From Brauer graph algebras to biserial weighted surface algebras
Abstract
We prove that the class of Brauer graph algebras coincides with the class of indecomposable idempotent algebras of biserial weighted surface algebras. These algebras are associated with triangulated surfaces with arbitrarily oriented triangles, investigated recently in Erdmann and Skowroński (J Algebra 505:490–558, 2018, Algebras of generalized dihedral type, Preprint. arXiv:1706.00688, 2017). Moreover, we prove that Brauer graph algebras are idempotent algebras of periodic weighted surface algebras, investigated in Erdmann and Skowroński (Algebras of generalized quaternion type, Preprint. arXiv:1710.09640, 2017).
Keywords
Brauer graph algebra Weighted surface algebra Biserial weighted surface algebra Symmetric algebra Special biserial algebra Tame algebra Periodic algebra Quiver combinatoricsMathematics Subject Classification
05E99 16G20 16G70 20C201 Introduction and the main results
Throughout this paper, K will denote a fixed algebraically closed field. By an algebra, we mean an associative, finitedimensional Kalgebra with an identity. For an algebra A, we denote by \({\text {mod}}A\) the category of finitedimensional right Amodules and by D the standard duality \({\text {Hom}}_K(,K)\) on \({\text {mod}}A\). An algebra A is called selfinjective if \(A_A\) is an injective module, or equivalently, the projective modules in \({\text {mod}}A\) are injective. Two selfinjective algebras A and B are said to be socle equivalent if the quotient algebras \(A / {\text {soc}}(A)\) and \(B / {\text {soc}}(B)\) are isomorphic. Symmetric algebras are an important class of selfinjective algebras. An algebra A is symmetric if there exists an associative, nondegenerate, symmetric, Kbilinear form \((,): A \times A \rightarrow K\). Classical examples of symmetric algebras include, in particular, blocks of group algebras of finite groups and Hecke algebras of finite Coxeter groups. In fact, any algebra A is the quotient algebra of its trivial extension algebra \({\text {T}}(A) = A < imes D(A)\), which is a symmetric algebra. By general theory, if e is an idempotent of a symmetric algebra A, then the idempotent algebra eAe also is a symmetric algebra.
Brauer graph algebras play a prominent role in the representation theory of tame symmetric algebras. Originally, R. Brauer introduced the Brauer tree, which led to the description of blocks of group algebras of finite groups of finite representation type, and they are the basis for their classification up to Morita equivalence [10, 25, 29], see also [2]. Relaxing the condition on the characteristic of the field, one gets Brauer tree algebras, and these occurred in the Morita equivalence classification of symmetric algebras of Dynkin type \(\mathbb {A}_n\) [22, 35]. If one allows arbitrary multiplicities, and also an arbitrary graph instead of just a tree, one obtains Brauer graph algebras. These occurred in the classification of symmetric algebras of Euclidean type \(\widetilde{\mathbb {A}}_n\) [7]. It was shown in [36] (see also [37]) that the class of Brauer graph algebras coincides with the class of symmetric special biserial algebras. Symmetric special biserial algebras occurred also in the Gelfand–Ponomarev classification of singular Harish–Chandra modules over the Lorentz group [23], and as well in the context of restricted Lie algebras, or more generally infinitesimal group schemes, [20, 21], and in classifications of tame Hecke algebras [3, 4, 14]. There are also results on derived equivalence classifications of Brauer graph algebras, and on the connection to Jacobian algebras of quivers with potential, we refer to [1, 11, 26, 31, 32, 34, 37].

the vertices in \(Q_{\varGamma }\) are the edges of \(\varGamma \);

there is an arrow \(i \rightarrow j\) in \(Q_{\varGamma }\) if and only if j is the consecutive edge of i in the cyclic ordering of edges adjacent to a vertex v of \(\varGamma \).
 (1)
all paths \(\alpha \beta \) of length 2 in \(Q_{\varGamma }\) which are not subpaths of \(C_{\alpha }\),
 (2)
\(C_{\alpha }^{e(v(\alpha ))}  C_{\bar{\alpha }}^{e(v(\bar{\alpha }))}\), for all arrows \(\alpha \) of \(Q_{\varGamma }\).
Algebras of generalized dihedral type (see [18, Theorem 1]), which contain blocks with dihedral defect groups, turned out to be (up to socle deformation) idempotent algebras of biserial weighted surface algebras, for very specific idempotents. Biserial weighted surface algebras belong to the class of Brauer graph algebras. It is therefore a natural question to ask which other Brauer graph algebras occur as idempotent algebras of biserial weighted surface algebras. This is answered by our first main result.
Theorem 1
 (i)
A is a Brauer graph algebra.
 (ii)
A is isomorphic to the idempotent algebra eBe for a biserial weighted surface algebra B and an idempotent e of B.
The main ingredient for this is Theorem 4.1. This gives a canonical construction, which we call \(*\)construction. A byproduct of the proof of Theorem 1 is the following fact.
Corollary 2
Let A be a Brauer graph algebra over an algebraically closed field K. Then, A is isomorphic to the idempotent algebra eBe of a biserial weighted surface algebra \(B = B(S,\vec {T},m_{\bullet })\), for a surface S without boundary, a triangulation T of S without selffolded triangles, and an idempotent e of B.
Moreover, we can adapt the \(*\)construction to algebras socle equivalent to Brauer graph algebras and prove an analog for the main part of Theorem 1:
Theorem 3
 (i)
\({\text {char}}(K)=2\), and
 (ii)
A is isomorphic to an idempotent algebra \(\bar{e} \bar{B} \bar{e}\), where \(\bar{B}\) is a socle deformed biserial weighted surface algebra \(\bar{B} = B(S,\vec {T},m_{\bullet },b_{\bullet })\). Here, S is a surface with boundary, T is a triangulation of S without selffolded triangles, and \(b_{\bullet }\) is a border function.
Recall that an algebra A is called periodic if it is periodic with respect to action of the syzygy operator \(\varOmega _{A^e}\) in the module category \({\text {mod}}A^e\), where \(A^e = A^{{\text {op}}} \otimes _K A\) is its enveloping algebra. If A is a periodic algebra of period n, then all indecomposable nonprojective right Amodules are periodic of period dividing n, with respect to the syzygy operator \(\varOmega _A\) in \({\text {mod}}A\). Periodic algebras are selfinjective and have connections with group theory, topology, singularity theory, and cluster algebras. In [17] and [19], we introduced and studied weighted surface algebras \(\varLambda (S, \vec {T}, m_{\bullet },c_{\bullet })\), which are tame, symmetric, and we showed that they are (with one exception) periodic algebras of period 4. They are defined by the quiver \(Q(S, \vec {T})\) and explicitly given relations, depending on a weight function \(m_{\bullet }\) and a parameter function \(c_{\bullet }\) (see Sect. 6). Most biserial weighted surface algebras occur as geometric degenerations of these periodic weighted surface algebras.
