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QPA

Quivers and Path Algebras

Version 1.34

August 2022

The QPA-team
Email: oyvind.solberg@ntnu.no
Homepage: https://folk.ntnu.no/oyvinso/QPA/
Address:
Department of Mathematical Sciences
NTNU
N-7491 Trondheim
Norway

Abstract

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.

Copyright

© 2011-2021 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 (https://www.gnu.org/licenses/gpl.html).

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 1.34; 2021 (https://folk.ntnu.no/oyvinso/QPA/)

Acknowledgements

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

Contents

1.1 General aims

1.2 Installation and system requirements

2.1 Example 1 -- quivers, path algebras and quotients of path algebras

2.2 Example 2 -- Introducing modules

2.3 Example 3 -- Constructing modules and module homomorphisms

3.1 Information class, Quivers

3.1-1 InfoQuiver

3.2 Constructing Quivers

  3.2-1 Quiver
  3.2-2 DynkinQuiver
  3.2-3 OrderedBy

3.3 Categories and Properties of Quivers

  3.3-1 IsQuiver
  3.3-2 IsAcyclicQuiver
  3.3-3 IsUAcyclicQuiver
  3.3-4 IsConnectedQuiver
  3.3-5 IsTreeQuiver
  3.3-6 IsDynkinQuiver

3.4 Orderings of paths in a quiver

3.5 Attributes and Operations for Quivers

  3.5-1 .
  3.5-2 VerticesOfQuiver
  3.5-3 ArrowsOfQuiver
  3.5-4 AdjacencyMatrixOfQuiver
  3.5-5 GeneratorsOfQuiver
  3.5-6 NumberOfVertices
  3.5-7 NumberOfArrows
  3.5-8 OrderingOfQuiver
  3.5-9 OppositeQuiver
  3.5-10 FullSubquiver
  3.5-11 ConnectedComponentsOfQuiver
  3.5-12 DoubleQuiver
  3.5-13 SeparatedQuiver

3.6 Categories and Properties of Paths

  3.6-1 IsPath
  3.6-2 IsQuiverVertex
  3.6-3 IsArrow
  3.6-4 IsZeroPath

3.7 Attributes and Operations of Paths

  3.7-1 SourceOfPath
  3.7-2 TargetOfPath
  3.7-3 LengthOfPath
  3.7-4 WalkOfPath
  3.7-5 *
  3.7-6 =
  3.7-7 <

3.8 Attributes of Vertices

  3.8-1 IncomingArrowsOfVertex
  3.8-2 OutgoingArrowsOfVertex
  3.8-3 InDegreeOfVertex
  3.8-4 OutDegreeOfVertex
  3.8-5 NeighborsOfVertex

  3.9-1 Poset
  3.9-2 Size
  3.9-3 UnderlyingSet
  3.9-4 PartialOrderOfPoset

4 Path Algebras

4.1 Introduction

4.2 Constructing Path Algebras

4.2-1 PathAlgebra

4.3 Categories and Properties of Path Algebras

4.3-1 IsPathAlgebra

4.4 Attributes and Operations for Path Algebras

  4.4-1 AssociatedMonomialAlgebra
  4.4-2 QuiverOfPathAlgebra
  4.4-3 OrderingOfAlgebra
  4.4-4 .

4.5 Operations on Path Algebra Elements

  4.5-1 ElementOfPathAlgebra
  4.5-2 <
  4.5-3 IsLeftUniform
  4.5-4 IsRightUniform
  4.5-5 IsUniform
  4.5-6 LeadingTerm
  4.5-7 LeadingCoefficient
  4.5-8 LeadingMonomial
  4.5-9 MakeUniformOnRight
  4.5-10 MappedExpression
  4.5-11 VertexPosition
  4.5-12 RelationsOfAlgebra

4.6 Constructing Quotients of Path Algebras

4.6-1 AssignGeneratorVariables

4.7 Ideals and operations on ideals

  4.7-1 Ideal
  4.7-2 IdealOfQuotient
  4.7-3 PathsOfLengthTwo
  4.7-4 NthPowerOfArrowIdeal
  4.7-5 AddNthPowerToRelations
  4.7-6 \in

4.8 Categories and properties of ideals

  4.8-1 IsAdmissibleIdeal
  4.8-2 IsIdealInPathAlgebra
  4.8-3 IsMonomialIdeal
  4.8-4 IsQuadraticIdeal

