On Extensions for Gentle Algebras

İlke Çanakçı,David Pauksztello,Sibylle Schroll

doi:10.4153/s0008414x2000005x

Abstract

AbstractWe give a complete description of a basis of the extension spaces between indecomposable string and quasi-simple band modules in the module category of a gentle algebra.

Highlights

The representation theory of finite-dimensional algebras plays an important role in many different areas of mathematics, such as in Lie theory, in number theory in connection with the Langlands program and automorphic forms, in geometry ranging from invariant theory to non-commutative resolutions of singularities and as far afield as harmonic analysis where the representation theory of S1 appears in the guise of Fourier analysis
Most finite-dimensional algebras are of wild representation type, that is their representation theory is at least as complicated as that of the free associative algebra in two generators
For ease of the already somewhat heavy notation, in the proofs in Section 3 and 4, whenever we have a map between two band complexes or an extension between two band modules, implicitly and without loss of generality we assume that the parameters of the corresponding band complexes or band modules are equal to one, see [9, §2.3] for more details on the placement of parameters with respect to mapping cones

Summary

Introduction

The representation theory of finite-dimensional algebras plays an important role in many different areas of mathematics, such as in Lie theory, in number theory in connection with the Langlands program and automorphic forms, in geometry ranging from invariant theory to non-commutative resolutions of singularities and as far afield as harmonic analysis where the representation theory of S1 appears in the guise of Fourier analysis. We explicitly determine the cohomology of the indecomposable objects in the bounded derived category of a gentle algebra given in terms of homotopy strings and bands. Building on this we give a complete description of the extension space between string and quasi-simple band modules by giving a combinatorial description of a basis. We do this by working not in the module category of a gentle algebra, but we transfer the problem into the derived category, where we are able to use the graphical mapping cone calculus developed in [9, 10]. For ease of the already somewhat heavy notation, in the proofs in Section 3 and 4, whenever we have a map between two band complexes or an extension between two band modules, implicitly and without loss of generality we assume that the parameters of the corresponding band complexes or band modules are equal to one, see [9, §2.3] for more details on the placement of parameters with respect to mapping cones

Background

Cohomology of string and band complexes

Determining extensions in the module category

Surjectivity of Φ onto overlap and arrow extensions