Algebras from surfaces without punctures (Q420725)

In the paper under review, the authors introduce a new class of finite dimensional gentle algebras, the \textit{surface algebras}, which includes the hereditary, the tilted, and the cluster-tilted algebras of Dynkin type \(\mathbb{A}\) and Euclidean type \(\tilde{\mathbb{A}}\). These algebras are constructed from an unpunctured Riemann surface with boundary and marked points by introducing cuts in internal triangles of an arbitrary triangulation of the surface. To be more precise, let \(T\) be a triangulation of a bordered unpunctured Riemann surface \(S\) with a set of marked points \(M\), and let \((Q_T,I_T)\) be the bound quiver associated to \(T\) as in [\textit{P. Caldero}, \textit{F. Chapoton} and \textit{R. Schiffler}, Trans. Am. Math. Soc. 358, No. 3, 1347--1364 (2006; Zbl 1137.16020)]. The corresponding algebra \(B_T=kQ_T/I_T\), over an algebraically closed field \(k\), is a finite dimensional gentle algebra. Moreover, \(B_T\) is the endomorphism algebra of the cluster-tilting object corresponding to \(T\) in the generalized cluster category associated to \((S,M)\).NEWLINENEWLINEIf the surface is a disc or an annulus then the corresponding cluster algebra, as defined in [\textit{S. Fomin}, \textit{M. Shapiro} and \textit{D. Thurston}, Acta Math. 201, No. 1, 83--146 (2008; Zbl 1263.13023)], is acyclic, and, in this case, the algebra \(B_T\) is cluster-tilted of type \(\mathbb{A}\), if \(S\) is a disc; and of type \(\tilde{\mathbb{A}}\), if \(S\) is an annulus. It is then natural to ask, what kind of algebras we can get from admissible cuts of algebras \(B_T\) coming from other surfaces. This motivates the definition of a {surface algebra}, which is constructed by cutting a triangulation \(T\) of a surface \((S,M)\) at internal triangles. Cutting an internal triangle \(\triangle\) means replacing the triangle \(\triangle\) by a quadrilateral \(\triangle^\dagger\) with one side on the boundary of the same surface \(S\) with an enlarged set of marked points \(M^\dagger\). Cutting as many internal triangles as we please, we obtain a partial triangulation \(T^\dagger\) of a surface with marked points \((S,M^\dagger)\), to which we can associate an algebra \(B_{T^\dagger}=kQ_{T^\dagger}/I_{T^\dagger}\) in a very similar way to the construction of \(B_T\) from \(T\). This algebra \(B_{T^\dagger}\) is called a {surface algebra of type} \((S,M)\). (A surface algebra is called an \textit{admissible cut} if it is obtained by cutting every internal triangle exactly once.)NEWLINENEWLINEThe first main results of the paper are the following.NEWLINENEWLINETheorem 1. Every surface algebra is isomorphic to the endomorphism algebra of a partial cluster-tilting object in a generalized cluster category. More precisely, if the surface algebra \(B_{T^\dagger} \) is given by the cut \((S,M^\dagger,T^\dagger)\) of the triangulated surface \((S,M,T)\), then NEWLINE\[NEWLINEB_{T^\dagger}\cong \text{End}_{\mathcal{C}_{(S,M^\dagger)}}{\mathfrak{T}}^\dagger,NEWLINE\]NEWLINE where \({\mathfrak{T}}^\dagger\) denotes the object in cluster category \(\mathcal{C}_{(S,M^\dagger)}\) corresponding to \(T^\dagger\).NEWLINENEWLINETheorem 2. If \((S,M^\dagger,T^\dagger)\) is an admissible cut of \((S,M,T)\) then (a) \(Q_{T^\dagger}\) is an admissible cut of \(Q_T\). (b) \(B_{T^\dagger}\) is of global dimension at most two. (c) the tensor algebra of \(B_{T^\dagger}\) with respect to the \(B_{T^\dagger}\)-bimodule NEWLINE\[NEWLINE\text{Ext}^2_{B_{T^\dagger}}(DB_{T^\dagger},B_{T^\dagger})NEWLINE\]NEWLINE is isomorphic to the algebra \(B_T\).NEWLINENEWLINEThe authors then studiy the module categories of the surface algebras. Since surface algebras are gentle, their indecomposable modules are either string modules or band modules. In the special case where the surface is not cut, the indecomposable modules and the irreducible morphisms in the module category of the algebras \(B_T\) have been described by \textit{T. Brüstle} and \textit{J. Zhang} in [Algebra Number Theory 5, No. 4, 529--566 (2011; Zbl 1250.16013)] in terms of the of generalized arcs on the surface \((S,M,T)\). One of the main tools used in the above paper is the description of irreducible morphisms between string modules by \textit{M. C. R. Butler} and \textit{C. M. Ringel} [Commun. Algebra 15, 145--179 (1987; Zbl 0612.16013)]. The authors generalized the results of Brüstle and Zhang to the case of arbitrary surface algebras \(B_{T^\dagger}\), and they described the indecomposable modules in terms of certain permissible generalized arcs in the surface \((S,M^\dagger,T^\dagger)\) and the irreducible morphisms in terms of pivots of these arcs in the surface. In this way, they constructed a combinatorial category \(\mathcal{E}^\dagger\) of permissible generalized arcs in \((S,M^\dagger,T^\dagger)\) and define a functor NEWLINE\[NEWLINE H: \mathcal{E}^\dagger \to {\mathsf{mod}} B_{T^\dagger}.NEWLINE\]NEWLINENEWLINENEWLINEThe result is the following.NEWLINENEWLINETheorem 3.NEWLINENEWLINE (a) The functor \(H\) is faithful and induces a dense, faithful functor from \(\mathcal{E}^{\dagger}\) to the category of string modules over \(B_{T^{\dagger}}\). Moreover, \(H\) maps irreducible morphisms to irreducible morphisms and commutes with Auslander-Reiten translations.NEWLINENEWLINE (b) If the surface \(S\) is a disc, then \(H\) is an equivalence of categories.NEWLINENEWLINE (c) If the algebra \(B_{T^{\dagger}}\) is of finite representation type, then \(H\) is an equivalence of categories.