Our third main result connects Brauer graph algebras with a large class of periodic weighted surface algebras.
Theorem 4
Let A be a Brauer graph algebra over an algebraically closed field K. Then, A is isomorphic to an idempotent algebra \(e \varLambda e\) of a periodic weighted surface algebra \(\varLambda = \varLambda (S,\vec {T},m_{\bullet },c_{\bullet })\), for a surface S without boundary, a triangulation T of S without selffolded triangles, and an idempotent e of \(\varLambda \).
There are many idempotent algebras of weighted surface algebras which are neither Brauer graph algebras nor periodic algebras. We give an example at the end of Sect. 6.
This paper is organized as follows. In Sect. 2, we recall basic facts on special biserial algebras and show that Brauer graph algebras, symmetric special biserial algebras, and symmetric algebras associated with weighted biserial quivers are essentially the same. In Sect. 3, we introduce biserial weighted surface algebras and present their basic properties. In Sect. 4, we prove Theorem 1. This contains an algorithmic construction which may be of independent interest. Sections 5 and 6 contain the proofs of Theorems 3 and 4 and related material. In Sect. 7, we present a diagram showing the relations between the main classes of algebras occurring in the paper.
For general background on the relevant representation theory, we refer to the books [5, 13, 38, 40], and we refer to [13, 15] for the representation theory of arbitrary selfinjective special biserial algebras.
2 Special biserial algebras
A quiver is a quadruple \(Q = (Q_0, Q_1, s, t)\) consisting of a finite set \(Q_0\) of vertices, a finite set \(Q_1\) of arrows, and two maps \(s,t : Q_1 \rightarrow Q_0\) which associate with each arrow \(\alpha \in Q_1\) its source \(s(\alpha ) \in Q_0\) and its target \(t(\alpha ) \in Q_0\). We denote by KQ the path algebra of Q over K whose underlying Kvector space has as its basis the set of all paths in Q of length \(\ge 0\), and by \(R_Q\) the arrow ideal of KQ generated by all paths in Q of length \(\ge 1\). An ideal I in KQ is said to be admissible if there exists \(m \ge 2\) such that \(R_Q^m \subseteq I \subseteq R_Q^2\). If I is an admissible ideal in KQ, then the quotient algebra KQ / I is called a bound quiver algebra and is a finitedimensional basic Kalgebra. Moreover, KQ / I is indecomposable if and only if Q is connected. Every basic, indecomposable, finitedimensional Kalgebra A has a bound quiver presentation \(A \cong K Q/I\), where \(Q = Q_A\) is the Gabriel quiver of A and I is an admissible ideal in KQ. For a bound quiver algebra \(A = KQ/I\), we denote by \(e_i\), \(i \in Q_0\), the associated complete set of pairwise orthogonal primitive idempotents of A. Then, the modules \(S_i = e_i A/e_i {\text {rad}}A\) (respectively, \(P_i = e_i A\)), \(i \in Q_0\), form a complete family of pairwise nonisomorphic simple modules (respectively, indecomposable projective modules) in \({\text {mod}}A\).
 (a)
each vertex of Q is a source and target of at most two arrows,
 (b)
for any arrow \(\alpha \) in Q, there are at most one arrow \(\beta \) and at most one arrow \(\gamma \) with \(\alpha \beta \notin I\) and \(\gamma \alpha \notin I\).
Proposition 2.1
Every special biserial algebra is tame.
If a special biserial algebra is in addition symmetric, there is a more convenient description. We propose the concept of a (weighted) biserial quiver algebra, which we will now define. Later, in Theorem 2.6 we will show that these algebras are precisely special biserial symmetric algebras.
Definition 2.2
 (a)
Q is 2regular, that is, every vertex of Q is the source and target of exactly two arrows,
 (b)
for each arrow \(\alpha \in Q_1\), we have \(s(f(\alpha )) = t(\alpha )\).
 (1)
\(\alpha f({\alpha })\), for all arrows \(\alpha \in Q_1\),
 (2)
\(B_{\alpha }  B_{\bar{\alpha }}\), for all arrows \(\alpha \in Q_1\).
The following describes basic properties of (weighted) biserial quiver algebras.
Proposition 2.3
Let \((Q,f,m_{\bullet })\) be a weighted biserial quiver and \(B = B(Q,f,m_{\bullet })\). Then, B is a basic, indecomposable, finitedimensional symmetric special biserial algebra with \(\dim _K {B} = \sum _{\mathcal {O}\in \mathcal {O}(g)} m_{\mathcal {O}} n_{\mathcal {O}}^2\).
Proof
It follows from the definition that B is the special biserial bound quiver algebra \(K Q_B / I_B\), where \(Q_B\) is obtained from Q by removing all virtual loops and \(I_B = J(Q,f,m_{\bullet }) \cap K Q_B\). Let i be a vertex of Q and \(\alpha , \bar{\alpha }\) the two arrows starting at i. Then, the indecomposable projective Bmodule \(P_i = e_i B\) has a basis given by \(e_i\) together with all initial proper subwords of \(B_{\alpha }\) and \(B_{\bar{\alpha }}\), and \(B_{\alpha } (= B_{\bar{\alpha }})\), and hence \(\dim _K P_i = m_{\alpha } n_{\alpha } + m_{\bar{\alpha }} n_{\bar{\alpha }}\). Note also that the union of these bases gives a basis of B consisting of paths in Q. We deduce that \(\dim _K {B} = \sum _{\mathcal {O}\in \mathcal {O}(g)} m_{\mathcal {O}} n_{\mathcal {O}}^2\). As well, the indecomposable projective module \(P_i\) has simple socle generated by \(B_{\alpha } ( = B_{\bar{\alpha }})\). We define a symmetrizing Klinear form \(\varphi : B \rightarrow K\) as follows. If u is a path in Q which belongs to the above basis, we set \(\varphi (u) = 1\) if \(u = B_{\alpha }\) for an arrow \(\alpha \in Q_1\), and \(\varphi (u) = 0\) otherwise. Then, \(\varphi (a b) = \varphi (b a)\) for all elements \(a,b \in B\) and \({\text {Ker}}\varphi \) does not contain any nonzero onesided ideal of B, and consequently, B is a symmetric algebra (see [40, Theorem IV.2.2]). \(\square \)
We wish to compare Brauer graph algebras and biserial quiver algebras. For this, we start analyzing the combinatorial data. Let Q be a connected 2regular quiver. We call a permutation g of the arrows of Q admissible if for every arrow \(\alpha \) we have \(t(\alpha )= s(g(\alpha ))\). That is, the arrows along a cycle of g can be concatenated in Q. The multiplicity function of a Brauer graph \(\varGamma \) taking only value 1 is said to be trivial.
