^{1 2}

1 Introduction↩︎

Derived equivalence is a classical topic in the representation theory of algebras. Two derived equivalent algebras share similar homological properties. By [1], [2], a fundamental method for determining whether two algebras are derived equivalent is to find a suitable tilting complex and compute its endomorphism algebra. Although this approach always works in theory, in practice it often involves extensive calculations. Therefore, we generally hope that within special classes of algebras, derived equivalence can be characterized by certain complete invariants that are easy to verify.

In recent years, the study of certain classical finite-dimensional algebras using combinatorial structures and geometric models has become increasingly popular. For these algebras, derived equivalence can often be completely described by combinatorial or geometric data. For example, derived equivalence of gentle algebras can be determined by their geometric models [3]–[5], and derived equivalence of Brauer graph algebras (abbr. BGAs) can be determined by their ribbon graphs and multiplicity functions [6]–[8].

Although many deep results are known for well-structured symmetric algebras such as BGAs, analogous characterizations have long been unavailable for self-injective algebras in general.

In order to study self-injective algebras via a similar combinatorial approach, [9] generalized BGAs to fractional BGAs. In general, fractional BGAs are neither special biserial nor of tame representation type [10]. Consequently, in [11], the authors considered a subclass, namely fractional BGAs of type MS on which the Nakayama automorphism acts admissibly; these algebras are self-injective and special biserial. Here, “MS” stands for “multiserial and self-injective” [12]. In this article, for simplicity, we refer to such algebras as admissible fractional BGAs (abbr. AFBGAs). As in the case of BGAs, AFBGs are defined from combinatorial data — a ribbon graph \(\Gamma\) equipped with a (fractional) multiplicity function \(m=\frac{d}{val}\) (see Definition 2 for details) — called an admissible fractional Brauer graph (abbr. AFBG).

In fact, we will prove in Proposition 5 that AFBGAs are uniquely determined by their combinatorial data. We illustrate these concepts with the following example.

Similar to the case of gentle algebras [13] and BGAs [7], AFBGAs are expected to be closed under derived equivalence (which is left as a conjecture in [14]). In particular, the representation-finite part [11] and the tilting-discrete part [14] of such algebras are closed under derived equivalence.

Therefore, in this paper, we aim to provide some easily checkable invariants for derived equivalences within this class of algebras with combinatorial structures. The proofs of these invariants differ from the classical case of BGAs in [6], [7], because the existence of a non-trivial Nakayama automorphism makes the classical arguments more complicated (in some cases, they even fail to work). Since in [11] each AFBGA \(A\) is associated with a BGA \(A_{\mathrm{red}}\) (called the reduced form of \(A\)), our strategy is to use covering techniques on derived categories [15], which were used in [16] to classify representation-finite self-injective algebras. This allows us to construct derived equivalences in the BGA setting and, consequently, obtain combinatorial invariants as follows (since derived equivalence coincides with Morita equivalence for local algebras [17], we may restrict our discussion to the non-local case).

Note that in Remark 9 we point out that the above invariants are not complete, and in Remark 10 we propose a potential complete invariant for derived equivalence.

It is often observed that many results for gentle algebras also extend to BGAs, mainly because they are related via trivial extensions [18], [19]. In this paper, we also show that AFBGAs exhibit a similar connection with gentle algebras. Indeed, the relationships among AFBGAs, BGAs, and gentle algebras can be illustrated by the following diagram.

Outline. In Section 2, we recall some classical results on AFBGAs and prove that, up to isomorphism, AFBGAs are uniquely determined by their AFBGs. In Section 3, we construct the repetitive algebras and the \(r\)-fold trivial extensions of gentle algebras, and show that they are all AFBGAs, that is, they can be described by ribbon graphs together with multiplicity functions. In Section 4, we present invariants for derived equivalences between AFBGAs.

Acknowledgments↩︎

The author sincerely thanks Aaron Chan for his supervision at Nagoya University, many fruitful discussions and comments on this work. The author also thanks Pengyun Chen, Nengqun Li and Yuming Liu for helpful discussions. This work is supported by the China Scholarship Council (No. 202506040127).

Notation↩︎

Throughout we assume that \(k\) is an algebraically closed field and all algebras considered are \(k\)-algebras. Unless stated otherwise, all modules will be finitely generated left modules. Furthermore, we say that \(A\) is a bound quiver algebra, if \(A\cong kQ/I\), where \(Q\) is a finite quiver and \(I\) is an admissible ideal in the path algebra \(kQ\). We denote by \(s(p)\) the source vertex of a path \(p\) and by \(t(p)\) its terminus vertex. We will write paths from right to left, for example, \(p=\alpha_{n}\alpha_{n-1}\cdots\alpha_{1}\) is a path with starting arrow \(\alpha_{1}\) and ending arrow \(\alpha_{n}\). A path is called a cycle if \(s(p)=t(p)\). By abuse of notation we sometimes view an element in \(kQ\) as an element in the quotient \(kQ/I\) if no confusion can arise.