4.9 Operations on ideals

4.9-1 ProductOfIdeals
4.9-2 QuadraticPerpOfPathAlgebraIdeal

4.10 Attributes of ideals

4.10-1 GroebnerBasisOfIdeal

4.11 Categories and Properties of Quotients of Path Algebras

4.12 Operations on String Algebras

  4.12-1 IsValidString
  4.12-2 StringsLessThan
  4.12-3 IsABand
  4.12-4 BandsLessThan
  4.12-5 BandRepresentativesLessThan
  4.12-6 IsDomesticStringAlgebra
  4.12-7 BridgeQuiver
  4.12-8 LocalARQuiver

4.13 Attributes and Operations (for Quotients) of Path Algebras

  4.13-1 CartanMatrix
  4.13-2 Centre/Center
  4.13-3 ComplexityOfAlgebra
  4.13-4 CoxeterMatrix
  4.13-5 CoxeterPolynomial
  4.13-6 Dimension
  4.13-7 FrobeniusForm
  4.13-8 FrobeniusLinearFunctional
  4.13-9 GlobalDimension
  4.13-10 LoewyLength
  4.13-11 NakayamaAutomorphism
  4.13-12 NakayamaPermutation
  4.13-13 OrderOfNakayamaAutomorphism
  4.13-14 RadicalSeriesOfAlgebra

4.14 Attributes and Operations on Elements of Quotients of Path Algebra

  4.14-1 IsElementOfQuotientOfPathAlgebra
  4.14-2 Coefficients
  4.14-3 IsNormalForm
  4.14-4 <
  4.14-5 ElementOfQuotientOfPathAlgebra
  4.14-6 OriginalPathAlgebra

4.15 Predefined classes and classes of (quotients of) path algebras

  4.15-1 BrauerConfigurationAlgebra
  4.15-2 CanonicalAlgebra
  4.15-3 KroneckerAlgebra
  4.15-4 NakayamaAlgebra
  4.15-5 AdmissibleSequenceGenerator
  4.15-6 PosetAlgebra
  4.15-7 PosetOfPosetAlgebra
  4.15-8 TruncatedPathAlgebra
  4.15-9 IsSpecialBiserialQuiver

4.16 Opposite algebras

  4.16-1 OppositePath
  4.16-2 OppositePathAlgebra
  4.16-3 OppositePathAlgebraElement
  4.16-4 OppositeAlgebraHomomorphism

4.17 Tensor products of path algebras

  4.17-1 QuiverProduct
  4.17-2 QuiverProductDecomposition
  4.17-3 IsQuiverProductDecomposition
  4.17-4 IncludeInProductQuiver
  4.17-5 ProjectFromProductQuiver
  4.17-6 TensorProductOfAlgebras
  4.17-7 TensorAlgebrasInclusion
  4.17-8 SimpleTensor
  4.17-9 TensorProductDecomposition
  4.17-10 EnvelopingAlgebra
  4.17-11 EnvelopingAlgebraHomomorphism
  4.17-12 IsEnvelopingAlgebra
  4.17-13 AlgebraAsModuleOverEnvelopingAlgebra
  4.17-14 DualOfAlgebraAsModuleOverEnvelopingAlgebra
  4.17-15 TrivialExtensionOfQuiverAlgebra
  4.17-16 TrivialExtensionOfQuiverAlgebraProjection

4.18 Operations on quiver algebras

4.18-1 QuiverAlgebraOfAmodAeA
4.18-2 QuiverAlgebraOfeAe

4.19 Finite dimensional algebras over finite fields

  4.19-1 AlgebraAsQuiverAlgebra
  4.19-2 BlocksOfAlgebra
  4.19-3 IsBasicAlgebra
  4.19-4 IsElementaryAlgebra
  4.19-5 PreprojectiveAlgebra
  4.19-6 PrimitiveIdempotents

4.20-1 LiftingCompleteSetOfOrthogonalIdempotents
4.20-2 LiftingIdempotent

4.21 Saving and reading quotients of path algebras to and from a file

4.21-1 ReadAlgebra
4.21-2 SaveAlgebra

6 Right Modules over Path Algebras

6.1 Modules of matrix type

  6.1-1 RightModuleOverPathAlgebra
  6.1-2 RightAlgebraModuleToPathAlgebraMatModule
  6.1-3 \=

