwise, we have $\mathop {\text {Ext}}\nolimits _T^1(M,N) = 0$.

Proof

We may deduce the assertion by a combinatorial thought using the formula (2.11) as in Hernandez and Leclerc [24] or Fujita [14]. But, here we shall give another proof using the algebra $\Pi _T$.

For any $t^\mathbb {Z}$-graded T-module M, we have a decomposition $M = \bigoplus _{u \in \mathbb {Z}} M^{[u]}$, where $M^{[u]} \,{:=}\,\bigoplus _{i \in I}e_iM_{u - \xi (i)}$. Note that each $M^{[u]}$ is an T-submodule of M, since $\xi$ is a height function satisfying $\Omega _\xi = \Omega$. We have the following isomorphism

$$\begin{aligned} {}_T(P_i^{\Pi _T})^{[u]} \cong {\left\{ \begin{array}{ll} \tau _T^{-k}P_i^{T} &{}\quad \text {if } u = \xi (i) + 2k \text { for } k \in \mathbb {Z}_{\ge 0}, \\ 0 &{}\quad \text {otherwise} \end{array}\right. } \end{aligned}$$

(4.11)

as (ungraded) T-modules. Now, we have for each $M \simeq \tau _T^{-k}P^T_i$ and $N \simeq \tau _T^{-l}P_j^T$

$$\begin{aligned} \dim _F \mathop {\text {Ext}}\nolimits _T^1(M,N)&= \dim _F \mathop {\text {Ext}}\nolimits _T^1(\tau _T^{-k}P^T_i, \tau _T^{-l}P^T_j) \\&= \dim _F e_j\tau _T^{(k-l-1)}P^T_i\\&\qquad \qquad \qquad \text {(cf. [3, Section IV 2.13])} \\&= \dim _F e_j(P_i^{\Pi _T})^{[\xi (i)+2(k-l-1)]}{} & {} (3.11) \\&= \dim _{F} (e_j{P}^{\Pi _T}_i)_{(\xi (i)+2k)-(\xi (j)+2l)-2}. \end{aligned}$$

When C is of infinite type, we deduce the desired Eq. (4.9) from Corollary 4.7 (2). When C is of finite type, we can find that $(e_j {P}^{\Pi _T}_i)_{(\xi (i)+2k)-(\xi (j)+2\,l)-2}$ is non-zero only if the condition (4.10) is satisfied by an analogue of Fujita and Murakami [15, Corollary 3.9]. When (4.10) is satisfied, we get (4.9) by Corollary 4.7 (1). $\square$

Remark 4.11

In Geiss et al. [20], they also introduced the 1-Iwanaga–Gorenstein algebra H over any field $\Bbbk$ associated with a GCM C, its symmetrizer D, and an orientation $\Omega$. The algebra H has quite similar features to our algebra T, and we can also show a version of Proposition 4.10 for the algebra H with $t^{\mathbb {Z}}$-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 Geiß et al. [19, Proposition 5.5]:

We keep the convention in Proposition 4.10. Let $X \simeq \tau _H^{-k}P^H_i$ and $Y \simeq \tau _H^{-l}P_j^H$ be any two indecomposable preprojective H-modules. Then we have

$$\begin{aligned} \dim _{\Bbbk } \mathop {\text {Ext}}\nolimits ^1_H(X, Y) = \dim _F \mathop {\text {Ext}}\nolimits ^1_T(M, N). \end{aligned}$$

Thanks to this common dimension property between the algebras H and T, Corollary 3.15 specializes to Corollary 4.7 after the specialization $q\rightarrow 1$ and $\underline{\mu }\rightarrow 1$ with Remark 3.4.

Remark 4.12

When the authors almost finished writing this paper, a preprint [26] 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 4.10 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 $\mathfrak {g}$ 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 $U_q(\mathfrak {g})$. We are interested in the $\mathbb {Z}_{\ge 0}$-valued invariant $\mathfrak {d}(S, S')$ 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 $\Omega$ of C, we have an affreal R-module $S_\Omega (\alpha )$ for each positive root $\alpha$ of $\mathfrak {g}$, called a cuspidal module. See Kashiwara and Oh [26] for details.

On the other hand, we have a generalization of the Gabriel theorem for F-species (see [9, 10, 37]). In particular, for each positive root $\alpha$ of $\mathfrak {g}$, there exists an indecomposable module $M_\Omega (\alpha )$ over the algebra $T = {T(C,D,\Omega )}$ satisfying $\sum _{i \in I} (\dim _{F_i} e_i M_{\Omega }(\alpha )) \alpha _i = \alpha$, uniquely up to isomorphism. Note that every indecomposable T-module is a preprojective module when C is of finite type.

Then, Kashiwara and Oh [26, Main Theorem] and Proposition 4.10 tell us that the equality

$$\begin{aligned} \mathfrak {d}\left( S_{{\Omega }^*}(\alpha ), S_{{\Omega }^*}(\beta )\right) = \dim _F \mathop {\text {Ext}}\nolimits _T^1(M_\Omega (\alpha ), M_\Omega (\beta )) + \dim _F \mathop {\text {Ext}}\nolimits _T^1(M_\Omega (\beta ), M_\Omega (\alpha ))\nonumber \\ \end{aligned}$$

(4.12)

holds for any positive roots $\alpha$ and $\beta$, where ${\Omega }^*$ denotes the orientation of C opposite to $\Omega$. In particular, (4.12) implies that the following three conditions are mutually equivalent for any positive roots $\alpha$ and $\beta$:

• The convolution product $S_{{\Omega }^*}(\alpha ) \circ S_{{\Omega }^*}(\beta )$ is simple;

• We have an isomorphism $S_{{\Omega }^*}(\alpha ) \circ S_{{\Omega }^*}(\beta ) \simeq S_{{\Omega }^*}(\beta ) \circ S_{{\Omega }^*}(\alpha )$ of R-modules;

• We have $\mathop {\text {Ext}}\nolimits _T^1(M_\Omega (\alpha ), M_\Omega (\beta )) = \mathop {\text {Ext}}\nolimits _T^1(M_\Omega (\beta ), M_\Omega (\alpha )) =0.$

Note that an analogous statement in the case of fundamental modules over the quantum loop algebra of type $\text {ADE}$ is obtained in Fujita [14].