In this paper, we study the indecomposable \(A\)-modules \(M\) via their Loewy structure, which is represented by a diagram where the \(i\)-th row corresponds to the simple summands of the completely reducible module \(\mathrm{rad}^{i-1}(A)M/\mathrm{rad}^{i}(A)M\) with \(\mathrm{rad}(A)\) the Jacobson radical of \(A\). Each number in the diagram denotes a distinct simple module in \(A\). For further details, see for example in [20].

Recall that for an algebra \(A\), the derived category \(\mathcal{D}(A)\) is the localization of homotopy category \(\mathcal{K}(A)\) by inverting quasi-isomorphisms. We say that two algebras are derived equivalent if their derived categories are equivalent as triangulated categories.

A \(k\)-algebra \(A\) is called self-injective if \(A_A\) is an injective \(A\)-module; and symmetric if \(_AA_A\cong DA\) as \(A\)-\(A\)-bimodules. Indeed, symmetric algebras are naturally self-injective (see for example in [21]).

A \(k\)-algebra \(A\) is special biserial if it is isomorphic to an algebra of the form \(kQ/I\) where \(kQ\) is a path algebra and \(I\) is an admissible ideal such that the following properties hold.

(1) At every vertex \(i\) in \(Q\), there are at most two arrows starting at \(i\) and there are at most two arrows ending at \(i\).

(2) For every arrow \(\alpha\) in \(Q\), there exists at most one arrow \(\beta\) such that \(\beta\alpha\notin I\) and there exists at most one arrow \(\gamma\) such that \(\alpha\gamma\notin I\).

2 Admissible fractional Brauer graph algebras↩︎

2.1 Ribbon graphs↩︎

Ribbon graphs combinatorially encode the structure of oriented surfaces with boundary (see for example in [8]). A key feature of ribbon graphs is the cyclic ordering of (half-)edges at each vertex, which captures the orientation data of the underlying surface. We begin this section by recalling their formal definition.

For a ribbon graph \(\Gamma=(V,H,s,\iota,\rho)\), we introduce the following notation. For each half-edge \(h \in H\), we write \[h^+ := \rho(h) \quad \text{and} \quad h^- := \rho^{-1}(h)\] for the successor and predecessor of \(h\), respectively. We denote by \[\bar{h} := \{h, \iota(h)\}\] the edge associated with \(h\) in the underlying combinatorial graph of \(\Gamma\). For each vertex \(\mathsf{v}\in V\), the valency of \(\mathsf{v}\) is defined by \[val(\mathsf{v}) := \bigl|\{h \in H \mid s(h) = \mathsf{v}\}\bigr|.\] In particular, a loop (that is, an edge \(\{h, \iota(h)\}\) with \(s(h) = s(\iota(h))\)) contributes twice to \(val(\mathsf{v})\).

Unless stated otherwise, we will assume that \(\Gamma\) is connected, i.e. its underlying graph is connected.

2.2 Admissible fractional Brauer graph algebras↩︎

In this section, we review some basic knowledge about admissible fractional Brauer graph algebras, which is constructed in [9].

We often omit \(d\) from the notation and simply refer to \(\Gamma\) as an AFBG when no confusion arises. For a vertex \(\mathsf{v}\), we define the multiplicity of \(\mathsf{v}\) (also called the fractional degree in [9]) to be the rational number \[m(\mathsf{v}) := \frac{d(\mathsf{v})}{val(\mathsf{v})}.\] It follows immediately from the definition that an AFBG is a (classical) Brauer graph (abbr. BG) (for an explicit definition, see for example in [8], [22]) if and only if the multiplicity of every vertex is an integer. A vertex \(\mathsf{v}\) is called truncated if \(d(\mathsf{v})=1\). Denote by \(\nu\) the map \[\nu: H \longrightarrow H, \qquad h \longmapsto \rho^{d(s(h))}h,\] which is called the Nakayama automorphism of an AFBG \((\Gamma,d)\).

To each AFBG \((\Gamma,d)\), one can associate a (\(2\)-regular) quiver \(Q=Q_\Gamma\) and an ideal of relations \(I=I_{\Gamma,d}\) in the path algebra \(kQ\) as follows.

(1) The vertices of \(Q\) correspond to the edges of \(\Gamma\) and for every \(h\in H\), there is an arrow \(\alpha_h:\bar{h}\rightarrow\overline{h^+}\). The assignment \(\alpha_h\mapsto \alpha_{h^+}\) defines a permutation \(\rho=\rho_\Gamma\) of the arrows of \(Q\) whose orbits are in bijection with vertices in \(\Gamma\). Hence every arrow \(\alpha\) defines a cycle \[C_\alpha=\alpha\rho(\alpha)\cdots\rho^l(\alpha)\] where \(l+1\) denotes the cardinality of the \(\pi\)-orbit of \(\alpha\). Every vertex of \(Q\) is the starting point of at most two cycles of form \(C_\alpha\). If \(\alpha=\alpha_h\), set \(d(\alpha):=d(s(h))\).

