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Quivers and Path Algebras

Version 1.31

September 2020

The QPA-team
Department of Mathematical Sciences
N-7491 Trondheim


The GAP4 deposited package QPA extends the GAP functionality for computations with finite dimensional quotients of path algebras. QPA has data structures for quivers, quotients of path algebras, representations of quivers with relations and complexes of modules. Basic operations on representations of quivers are implemented as well as contructing minimal projective resolutions of modules (using using linear algebra). A not necessarily minimal projective resolution constructed by using Groebner basis theory and a paper by Green-Solberg-Zacharia, "Minimal projective resolutions", has been implemented. A goal is to have a test for finite representation type. This work has started, but there is a long way left. Part of this work is to implement/port the functionality and data structures that was available in CREP.


© 2010-2020 by The QPA-team.

QPA is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. For details, see the FSF’s own site (

If you obtained QPA, we would be grateful for a short notification sent to one of members of the QPA-team. If you publish a result which was partially obtained with the usage of QPA, please cite it in the following form:

The QPA-team, QPA - Quivers, path algebras and representations, Version 1.31; 2018 (


The system design of QPA was initiated by Edward L. Green, Lenwood S. Heath, and Craig A. Struble. It was continued and completed by Randall Cone and Edward Green. We would like to thank the following people for their contributions:

Chain complexes Kristin Krogh Arnesen and Øystein Skartsæterhagen
Degeneration order for modules in finite type Andrzej Mroz
GBNP interface (for Groebner bases) Randall Cone
Homomorphisms of modules Øyvind Solberg and Anette Wraalsen
Koszul duals Stephen Corwin
Matrix representations of path algebras Øyvind Solberg and George Yuhasz
Opposite algebra and tensor products of algebras Øystein Skartsæterhagen
Predefined classes of algebras Andrzej Mroz and Øyvind Solberg
Projective resolutions (using Groebnar basis) Randall Cone and Øyvind Solberg
Projective resolutions (using linear algebra) Øyvind Solberg
Quickstart Kristin Krogh Arnesen
Quivers, path algebras Gerard Brunick
The bounded derived category Kristin Krogh Arnesen and Øystein Skartsæterhagen
Unitforms Øyvind Solberg



1 Introduction
2 Quickstart
3 Quivers
4 Path Algebras
5 Groebner Basis
6 Right Modules over Path Algebras
7 Homomorphisms of Right Modules over Path Algebras
8 Homological algebra
 8.1 Homological algebra

  8.1-1 1stSyzygy

  8.1-2 AllComplementsOfAlmostCompleteTiltingModule

  8.1-3 CotiltingModule

  8.1-4 DominantDimensionOfAlgebra

  8.1-5 DominantDimensionOfModule

  8.1-6 ExtAlgebraGenerators

  8.1-7 ExtOverAlgebra

  8.1-8 FaithfulDimension

  8.1-9 GlobalDimensionOfAlgebra

  8.1-10 GorensteinDimension

  8.1-11 GorensteinDimensionOfAlgebra

  8.1-12 HaveFiniteCoresolutionInAddM

  8.1-13 HaveFiniteResolutionInAddM

  8.1-14 InjDimension

  8.1-15 InjDimensionOfModule

  8.1-16 InjectiveEnvelope

  8.1-17 IsCotiltingModule

  8.1-18 IsNthSyzygy

  8.1-19 IsOmegaPeriodic

  8.1-20 IsTtiltingModule

  8.1-21 IyamaGenerator

  8.1-22 LeftApproximationByAddTHat

  8.1-23 LeftFacMApproximation

  8.1-24 LeftMutationOfTiltingModuleComplement

  8.1-25 LeftSubMApproximation

  8.1-26 LiftingInclusionMorphisms

  8.1-27 LiftingMorphismFromProjective

  8.1-28 LeftApproximationByAddM

  8.1-29 RightApproximationByAddM

  8.1-30 RadicalRightApproximationByAddM

  8.1-31 MorphismOnKernel

  8.1-32 NthSyzygy

  8.1-33 NumberOfComplementsOfAlmostCompleteTiltingModule

  8.1-34 ProjDimension

  8.1-35 ProjDimensionOfModule

  8.1-36 ProjectiveCover

  8.1-37 ProjectiveResolutionOfPathAlgebraModule

  8.1-38 ProjectiveResolutionOfSimpleModuleOverEndo

  8.1-39 PullBack

  8.1-40 PushOut

  8.1-41 RightApproximationByPerpT

  8.1-42 RightFacMApproximation

  8.1-43 RightMutationOfTiltingModuleComplement

  8.1-44 RightSubMApproximation

  8.1-45 N_RigidModule

  8.1-46 TiltingModule
9 Auslander-Reiten theory
10 Chain complexes
11 Projective resolutions and the bounded derived category
12 Combinatorial representation theory
13 Degeneration order for modules in finite type

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