6.2 Categories Of Matrix Modules

6.2-1 IsPathAlgebraMatModule

6.3 Acting on Module Elements

6.3-1 ^

6.4 Operations on representations

  6.4-1 AnnihilatorOfModule
  6.4-2 BasicVersionOfModule
  6.4-3 BlockDecompositionOfModule
  6.4-4 BlockSplittingIdempotents
  6.4-5 CommonDirectSummand
  6.4-6 ComplexityOfModule
  6.4-7 DecomposeModule
  6.4-8 DecomposeModuleProbabilistic
  6.4-9 DecomposeModuleViaCharPoly
  6.4-10 DecomposeModuleViaTop
  6.4-11 DecomposeModuleWithMultiplicities
  6.4-12 Dimension
  6.4-13 DimensionVector
  6.4-14 DirectSumOfQPAModules
  6.4-15 DirectSumInclusions
  6.4-16 DirectSumProjections
  6.4-17 FromIdentityToDoubleStarHomomorphism
  6.4-18 IntersectionOfSubmodules
  6.4-19 IsDirectSummand
  6.4-20 IsDirectSumOfModules
  6.4-21 IsExceptionalModule
  6.4-22 IsIndecomposableModule
  6.4-23 IsInAdditiveClosure
  6.4-24 IsInjectiveModule
  6.4-25 IsomorphicModules
  6.4-26 IsProjectiveModule
  6.4-27 IsRigidModule
  6.4-28 IsSemisimpleModule
  6.4-29 IsSimpleQPAModule
  6.4-30 IsTauRigidModule
  6.4-31 LoewyLength
  6.4-32 IsZero
  6.4-33 MatricesOfPathAlgebraModule
  6.4-34 MaximalCommonDirectSummand
  6.4-35 NumberOfNonIsoDirSummands
  6.4-36 MinimalGeneratingSetOfModule
  6.4-37 RadicalOfModule
  6.4-38 RadicalSeries
  6.4-39 SocleSeries
  6.4-40 SocleOfModule
  6.4-41 SubRepresentation
  6.4-42 SumOfSubmodules
  6.4-43 SupportModuleElement
  6.4-44 TopOfModule

6.5 Special representations

  6.5-1 BasisOfProjectives
  6.5-2 ElementInIndecProjective
  6.5-3 IndecInjectiveModules
  6.5-4 IndecProjectiveModules
  6.5-5 SimpleModules
  6.5-6 ZeroModule

6.6 Functors on representations

  6.6-1 DualOfModule
  6.6-2 DualOfModuleHomomorphism
  6.6-3 DTr
  6.6-4 DTr
  6.6-5 NakayamaFunctorOfModule
  6.6-6 NakayamaFunctorOfModuleHomomorphism
  6.6-7 OppositeNakayamaFunctorOfModule
  6.6-8 OppositeNakayamaFunctorOfModuleHomomorphism
  6.6-9 RestrictionViaAlgebraHomomorphism
  6.6-10 RestrictionViaAlgebraHomomorphismMap
  6.6-11 StarOfModule
  6.6-12 StarOfModuleHomomorphism
  6.6-13 TensorProductOfModules
  6.6-14 TrD
  6.6-15 TrD
  6.6-16 TransposeOfModule
  6.6-17 TransposeOfModuleHomomorphism

7 Homomorphisms of Right Modules over Path Algebras

7.1 Categories and representation of homomorphisms

7.1-1 IsPathAlgebraModuleHomomorphism
7.1-2 RightModuleHomOverAlgebra

7.3 Homomorphisms and modules constructed from homomorphisms and modules

  7.3-1 AllIndecModulesOfLengthAtMost
  7.3-2 AllModulesOfLengthAtMost
  7.3-3 AllSimpleSubmodulesOfModule
  7.3-4 AllSubmodulesOfModule
  7.3-5 CoKernel
  7.3-6 CoKernelProjection
  7.3-7 EndModuloProjOverAlgebra
  7.3-8 EndOfModuleAsQuiverAlgebra
  7.3-9 EndOverAlgebra
  7.3-10 FromEndMToHomMM
  7.3-11 FromHomMMToEndM
  7.3-12 HomFactoringThroughProjOverAlgebra
  7.3-13 HomFromProjective
  7.3-14 HomOverAlgebra
  7.3-15 Image
  7.3-16 ImageInclusion
  7.3-17 ImageProjection
  7.3-18 ImageProjectionInclusion
  7.3-19 IsomorphismOfModules
  7.3-20 Kernel
  7.3-21 LeftMinimalVersion
  7.3-22 MatrixOfHomomorphismBetweenProjectives
  7.3-23 RightMinimalVersion
  7.3-24 RadicalOfModuleInclusion
  7.3-25 RejectOfModule
  7.3-26 SocleOfModuleInclusion
  7.3-27 SubRepresentationInclusion
  7.3-28 TopOfModuleProjection
  7.3-29 TraceOfModule

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

9.1 Almost split sequences and AR-quivers

  9.1-1 AlmostSplitSequence
  9.1-2 AlmostSplitSequenceInPerpT
  9.1-3 IrreducibleMorphismsEndingIn
  9.1-4 IsTauPeriodic
  9.1-5 PredecessorOfModule