(2) The ideal \(I\) is generated by the following set of relations:

(i) $$\alpha\rho(\alpha)\cdots\rho^{d(\alpha)-1}(\alpha)=\beta\rho(\beta)\cdots\rho^{d(\beta)-1}(\beta),$$
    where $\alpha,\beta\in Q_1$ and $s(\alpha)=s(\beta)$, that is
    $\alpha$ and $\beta$ start at the same edge of $\Gamma$.

(ii) $$\alpha\beta=0,$$ where $\alpha,\beta\in Q_1$ are composable
     and $\rho(\alpha)\neq \beta$;

Note that \(A\) is indecomposable as a ring if and only if \(\Gamma\) is connected, and \(A\) is a (classical) Brauer graph algebra (abbr. BGA) if and only if \(m(\mathsf{v})\) is an integer for every vertex \(\mathsf{v}\in V(\Gamma)\). In fact, BGAs are coincide with symmetric special biserial algebras [19]. Moreover, AFBGAs satisfy the following elementary properties; see [9] if one needs details.

2.3 Reduced forms↩︎

For each AFBG \((\Gamma = (V, H, s, \iota, \rho), d)\), define its reduced form to be \[(\Gamma_{\mathrm{red}},d)=(\Gamma / \langle \nu \rangle,d) := ((V, H', s', \iota', \rho'),d),\] where \(V\) coincides with the same vertex set of \(\Gamma\) and \(d\) is the same function defined on \(V\), and

1. \(H' = H / \langle \nu \rangle = \{ [h] \mid h \in H \}\) is the set of \(\langle \nu \rangle\)-orbits in \(H\);

2. \(s'([h]) = s(h)\);

3. \(\iota'([h]) = [\iota(h)]\);

4. \(\rho'([h]) = [\rho(h)]\).

In fact, \((\Gamma_{\mathrm{red}},d)\) is a Brauer graph (see for example [11]).

Let \(A\) denote the AFBGA associated with \((\Gamma,d)\). We define \(A_{\mathrm{red}}\) to be the BGA associated with \((\Gamma_{\mathrm{red}},d)\), which we also call the reduced form of \(A\).

We note that the reduced form of the algebra \(A\) in Example 1 is the Brauer graph algebra associated with the ribbon graph consisting of a single edge connecting two distinct vertices, both of multiplicity two.

For the following statement, we recall that a Brauer graph is called a Brauer tree if its associated ribbon graph \(\Gamma\) is a tree and the multiplicity function \(m\) assigns the value \(1\) to all vertices, except possibly for a single vertex \(\mathsf{v}\), called the exceptional vertex, for which \(m(\mathsf{v})\) is referred to as the exceptional multiplicity. The BGA associated with a Brauer tree is called a Brauer tree algebra. In fact, a BGA is representation-finite if and only if it is a Brauer tree algebra (see for example [22]).

It is shown in [11] that an AFBGA is representation-finite (resp.domestic) if and only if its reduced form is representation-finite (resp.domestic). Moreover, in the representation-finite case, we have the following theorem.

2.4 Determination by admissible fractional Brauer graphs↩︎

In this section, we prove that, similarly to the case of BGA [7], an AFBGA is uniquely determined by its associated AFBG.

3 Repetitive algebras and \(r\)-fold trivial extensions of gentle algebras↩︎

We begin by recall some basic definitions in representation theory.

Since a gentle algebra \(A=kQ/I\) is monomial, there exists a set \(\mathcal{M}\) of maximal paths in \(A\), that is, for each \(p \in \mathcal{M}\) and each arrow \(\alpha \in Q_1\), one has \(\alpha p = 0 = p \alpha\) in \(A\). The set \(\mathcal{M}\) is uniquely determined by \(A = kQ/I\), since, by [23], the ideal \(I\) admits a unique minimal generating set \(\mathcal{G}\) consisting of paths. Then there exists a ribbon graph \(\Gamma_A\) (see [19], [24]) whose vertices correspond to \[\overline{\mathcal{M}} := \mathcal{M} \cup \left\{\, i \in Q_0 \;\middle|\; \begin{array}{l} i \text{ is a source with a single arrow starting at } i, \\ \text{or } i \text{ is a sink with a single arrow ending at } i, \\ \text{or there is a single arrow } \alpha \text{ ending at } i \\ \text{and a single arrow } \beta \text{ starting at } i \text{ with } \beta\alpha \notin I \end{array} \right\},\] and whose edges correspond to the vertices of \(Q\). An edge \(e\) is incident to a vertex \(\mathsf{v}\in \overline{\mathcal{M}}\) in \(\Gamma_A\) if the corresponding path \(p_\mathsf{v}\) passes through \(e \in Q_0\). The cyclic order around each vertex is naturally induced by the corresponding path.