Lemma 2.4
There is a bijection between Brauer graphs \(\varGamma \) with trivial multiplicity function and pairs (Q, g) where Q is a connected 2regular quiver and g is an admissible permutation of the arrows of Q.
Proof
(1) Given \(\varGamma \), we take the quiver \(Q=Q_{\varGamma }\), as defined in “Introduction.”
(1a) We show that \(Q_{\varGamma }\) is 2regular. Take an edge i of \(\varGamma \), it is adjacent to vertices v, w (which may be equal). If \(v\ne w\), then the edge i occurs both in the cyclic ordering around v and of w, so there are two arrows starting at i and there are two arrows ending at i. If \(v=w\), then the edge i occurs twice in the cyclic ordering of edges adjacent to v, so again there are two arrows starting at i and two arrows ending at i.
(1b) We define an (admissible) permutation g on the arrows. Given \(\alpha : i\rightarrow j\), let v be the vertex such that \(\alpha \) is attached to v, and then there are a unique edge k adjacent to v such that i, j, k are consecutive edges in the ordering around v, and hence a unique arrow \(\beta : j\rightarrow k\), also ‘attached’ to v, and we set \(g(\alpha ):= \beta \). This defines an admissible permutation on the arrows. Writing g as a product of disjoint cycles, gives a bijection between the cycles of g and the vertices of \(\varGamma \). Namely, let the cycle of g correspond to v if it consists of the arrows attached to v.
(2) Suppose we are given a connected 2regular quiver Q and an admissible permutation g, written as a product of disjoint cycles. Define a graph \(\varGamma \) with vertices the cycles of g and edges the vertices of Q. Each cycle of g defines a cyclic ordering of the edges adjacent to the vertex corresponding to this cycle. Hence, we get a Brauer graph.
(3) It is clear that these give a bijection. \(\square \)
Remark 2.5
In part (1b) of the above proof, we may have \(i=j\). There are two such cases. If the edge i is adjacent to two distinct vertices of \(\varGamma \), then i is the only edge adjacent to a vertex v, and we have \(g(\alpha )=\alpha \). We call \(\alpha \) an external loop. Otherwise, the edge i is a loop of \(\varGamma \), and then \(g(\alpha )\ne \alpha \). In this case, the cycle of g passes twice through vertex i of the quiver. We call \(\alpha \) an internal loop.
The Brauer graph \(\varGamma \) comes with a multiplicity function e defined on the vertices. Given (Q, g), we take the same multiplicity function, defined on the cycles of g, which gives the function \(m_{\bullet }\) which we have called a weight function. The permutation g determines the permutation f of the arrows where \(f(\alpha ) = \overline{g(\alpha )}\) for any arrow \(\alpha \). Clearly f is also admissible, and f and g determine each other.
We have seen that the combinatorial data for \(B_{\varGamma }\) are the same as the combinatorial data for \(B(Q, f, m_{\bullet })\). Therefore, \(B_{\varGamma }\) is in fact equal to \(B(Q, f, m_{\bullet })\).
In the definition of a biserial quiver we focus on (Q, f), this is motivated by the connection to biserial weighted surface algebras, which we will define later.
The following compares various algebras. The equivalence of the statements (i) and (iii) was already obtained by Roggenkamp in [36, Sections 2 and 3] (see also [1, Proposition 1.2] and [37, Theorem 1.1]). We include it, for completeness.
Theorem 2.6
 (i)
A is a Brauer graph algebra.
 (ii)
A is isomorphic to an algebra \(B(Q,f,m_{\bullet })\) where \((Q,f,m_{\bullet })\) is a (weighted) biserial quiver.
 (iii)
A is a symmetric special biserial algebra.
Proof
As we have just seen, (i) and (ii) are equivalent. The implication (ii) \(\Rightarrow \) (iii) follows from Proposition 2.3.
We prove now (iii) \(\Rightarrow \) (ii). Assume that A is a basic symmetric special biserial algebra, let \(A= KQ_A/I\) where \(Q_A\) is the Gabriel quiver of A. We will define a (weighted) biserial quiver \((Q,f,m_{\bullet })\) and show that A is isomorphic to \(B(Q,f,m_{\bullet })\). Since A is special biserial, for each vertex i of \(Q_A\), we have \(s^{1}(i) \le 2\) and \(t^{1}(i) \le 2\). The algebra A is symmetric; therefore, for each vertex \(i \in Q_0\), we have \(s^{1}(i)= t^{1}(i)\): Namely, if \(s^{1}(i) = 1\), then by the special biserial relations, the projective module \(e_iA\) is uniserial. It is isomorphic to the injective hull of the simple module \(S_i\), and hence, \(t^{1}(i)=1\). If \(t^{1}(i)=1\), then by the same reasoning, applied to \(D(Ae_i) \cong e_iA\) it follows that \(s^{1}(i) = 1\).
Let \(\varDelta := \{ i\in (Q_A)_0 \mid s^{1}(i)= 1 \}\); to each \(i\in \varDelta \), we adjoin a loop \(\eta _i\) at i to the quiver \(Q_A\), which then gives a 2regular quiver. Explicitly, let \(Q := (Q_0,Q_1,s,t)\) with \(Q_0 = (Q_A)_0\) and \(Q_1\) is the disjoint union \((Q_A)_1\bigcup \{ \eta _i: i\in \varDelta \}.\)
We define a permutation f of \(Q_1\). For each \(i \in \varDelta \), there are unique arrows \(\alpha _i\) and \(\beta _i\) in \(Q_A\) with \(t(\alpha _i) = i = s(\beta _i)\), and we set \(f(\alpha _i) = \eta _i\) and \(f(\eta _i) = \beta _i\). If \(\alpha \) is any arrow of \(Q_A\) with \(t(\alpha )\) not in \(\varDelta \), we define \(f(\alpha )\) to be the unique arrow in \((Q_A)_1\) with \(\alpha f(\alpha ) \in I\). With this, (Q, f) is a biserial quiver.