10 Chain complexes

10.1 Introduction

10.2 Infinite lists

  10.2-1 IsInfiniteNumber
  10.2-2 PositiveInfinity
  10.2-3 NegativeInfinity
  10.2-4 IsInfList
  10.2-5 IsHalfInfList
  10.2-6 \^
  10.2-7 MakeHalfInfList
  10.2-8 StartPosition
  10.2-9 Direction
  10.2-10 InfListType
  10.2-11 RepeatingList
  10.2-12 ElementFunction
  10.2-13 IsStoringValues
  10.2-14 NewValueCallback
  10.2-15 IsRepeating
  10.2-16 InitialValue
  10.2-17 LowestKnownPosition
  10.2-18 HighestKnownValue
  10.2-19 Shift
  10.2-20 Cut
  10.2-21 HalfInfList
  10.2-22 MakeInfListFromHalfInfLists
  10.2-23 MakeInfList
  10.2-24 FunctionInfList
  10.2-25 ConstantInfList
  10.2-26 FiniteInfList
  10.2-27 MiddleStart
  10.2-28 MiddleEnd
  10.2-29 MiddlePart
  10.2-30 PositivePart
  10.2-31 NegativePart
  10.2-32 HighestKnownPosition
  10.2-33 LowestKnownPosition
  10.2-34 UpperBound
  10.2-35 LowerBound
  10.2-36 FinitePartAsList
  10.2-37 PositivePartFrom
  10.2-38 NegativePartFrom
  10.2-39 Shift
  10.2-40 Splice
  10.2-41 InfConcatenation
  10.2-42 InfList
  10.2-43 IntegersList

10.3 Representation of categories

10.3-1 IsCat
10.3-2 CatOfRightAlgebraModules

10.4 Making a complex

  10.4-1 IsQPAComplex
  10.4-2 IsZeroComplex
  10.4-3 Complex
  10.4-4 ZeroComplex
  10.4-5 FiniteComplex
  10.4-6 StalkComplex
  10.4-7 ShortExactSequence

10.5 Information about a complex

  10.5-1 CatOfComplex
  10.5-2 ObjectOfComplex
  10.5-3 DifferentialOfComplex
  10.5-4 DifferentialsOfComplex
  10.5-5 CyclesOfComplex
  10.5-6 BoundariesOfComplex
  10.5-7 HomologyOfComplex
  10.5-8 IsFiniteComplex
  10.5-9 UpperBound
  10.5-10 LowerBound
  10.5-11 LengthOfComplex
  10.5-12 HighestKnownDegree
  10.5-13 LowestKnownDegree
  10.5-14 IsExactSequence
  10.5-15 IsExactInDegree
  10.5-16 IsShortExactSequence
  10.5-17 ForEveryDegree

10.6 Transforming and combining complexes

  10.6-1 Shift
  10.6-2 ShiftUnsigned
  10.6-3 YonedaProduct
  10.6-4 BrutalTruncationBelow
  10.6-5 BrutalTruncationAbove
  10.6-6 BrutalTruncation
  10.6-7 SyzygyTruncation
  10.6-8 CosyzygyTruncation
  10.6-9 SyzygyCosyzygyTruncation

10.7 Chain maps

  10.7-1 IsChainMap
  10.7-2 ChainMap
  10.7-3 ZeroChainMap
  10.7-4 FiniteChainMap
  10.7-5 ComplexAndChainMaps
  10.7-6 MorphismOfChainMap
  10.7-7 MorphismsOfChainMap
  10.7-8 ComparisonLifting
  10.7-9 ComparisonLiftingToProjectiveResolution
  10.7-10 MappingCone

11 Projective resolutions and the bounded derived category

11.1 Projective and injective complexes

  11.1-1 InjectiveResolution
  11.1-2 IsProjectiveComplex
  11.1-3 IsInjectiveComplex
  11.1-4 ProjectiveResolution

11.2 The bounded derived category

  11.2-1 ProjectiveResolutionOfComplex
  11.2-2 ProjectiveToInjectiveComplex
  11.2-3 TauOfComplex
  11.2-4 Example
  11.2-5 StarOfMapBetweenProjectives

12 Combinatorial representation theory

12.1 Introduction

12.2 Different unit forms

  12.2-1 IsUnitForm
  12.2-2 BilinearFormOfUnitForm
  12.2-3 IsWeaklyNonnegativeUnitForm
  12.2-4 IsWeaklyPositiveUnitForm
  12.2-5 PositiveRootsOfUnitForm
  12.2-6 QuadraticFormOfUnitForm
  12.2-7 SymmetricMatrixOfUnitForm
  12.2-8 TitsUnitFormOfAlgebra
  12.2-9 EulerBilinearFormOfAlgebra
  12.2-10 UnitForm

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