It is well known that the BGA associated with \((\Gamma_A,{val})\) is the trivial extension of \(A\) [19]. Moreover, a cutting set \(D\) of a BGA is defined by choosing exactly one arrow from each special cycle (up to cyclic permutation) \(C_\alpha\) (see Subsection 2.2). Furthermore, each gentle algebra corresponds bijectively to a BGA with BG \((\Gamma, {val})\), together with a cutting set \(D\) (see for example in [25]).

We omit some basic material on covering theory and refer the reader to [26]–[28]. Now we begin by constructing the \(r\)-covering of a BGA with multiplicity one. Let \((\Gamma = (V, H, s, \iota, \rho), d=val)\) be a BG with multiplicity one, meaning that for every vertex \(\mathsf{v}\in V(\Gamma)\), the value \(d(\mathsf{v})={val}(\mathsf{v})\). Denote by \(A = kQ/I\) the Brauer graph algebra associated with \((\Gamma,val)\). For any cyclic \(\langle \rho \rangle\)-orbit in \(H\), choose an angle from \(h_n\) to \(h_1\) (so that \(\rho(h_n) = h_1\)), and fix an ordering \(\{h_1, \dots, h_n\}\) satisfying

• \(\rho(h_i) = h_{i+1} \quad (1 \le i \le n-1)\);

• \(h_i\neq h_j\), for all \(i\neq j\).

Such a selection of angles determines a cutting set \(D\) of the quiver \(Q\). We call a \(\langle \rho \rangle\)-orbit \(\{h_1, \dots, h_n\}\) a \(\langle \rho \rangle\)-orbit with respect to \(D\). We define two new ribbon graphs associated with \(\Gamma\) and \(D\).

(1) The infinite covering graph \(\Gamma_D^{\mathbb{Z}}\) is defined by \[\Gamma_D^{\mathbb{Z}} = \bigl(V, \bigsqcup_{j \in \mathbb{Z}} H^{(j)}, s, \iota, \rho \bigr),\] where \(H^{(j)} = H\) for each \(j \in \mathbb{Z}\), and elements of \(H^{(j)}\) are denoted by \(h^{(j)}\) for \(h \in H\).

The structure maps are given by \[s(h^{(j)}) = s(h), \qquad \iota(h^{(j)}) = \iota(h)^{(j)}.\] For each \(\langle \rho \rangle\)-orbit \(\{h_1, \dots, h_n\}\) with respect to \(D\) in \(H\), define the permutation \(\rho\) on \(\bigsqcup_{j \in \mathbb{Z}} H^{(j)}\) by \[\rho(h_i^{(j)}) = h_{i+1}^{(j)} \quad (1 \le i \le n-1), \qquad \rho(h_n^{(j)}) = h_1^{(j+1)},\] for all \(j \in \mathbb{Z}\).

(2) For a positive integer \(r\), the finite graph \(\Gamma_D^{(r)}\) is defined as \[\Gamma_D^{(r)} = \bigl(V, \bigsqcup_{j \in \mathbb{Z}/r\mathbb{Z}} H^{(j)}, s, \iota, \rho \bigr),\] where \(H^{(j)} = H\) for each \(j \in \mathbb{Z}/r\mathbb{Z}\).

The maps \(s\), \(\iota\), and \(\rho\) are defined in the same way as above, namely, \[s(h^{(j)}) = s(h), \qquad \iota(h^{(j)}) = \iota(h)^{(j)},\] and for each \(\langle \rho \rangle\)-orbit \(\{h_1, \dots, h_n\}\) with respect to \(D\) in \(H\), \[\rho(h_i^{(j)}) = h_{i+1}^{(j)} \quad (1 \le i \le n-1), \qquad \rho(h_n^{(j)}) = h_1^{(j+1)},\] for all \(j \in \mathbb{Z}/r\mathbb{Z}\).

Then we have the following proposition.

Now we show the main theorem of this section.

We end this section with some examples.

4 Invariants under derived equivalences↩︎

In the case of BGAs, [7] and [8] provide a complete derived invariant characterizing derived equivalences between BGAs. In this section, we study invariants under derived equivalences for AFBGAs. The situation is analogous to that of BGAs.

By [17], two local algebras are derived equivalent if and only if they are Morita equivalent. By Proposition 5, AFBGAs are completely determined by their associated AFBGs, except for two exceptional local cases. Therefore, we restrict our discussion to the non-local case. We also remark that the following theorem also holds for infinite-dimensional AFBGAs.

We end this section with the following remarks.

References↩︎