We will from now suppress the word ’weighted,’ in analogy to the convention for Brauer graph algebras, where the multiplicity function is part of the definition but is not explicitly mentioned.
We will study idempotent algebras, and it is important that any idempotent algebra of a special biserial symmetric algebra is again special biserial symmetric.
Proposition 2.7
Let A be a symmetric special biserial algebra. Assume e is an idempotent of A which is a sum of some of the \(e_i\) associated with vertices of \(Q_A\). Then, eAe also is a symmetric special biserial algebra.
Proof
We may assume that \(A = B(Q,f,m_{\bullet })\) for a weighted biserial quiver \((Q,f,m_{\bullet })\) and eAe is indecomposable, and let \(Q = (Q_0,Q_1,s,t)\). We will show that \(eAe = (\tilde{Q},\tilde{f},\tilde{m}_{\bullet }) = K\tilde{Q}/J(\tilde{Q},\tilde{f},\tilde{m}_{\bullet })\) for a weighted biserial quiver \((\tilde{Q},\tilde{f},\tilde{m}_{\bullet })\). We define \(\tilde{Q}_0\) to be the set of all vertices \(i \in Q_0\) such that e is the sum of the primitive idempotents \(e_i\). For each arrow \(\alpha \in Q_1\) with \(s(\alpha ) \in \tilde{Q}_0\), we denote by \(\tilde{\alpha }\) the shortest path in Q of the form \(\alpha g(\alpha ) \dots g^p(\alpha )\) with \(p \in \{0,1, \ldots ,n_{\alpha }1\}\) and \(t(g^p(\alpha )) \in \tilde{Q}_0\). Such a path exists because \(\alpha g(\alpha ) \dots g^{n_{\alpha }1}(\alpha )\) is a cycle around vertex \(s(\alpha ) =t(g^{n_{\alpha }1}(\alpha ))\) in \(\tilde{Q}_0\). Then, we define \(\tilde{Q}_1\) to be set of paths \(\tilde{\alpha }\) in Q for all arrows \(\alpha \in Q_1\) with \(s(\alpha ) \in \tilde{Q}_0\). Moreover, for \(\tilde{\alpha } = \alpha g(\alpha ) \dots g^p(\alpha )\), we set \(\tilde{s} (\tilde{\alpha }) = s(\alpha )\) and \(\tilde{t} (\tilde{\alpha }) = t(g^p(\alpha ))\). This defines a 2regular quiver \(\tilde{Q} = (\tilde{Q}_0,\tilde{Q}_1,\tilde{s},\tilde{t})\). Further, for each arrow \(\tilde{\alpha } = \alpha g(\alpha ) \dots g^p(\alpha )\) in \(\tilde{Q}_1\), there is exactly one arrow \(\tilde{\beta } = \beta g(\beta ) \dots g^r(\beta )\) in \(\tilde{Q}_1\) such that \(\tilde{t} (\tilde{\alpha }) = t(g^p(\alpha )) = s(\beta ) = \tilde{s} (\tilde{\beta })\) and \(f(\alpha ) = \beta \), and we set \(\tilde{f} (\tilde{\alpha }) = \tilde{\beta }\). This defines a biserial quiver \((\tilde{Q},\tilde{f})\). Let \(\tilde{g}\) be the permutation of \(\tilde{Q}_1\) associated with \(\tilde{f}\), and \(\mathcal {O}(\tilde{g})\) the set of \(\tilde{g}\)orbits in \(\tilde{Q}_1\). Then, we define the weight function \(\tilde{m}_{\bullet } : \mathcal {O}(\tilde{g}) \rightarrow \mathbb {N}^*\) of \((\tilde{Q},\tilde{f})\) by setting \(\tilde{m}_{\mathcal {O}(\tilde{\alpha })} = m_{\mathcal {O}(\alpha )}\) for each arrow \(\tilde{\alpha } \in \tilde{Q}_1\). With these, the biserial quiver algebra \(B(\tilde{Q},\tilde{f},\tilde{m}_{\bullet }) = K \tilde{Q}/J(\tilde{Q},\tilde{f},\tilde{m}_{\bullet })\) is isomorphic to eAe. \(\square \)
We end this section with an example illustrating Theorem 2.6. This also shows that an idempotent algebra of a Brauer graph algebra need not be indecomposable, by taking \(e = 1_{B_{\varGamma }}  e_4\).
Example 2.8
3 Biserial weighted surface algebras
In this section, we introduce biserial weighted surface algebras and describe their basic properties.
In this paper, by a surface we mean a connected, compact, twodimensional real manifold S, orientable or nonorientable, with boundary or without boundary. It is well known that every surface S admits an additional structure of a finite twodimensional triangular cell complex and hence a triangulation (by the deep Triangulation Theorem (see, e.g., [9, Section 2.3])).
For a positive natural number n, we denote by \(D^n\) the unit disk in the ndimensional Euclidean space \(\mathbb {R}^n\), formed by all points of distance \(\le 1\) from the origin. Then, the boundary \(\partial D^n\) of \(D^n\) is the unit sphere \(S^{n1}\) in \(\mathbb {R}^n\), formed by all points of distance 1 from the origin. Further, by an ncell we mean a topological space homeomorphic to the open disk \({\text {int}}D^n = D^n {\setminus } \partial D^n\). In particular, \(S^0 = \partial D^1\) consists of two points. Moreover, we define \(D^0 = {\text {int}}D^0\) to be a point.
We refer to [24, Appendix] for some basic topological facts about cell complexes.
 (1)
Each \(\varphi _i^n\) restricts to a homeomorphism from \({\text {int}}D_i^n\) to the ncell \(e_i^n = \varphi _i^n({\text {int}}D_i^n)\) of S, and these cells are all disjoint and their union is S.
 (2)
For each twodimensional cell \(e_i^2\), \(\varphi _i^2(\partial D_i^2)\) is the union of k 1cells and m 0cells, with \(k \in \{2,3\}\) and \(m \in \{1,2,3\}\).
 (1)for any oriented triangle \(\varDelta = (a b c)\) in \(\vec {T}\) with pairwise different edges a, b, c, we have the cycle
 (2)for any selffolded triangle \(\varDelta = (a a b)\) in \(\vec {T}\), we have the quiver
 (3)for any boundary edge a in T, we have the loop
 (a)
every vertex \(i \in Q_0\) is the source and target of exactly two arrows in \(Q_1\),
 (b)
for each arrow \(\alpha \in Q_1\), we have \(s(f(\alpha )) = t(\alpha )\),
 (c)
\(f^3\) is the identity on \(Q_1\).
