Schemes of modules over gentle algebras and laminations of surfaces. (English) Zbl 1491.16021
The class of gentle algebras was defined by I. Assem and A. Skowroński [Math. Z. 195, 269–290 (1987; Zbl 0601.16022)]. Module categories over gentle algebras can be described combinatorially [B. Wald and J. Waschbüsch, J. Algebra 95, 480–500 (1985; Zbl 0567.16017); M. C. R. Butler and C. M. Ringel, Commun. Algebra 15, 145–179 (1987; Zbl 0612.16013)]. The class of Jacobian algebras associated to triangulations of unpunctured marked surfaces is a special class of gentle algebras. In this paper, the authors study some geometric properties of the representation theory of gentle algebras. First, they classify the irreducible components of the affine schemes of modules over gentle algebras and describe all smooth points of these schemes. They show that most of irreducible components are generically reduced and if a gentle algebra \(A\) has no loops, then each irreducible component is generically reduced. For the class of Jacobian algebras associated to triangulations of unpunctured marked surfaces they give a bijection between the set of generically \(\tau\)-reduced decorated irreducible components and the set of laminations of the surface, where a lamination of an unpunctured marked surface \((S,M)\) is a set of homotopy classes of curves and loops in \((S,M)\), which do not intersect each other, together with a positive integer attached to each class [G. Musiker et al., Compos. Math. 149, No. 2, 217–263 (2013; Zbl 1263.13024)]. This bijection has some application to cluster algebras were defined by S. Fomin and A. Zelevinsky [J. Am. Math. Soc. 15, No. 2, 497–529 (2002; Zbl 1021.16017)]. C. Geiß et al. [J. Am. Math. Soc. 25, No. 1, 21–76 (2012; Zbl 1236.13020)] proved that the generic Caldero-Chapoton functions form a basis, called the generic basis, of the coefficient-free upper cluster algebra \(U_{(S,M)}\) associated with an unpunctured marked surface \((S,M)\). By using the above bijection, the authors prove that the generic basis coincides with Musiker-Schiffler-Williams’ bangle basis [G. Musiker et al., Compos. Math. 149, No. 2, 217–263 (2013; Zbl 1263.13024)] of the coefficient-free cluster algebra \(A_{(S,M)}\) associated with an unpunctured marked surface \((S,M)\).
Reviewer: Alireza Nasr-Isfahani (Isfahan)
MSC:
| 16P10 | Finite rings and finite-dimensional associative algebras |
| 16G20 | Representations of quivers and partially ordered sets |
| 13F60 | Cluster algebras |
Citations:
Zbl 0601.16022; Zbl 0567.16017; Zbl 0612.16013; Zbl 1263.13024; Zbl 1021.16017; Zbl 1236.13020References:
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