On the Combinatorics of Gentle Algebras

By T. Brüstle, Guillaume Douville, Kaveh Mousavand, Hugh Thomas, Emine Yıldırım. Canadian Journal of Mathematics - Journal Canadien de Mathematiques, 2017

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Abstract For $A$ a gentle algebra, and $X$ and $Y$ string modules, we construct a combinatorial basis for $\operatorname{Hom}(X,\unicode[STIX]{x1D70F}Y)$. We use this to describe support $\unicode[STIX]{x1D70F}$-tilting modules for $A$. We give a combinatorial realization of maps in both directions realizing the bijection between support $\unicode[STIX]{x1D70F}$-tilting modules and functorially finite torsion classes. We give an explicit basis of $\operatorname{Ext}^{1}(Y,X)$ as short exact sequences. We analyze several constructions given in a more restricted, combinatorial setting by McConville, showing that many but not all of them can be extended to general gentle algebras.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a72274d902666bc01fb977018b42bd799c655977

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Citations in the corpus

A complete derived invariant for gentle algebras via winding numbers and Arf invariants (2019, background): “It turns out that any gentle algebra can be obtained from a dissection of a surface; this has led to geometric models for their module categories [32] (building on [41]) and τ -tilting theory [39] (see also [15, 40]).”
A Geometric Model for the Module Category of a Gentle Algebra (2018, background): “s such as Fukaya categories [27], dimer models [8], enveloping algebras of Lie algebras [29] and cluster theory. As such, there has been widespread interest in this class of algebras; see for example [10, 14, 15, 37] for recent developments in this area. In the context of cluster theory, gentle alge”