 (1)
Open image in new window \(f(\alpha ) = \beta \), \(f(\beta ) = \gamma \), \(f(\gamma ) = \alpha \), for an oriented triangle \(\varDelta = (a b c)\) in \(\vec {T}\), with pairwise different edges a, b, c,
 (2)
Open image in new window \(f(\alpha ) = \beta \), \(f(\beta ) = \gamma \), \(f(\gamma ) = \alpha \), for a selffolded triangle \(\varDelta = (a a b)\) in \(\vec {T}\),
 (3)
Open image in new window \(f(\alpha ) = \alpha \), for a boundary edge a of T.
The following theorem is a slightly stronger version of [17, Theorem 4.11] (see also [18, Example 8.2] for the case with two vertices).
Theorem 3.1
Let (Q, f) be a triangulation quiver with at least two vertices. Then, there exists a directed triangulated surface \((S,\vec {T})\) such that S is orientable, \(\vec {T}\) is a coherent orientation of triangles in T, and \((Q,f) = (Q(S,\vec {T}),f)\).
Proof
Remark 3.2
There is an alternative proof of Theorem 3.1. According to Lemma 2.4 and Theorem 2.6, we may associate with a triangulation quiver (Q, f) a Brauer graph \(\varGamma \) with trivial multiplicity function such that \(B_{\varGamma } \cong B(Q,f,\mathbb {1})\), where \(\mathbb {1}\) is the trivial weight function of (Q, f). In the Brauer graph \(\varGamma \), the vertices correspond to the gorbits in \(Q_1\) and the edges to the vertices of Q. Thickening the edges of \(\varGamma \), we obtain an oriented surface S whose border is given by the faces of \(\varGamma \), corresponding to the forbits in \(Q_1\). Since (Q, f) is a triangulation quiver, the faces are either triangles or (internal) loops. Capping now all triangle faces of S by disks \(D^2\), we obtain a directed triangulated surface \(\left( ({\mathcal {T}},\mathbf {T})\right) \) such that \((Q,f) = (Q(\mathcal {T},\vec {T}),f)\).
Remark 3.3
We would like to stress that the setting of directed triangulated surfaces is natural for the purposes of a selfcontained representation theory of symmetric tame algebras of nonpolynomial growth which we are currently developing. In particular, this gives the option of changing orientation of any triangle independently, keeping the same surface and triangulation.
Biserial weighted surface algebras belong to the class of algebras of generalized dihedral type, which generalize blocks of group algebras with dihedral defect groups. They are introduced and studied in [18]. We end this section by giving two examples of biserial weighted surface algebras.
Example 3.4
Example 3.5
4 Proof of Theorem 1
To prove the implication (ii) \(\Rightarrow \) (i), let B be a biserial weighted surface algebra. Then, by Theorem 3.1 we may assume \(B= B(Q, f, m_{\bullet })\) where (Q, f) is a biserial quiver and \(f^3\) is the identity. Then, in particular B is a biserial quiver algebra, and by Theorem 2.6, we see that B is a Brauer graph algebra. Now it follows from Theorem 2.6 and Proposition 2.7 that also eBe is a Brauer graph algebra, and (i) holds.
We consider the implication (i)\(\Rightarrow \) (ii). Assume A is a Brauer graph algebra, by Theorem 2.6 we may assume \(A=B(Q, f, m_{\bullet })\) where (Q, f) is a biserial quiver. To obtain (ii), we must find a biserial quiver \((Q^*, f^*)\) with \((f^*)^3 = 1\) such that \(A = e^*B^*e^*\) where \(B^* = B(Q^*, f^*, m_{\bullet }^*)\) and \(e^*\) an idempotent of \(B^*\).
The following shows that this can be done in a canonical way the construction gives an algorithm. Furthermore, applying the construction twice gives an interesting consequence.
Theorem 4.1
 (i)
B is isomorphic to the idempotent algebra \(e^* B^* e^*\) of the biserial triangulation algebra \(B^* = B(Q^*,f^*,m_{\bullet }^*)\) with respect to a canonically defined idempotent \(e^*\) of \(B^*\).
 (ii)
The triangulation quiver \((Q^*, f^*)\) has no loops fixed by \(f^*\).
 (iii)
The triangulation quiver \((Q^{**}, f^{**})\) has no loops and selffolded triangles.
 (iv)
B is isomorphic to the idempotent algebra \(e^{**} B^{**} e^{**}\) of the biserial triangulation algebra \(B^{**} = B(Q^{**},f^{**},m_{\bullet }^{**})\) with respect to a canonically defined idempotent \(e^{**}\) of \(B^{**}\).
Proof
Let \(B^* = B(Q^*,f^*,m_{\bullet }^*)\) be the biserial triangulation algebra associated with \((Q^*,f^*,m_{\bullet }^*)\) and let \(e^*\) be the sum of the primitive idempotents \(e_i^*\) in \(B^*\) associated with all vertices \(i \in Q_0\). Using the proof of Proposition 2.7, we see directly that the idempotent algebra \(e^*B^*e^*\) is isomorphic to B. It follows also from the definition of \(f^*\) that \(Q^*\) has no loops fixed by \(f^*\), and (ii) holds. In particular, we conclude that \(f^*(\varepsilon _{\alpha }) \ne \varepsilon _{\alpha }\) for any arrow \(\alpha \in Q_1\). Hence, the triangulation quiver \((Q^{**}, f^{**})\) has no loops, and consequently, it has also no selffolded triangles, and (iii) follows. Finally, by (i), \(B^*\) is isomorphic to an idempotent algebra \(\hat{e}B^{**}\hat{e}\) of \(B^{**} = B(Q^{**},f^{**},m_{\bullet }^{**})\) for the corresponding idempotent \(\hat{e}\) of \(B^{**}\). Taking \(e^{**} = e^*\hat{e}\), we obtain that B is isomorphic to the idempotent algebra \(e^{**} B^{**} e^{**}\), and hence (iv) also holds. \(\square \)
We give some illustrations for the \(*\)construction.
Remark 4.2
The statement (i) of the above theorem also holds if we replace the canonically defined weight function \(m_{\bullet }^*\) by a weight function \(\bar{m}_{\bullet }^*\) such that \(\bar{m}_{\mathcal {O}^*(\alpha ')} = m_{\alpha }\) and \(\bar{m}_{\mathcal {O}^*(\varepsilon _{\alpha })}\) is an arbitrary positive integer, for any arrow \(\alpha \in Q_1\).
Remark 4.3
The construction of the triangulation quiver \((Q^*, f^*)\) associated with (Q, f) is canonical, though a quiver with fewer vertices may often be sufficient. In fact, it would be enough to apply the construction only to the arrows in forbits of length different from 1 and 3. An algebra \(B(Q,f,m_{\bullet })\) may have many presentations as an idempotent algebra of some biserial triangulation algebra, even for a triangulation quiver \((Q', f')\) with fewer \(f'\)orbits than the number of \(f^*\)orbits in the triangulation quiver \((Q^*, f^*)\) (see Example 4.7).
Remark 4.4
The \(*\)construction described in Theorem 4.1 provides a special class of triangulation quivers. Namely, let (Q, f) be a biserial quiver, g the permutation of \(Q_1\) associated with (Q, f), and \(g^*\) the permutation of \(Q_1^*\) associated with \((Q^*,f^*)\). Then, for every arrow \(\alpha \in Q_1\), we have in \(Q_1^*\) the \(g^*\)orbit \(\mathcal {O}^*(\alpha ')\) of even length \(2\mathcal {O}(\alpha )\) and the \(g^*\)orbit \(\mathcal {O}^*(\varepsilon _{\alpha })\) whose length is the length of the forbit of \(\alpha \) in \(Q_1\). In particular, all triangulation quivers \((Q',f')\) having only \(g'\)orbits of odd length do not belong to this class of triangulation quivers. For example, it is the case for the tetrahedral quiver considered in Sect. 6. We refer also to [17, Example 4.9] for an example of triangulation quiver \((Q'',f'')\) for which all arrows in \(Q_1''\) belong to one \(g''\)orbit of length 18.
Example 4.5
Example 4.6
Example 4.7
We finish this section with a combinatorial interpretation of the \(*\)construction in terms of Brauer graphs.
4.1 Barycentric division of Brauer graphs
Let \(\varGamma \) be the Brauer graph so that \(B_{\varGamma } = B(Q, f, m_{\bullet })\), and then the algebra \(B(Q^*, f^*, m_{\bullet }^*)\) as in the \(*\)construction of Theorem 4.1 is again a Brauer graph algebra, \(B_{\varGamma ^*}\) say, by Theorem 2.6. The proof of Lemma 2.4 shows how to construct \(\varGamma ^*\): Its vertices are in bijection with the cycles of \(g^*\). First, each cycle of g is ‘augmented,’by replacing an arrow \(\alpha \) by \(\alpha ', \alpha ''\), and this gives a cycle of \(g^*\); we call a corresponding vertex of \(\varGamma ^*\) an augmented vertex. Second, any other cycle of \(g^*\) consists of \(\varepsilon \)arrows, and these cycles correspond to fcycles of Q, as described in Theorem 4.1. Let \(F(\alpha )\) be the forbit of \(\alpha \) in Q, then we write \(v_{F(\alpha )}\) for the corresponding vertex of \(\varGamma ^*\), and then the arrows attached to this vertex are precisely the \(\varepsilon _{f^t(\alpha )}\).
The Brauer graph \(\varGamma ^*\) can be considered as a barycentric division of the Brauer graph \(\varGamma \) and has a triangular structure. Namely, every \(v_{F(\alpha )}\) is the vertex of \(F(\alpha )\) triangles in \(\varGamma ^*\) whose edges opposite to \(v_{F(\alpha )}\) are the edges of \(\varGamma \) corresponding to the vertices in Q along \(F(\alpha )\).
In this way, we obtain an orientable surface \(S^*\) without boundary, the triangulation \(T^*\) of \(S^*\) indexed by the set of edges of \(\varGamma \), and the orientation \(\vec {T^*}\) of triangles in \(T^*\) such that the associated triangulation quiver \((Q(S^*,\vec {T^*}),f)\) is the quiver \((Q^*,f^*)\). The triangulated surface \((S^*,T^*)\) can be considered as a completion of the Brauer graph \(\varGamma \) to a canonically defined triangulated surface, by a finite number of pyramids whose peaks are the forbits and bases are given by the edges of \(\varGamma \). We also note that the surface \(S^*\) (without triangulation \(T^*\)) can be obtained as follows. We may embed the Brauer graph \(\varGamma \) into a surface S with boundary given by thickening the edges of \(\varGamma \). The components of the border \(\partial S\) of S are given by the ‘Green walks’ around \(\varGamma \) on S, corresponding to the forbits in \(Q_1\). Then, the surface \(S^*\) is obtained from S by capping all the boundary components of S by the disks \(D^2\).
Example 4.8
5 Proof of Theorem 3
This theorem describes algebras socle equivalent to Brauer graph algebras. By Theorem 2.6, this is the same as describing algebras socle equivalent to a biserial quiver algebra \(A=B(Q, f, m_{\bullet })\) where (Q, f) is a biserial quiver. We show that such algebras can be described using the methods of [18, Section 6]. Then, we show that the \(*\)construction for the biserial quiver algebras can be extended.
Let (Q, f) be a biserial quiver. A vertex \(i \in Q_0\) is said to be a border vertex of (Q, f) if there is a loop \(\alpha \) at i with \(f(\alpha ) = \alpha \). We denote by \(\partial (Q,f)\) the set of all border vertices of (Q, f), and call it the border of (Q, f). The terminology is motivated by the connection with surfaces: If (Q, f) is the triangulation quiver \((Q(S,\vec {T}),f)\) associated with a directed triangulated surface \((S,\vec {T})\), then the border vertices of (Q, f) correspond bijectively to the boundary edges of the triangulation T of S. If (Q, f) is the biserial quiver associated with a Brauer graph \(\varGamma \), then the border vertices of (Q, f) correspond bijectively to the internal loops of \(\varGamma \) (see Sect. 2).
Definition 5.1
 (1)
\(\alpha f({\alpha })\), for all arrows \(\alpha \in Q_1\) which are not border loops,
 (2)
\(\alpha ^2  b_{s(\alpha )} B_{\alpha }\), for all border loops \(\alpha \in Q_1\),
 (3)
\(B_{\alpha }  B_{\bar{\alpha }}\), for all arrows \(\alpha \in Q_1\).
We summarize the basic properties of these algebras.
Proposition 5.2
 (i)
\(\bar{B}\) is a basic, indecomposable, finitedimensional, symmetric, biserial algebra with \(\dim _K \bar{B} = \sum _{\mathcal {O}\in \mathcal {O}(g)} m_{\mathcal {O}} n_{\mathcal {O}}^2\).
 (ii)
\(\bar{B}\) is socle equivalent to B.
 (iii)
If K is of characteristic different from 2, then \(\bar{B}\) is isomorphic to B.
Proof
Part (ii) is clear from the definition and then part (i) follows from Proposition 2.3. For the last part, see arguments in the proof of Proposition 6.3 in [18]. \(\square \)
The following theorem gives a complete description of symmetric algebras socle equivalent to a biserial quiver algebra.
Theorem 5.3
 (i)
If \(\partial (Q, f)\) is empty, then A is isomorphic to \(B(Q,f,m_{\bullet })\).
 (ii)
Otherwise, A is isomorphic to \(B(Q,f,m_{\bullet },b_{\bullet })\) for some border function \(b_{\bullet }\) of (Q, f).
Proof
Let \(B= B(Q, f, m_{\bullet }) = KQ/J\) where \(J = J(Q, f, m_{\bullet })\). Since \(A/{\text {soc}}(A)\) is isomorphic to \(B/{\text {soc}}(B)\), we can assume that these are equal, using an isomorphism as identification. We assume A is symmetric; therefore, for each \(i \in Q_0\), the module \(e_iA\) has a onedimensional socle which is spanned by some \(\omega _i \in e_iAe_i\), and we fix such an element. Then, let \(\varphi \) be a symmetrizing linear form for A, and then \(\varphi (\omega _i)\) is nonzero. We may assume that \(\varphi (\omega _i) = 1\).
We claim that \({\text {soc}}(A) \subset ({\text {rad}}A)^2 (\subset {\text {rad}}A)\). If not, then for some j we have \(\omega _j \not \in ({\text {rad}}A)^2\). This means that \(e_jA = e_jAe_j\), which is not possible since A is indecomposable with at least two simple modules. It follows that A and B have the same Gabriel quiver. Recall that the quiver Q is the disjoint union of the Gabriel quiver of B with virtual loops. Any virtual loop of Q is then in the socle of B, and it is zero in \(B/{\text {soc}}(B)\) and is therefore zero in \(A/{\text {soc}}(A)\). We may therefore take A of the form \(A=KQ/I\) for the same quiver Q, and some ideal I of KQ such that any virtual loop lies in the socle of A.
(I) We may assume that \(A_{\alpha }\beta = B_{\alpha }\) in A (and hence is equal to \(B_{\alpha }\) in KQ).
If not, then we have \(A_{\alpha }\beta = 0\), and then \(A_{\alpha }\gamma \ne 0\). We will show that we may interchange \(\beta \) and \(\gamma \).
Since \(A_{\alpha }\gamma \ne 0\), in particular \(g^{2}(\alpha )\gamma \ne 0\) and also \(t(\gamma ) = i = s(\alpha )\). Since \(\gamma = f(g^{2}(\alpha ))\), we know that \(g^{2}(\alpha )\gamma \) belongs to the socle of A. It is nonzero, which implies that \(A_{\alpha } = g^{2}(\alpha )\) (and \(m_{\alpha }=1\)), and therefore, \(\alpha = g^{2}(\alpha )\), and \(\gamma = f(\alpha )\). We claim that \(g(\alpha )\ne \alpha \). Namely, if we had \(g(\alpha )=\alpha \), then both \(\alpha \) and \(f(\alpha )\) would be loops at vertex i and \(Q_0=1\), which contradicts our assumption. Hence, the cycle of g containing \(\alpha \) is \((\alpha \ g(\alpha ))\), of length two. We claim that also the fcycle of \(\alpha \) (in B) has length two. Namely, if \(\bar{\alpha }\) is the other arrow starting at i and \(\rho \) is the other arrow ending at \(j=t(\alpha )\), then we must have by the properties of f and g that \(f(\rho ) = \beta \) and \(f(\beta ) = \bar{\alpha }\). This implies that \(f(\gamma ) = \alpha \) and hence f has a cycle \((\alpha \ \gamma )\).
It follows that there is an algebra isomorphism from B to the biserial quiver algebra \(B'\) given by the weighted biserial quiver obtained from \((Q,f,m_{\bullet })\) by interchanging \(\beta \) and \(\gamma \) (which form a pair of double arrows) and fixing all other arrows of Q. We replace B by \(B'\) and the claim follows.
(II) We show that relation (1) holds in A. If \(\alpha \) is a virtual loop of B, then \(\alpha f(\alpha )=0\) since \(\alpha \in {\text {soc}}(A)\). We consider now an arrow \(\alpha \) which is not a virtual loop. Suppose \(\alpha \) is not fixed by f, then \(\alpha f(\alpha )\) belongs to the socle of A. We can write \(\alpha f(\alpha ) = a_{\alpha } B_{\alpha } = a_{\alpha }\alpha A_{g(\alpha )}\) for some \(a_{\alpha } \in K\) (here \(g(\alpha )\) is not a virtual loop).
(a) If \(s(\alpha )\ne t(f(\alpha ))\), then \(\alpha f(\alpha ) = \alpha f(\alpha ) e_{s(\alpha )}= 0\); in fact, this holds for any choice of \(\alpha , f(\alpha )\).
Assume first this cycle contains an arrow \(\alpha \) such that \(f^{r1}(\alpha )\alpha \) is not a cyclic path. We may start with \(\alpha \) and adjust \(f(\alpha ), f^2(\alpha ), \ldots , f^{r1}(\alpha )\) as described above. Then, \(f^{r1}(\alpha )'\cdot \alpha =0\), by (a) above.
(IV) We show that relation (2) holds in A. When the border \(\partial (Q,f)\) of (Q, f) is empty, there is nothing to do (and A is isomorphic to B). Assume now that \(\partial (Q,f)\) is not empty. Then, for any loop \(\alpha \) with \(i = s(\alpha ) \in \partial (Q,f)\), we have \(\alpha ^2 = \alpha f(\alpha ) = b_i \omega _i = b_i B_{\alpha }\) for some \(b_i \in K\). Hence, we have a border function \(b_{\bullet } : \partial (Q,f) \rightarrow K\), and A is isomorphic to the algebra \(B(Q, f, m_{\bullet }, b_{\bullet })\). \(\square \)
Recall that a selfinjective algebra A is biserial if the radical of any indecomposable nonuniserial projective, left or right, Amodule is a sum of two uniserial modules whose intersection is simple.
Theorem 3 follows from Theorems 2.6, 3.1, 5.3 and the following relative version of Theorem 4.1 (see Remark 4.3).
Theorem 5.4
 (i)
\(\partial (Q,f) = \partial (Q^\#,f^\#)\).
 (ii)
B is isomorphic to the idempotent algebra \(e^\# B^\# e^\#\) of the biserial weighted triangulation algebra \(B^\# = B(Q^\#,f^\#,m_{\bullet }^\#)\) with respect to a canonically defined idempotent \(e^\#\) of \(B^\#\).
 (iii)
For any border function \(b_{\bullet }\) of (Q, f) and the induced border function \(b_{\bullet }^\#\) of \((Q^\#,f^\#)\), the algebras \(B(Q,f,m_{\bullet },b_{\bullet })\) and \(e^\# B(Q^\#,f^\#,m_{\bullet }^\#,b_{\bullet }^\#) e^\#\) are isomorphic.
Proof
Let \(B^\# = B(Q^\#,f^\#,m_{\bullet }^\#)\) be the biserial weighted triangulation algebra associated with \((Q^\#,f^\#,m_{\bullet }^\#)\) and \(e^\#\) the sum of the primitive idempotents \(e_i^\#\) in \(B^\#\) associated with the vertices \(i \in Q_0\). Then, it follows from the arguments as in the proof of Proposition 2.7 that B is isomorphic to the idempotent algebra \(e^\# B^\# e^\#\). Moreover, let \(b_{\bullet }\) be a border function of (Q, f) and \(b_{\bullet }^\#\) be the induced border function of \((Q^\#,f^\#)\), that is, \(b_{i}^\# = b_{i}\) for any border vertex i. Then, it follows from the description of \(g^\#\)orbits in \(Q_1^\#\) and the definition of the weight function \({m}_{\bullet }^\#\) that \(B(Q,f,m_{\bullet },b_{\bullet })\) is isomorphic to the idempotent algebra \(e^\# B(Q^\#,f^\#,m_{\bullet }^\#,b_{\bullet }^\#) e^\#\). \(\square \)
Example 5.5
We observe now that \(B(Q,f,m_{\bullet },b_{\bullet })\) is isomorphic to the idempotent algebra \(e^\# B(Q^\#,f^\#,m_{\bullet }^\#,b_{\bullet }^\#) e^\#\) where the idempotent \(e^\#\) is the sum of the primitive idempotents at the vertices 1, 2, 3, 4. Moreover, the algebras \(B(Q,f,m_{\bullet })\) and \(e^\# B(Q^\#,f^\#,m_{\bullet }^\#) e^\#\) are also isomorphic. Finally, we note that if K has characteristic 2 and \(b_{\bullet }^\# = b_{\bullet }\) is nonzero, then the algebras \(B(Q^\#,f^\#,m_{\bullet }^\#,b_{\bullet }^\#)\) and \(B(Q^\#,f^\#,m_{\bullet }^\#)\) are not isomorphic.
6 Proof of Theorem 4
 (1)
\({\alpha } f({\alpha })  c_{\bar{\alpha }} A_{\bar{\alpha }}\), for all arrows \(\alpha \in Q_1\),
 (2)
\(\beta f(\beta ) g(f(\beta ))\), for all arrows \(\beta \in Q_1\).
We note that the Gabriel quiver of \(\varLambda \) is equal to Q, and this holds because we assume \(m_{\alpha } n_{\alpha } \ge 3\) for all arrows \(\alpha \in Q_1\).
We have the following proposition (see [17, Proposition 5.8]).
Proposition 6.1
Let (Q, f) be a triangulation quiver, \(m_{\bullet }\) and \(c_{\bullet }\) weight and parameter functions of (Q, f). Then, \(\varLambda = \varLambda (Q,f,m_{\bullet },c_{\bullet })\) is a finitedimensional tame symmetric algebra of dimension \(\sum _{\mathcal {O}\in \mathcal {O}(g)} m_{\mathcal {O}} n_{\mathcal {O}}^2\).
We have also the following theorem proved in [17, Theorem 1.2] (see also [6, Proposition 7.1] and [16, Theorem 5.9] for the case of two vertices).
Theorem 6.2
 (i)
All simple modules in \({\text {mod}}\varLambda \) are periodic of period 4.
 (ii)
\(\varLambda \) is a periodic algebra of period 4.
 (iii)
\(\varLambda \) is not isomorphic to a singular tetrahedral algebra.
The following theorem is an essential ingredient for the proof of Theorem 4.
Theorem 6.3
 (i)
\(\varLambda ^*\) is a periodic algebra of period 4.
 (ii)
B is isomorphic to the idempotent algebra \(e^{*} \varLambda ^{*} e^{*}\) for an idempotent \(e^{*}\) of \(\varLambda ^{*}\).
Proof
We may now complete the proof of Theorem 4. Let \(B = B(Q,f,m_{\bullet })\) be a biserial quiver algebra. Then, it follows from Theorem 4.1 that B is isomorphic to the idempotent algebra \(e^{**}B^{**}e^{**}\) of the biserial triangulation algebra \(B^{**} = B(Q^{**},f^{**},m_{\bullet }^{**})\) for some idempotent \(e^{**}\) of \(B^{*}\), and \(Q^{**}\) has no loops. Applying now Theorem 6.3, we conclude that \(B^{**}\) is isomorphic to the idempotent algebra \(e \varLambda e\) of a periodic weighted triangulation algebra, for an idempotent e of \(\varLambda \). Since \(e^{**}\) is a summand of e, we have \(B \cong e^{**}B^{**}e^{**} \cong e^{**}(e \varLambda e)e^{**} = e^{**} \varLambda e^{**}\). Then, Theorem 4 follows from Theorems 2.6 and 3.1.
Remark 6.4
Let \(\varLambda = \varLambda (Q,f,m_{\bullet },c_{\bullet })\) be a weighted triangulation algebra. Then, the biserial triangulation algebra \(B = B(Q,f,m_{\bullet })\) is not an idempotent algebra \(e \varLambda e\) of \(\varLambda \). On the other hand, if \(\varLambda \) is not a tetrahedral algebra, then B is a geometric degeneration of \(\varLambda \) (see [17, Proposition 5.8]).
Example 6.5
We present now an example of an idempotent algebra of a periodic weighted surface algebra which is neither a Brauer graph algebra nor a weighted surface algebra.
Example 6.6
7 Diagram of algebras
Notes
Acknowledgements
The results of the paper were partially presented during the Workshop on Brauer Graph Algebras held in Stuttgart in March 2016. The paper was completed during the visit of the first named author at the Faculty of Mathematics and Computer Science of Nicolaus Copernicus University in Toruń (June 2017). The authors thank the referees very much for valuable suggestions.
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