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TitleQuivers and Path Algebras Version 1.27
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Table of Contents
	General aims
	Installation and system requirements
	Example 1 – quivers, path algebras and quotients of path algebras
	Example 2 – Introducing modules
	Example 3 – Constructing modules and module homomorphisms
	Information class, Quivers
	Constructing Quivers
	Categories and Properties of Quivers
	Orderings of paths in a quiver
	Attributes and Operations for Quivers
	Categories and Properties of Paths
	Attributes and Operations of Paths
	Attributes of Vertices
Path Algebras
	Constructing Path Algebras
	Categories and Properties of Path Algebras
	Attributes and Operations for Path Algebras
	Operations on Path Algebra Elements
	Constructing Quotients of Path Algebras
	Ideals and operations on ideals
	Categories and properties of ideals
	Operations on ideals
	Attributes of ideals
	Categories and Properties of Quotients of Path Algebras
	Attributes and Operations (for Quotients) of Path Algebras
	Attributes and Operations on Elements of Quotients of Path Algebra
	Predefined classes and classes of (quotients of) path algebras
	Opposite algebras
	Tensor products of path algebras
	Finite dimensional algebras over finite fields
	Saving and reading quotients of path algebras to and from a file
Groebner Basis
	Constructing a Groebner Basis
	Categories and Properties of Groebner Basis
	Attributes and Operations for Groebner Basis
	Right Groebner Basis
Right Modules over Path Algebras
	Modules of matrix type
	Categories Of Matrix Modules
	Acting on Module Elements
	Operations on representations
	Special representations
	Functors on representations
	Vertex projective modules and submodules thereof
Homomorphisms of Right Modules over Path Algebras
	Categories and representation of homomorphisms
	Generalities of homomorphisms
	Homomorphisms and modules constructed from homomorphisms and modules
Homological algebra
	Homological algebra
Auslander-Reiten theory
	Almost split sequences and AR-quivers
Chain complexes
	Infinite lists
	Representation of categories
	Making a complex
	Information about a complex
	Transforming and combining complexes
	Chain maps
Projective resolutions and the bounded derived category
	Projective and injective complexes
	The bounded derived category
Combinatorial representation theory
	Different unit forms
Degeneration order for modules in finite type
	Basic definitions
	Defining Auslander-Reiten quiver in finite type
	Elementary operations
	Operations returning families of modules
Document Text Contents
Page 1

Quivers and Path Algebras

Version 1.27

December 2017

The QPA-team

The QPA-team Email: [email protected]
Address: Department of Mathematical Sciences

N-7491 Trondheim

Page 2


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 1.27 ; 2017
( )

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:

Page 78

QPA 78

retrieved by the command BasisVectors. The underlying module is likewise returned by
the command UnderlyingModule. The output of the function is satisfying the filter/category

6.7.17 TargetVertex

. TargetVertex(v) (operation)

Arguments: v – a PathAlgebraVector.
Returns: a vertex w such that v∗w = v, if such a vertex exists, and fail otherwise.
Given a PathAlgebraVector v , if v is right uniform, this function finds the vertex w such that

v ∗w = v whenever v is non-zero, and returns the zero path otherwise. If v is not right uniform it
returns fail.

6.7.18 UniformGeneratorsOfModule

. UniformGeneratorsOfModule(M) (attribute)

Arguments: M – a PathAlgebraModule.
Returns: a set of right uniform generators of the mdoule M . If M is the zero module, then it

returns an empty list.

6.7.19 Vectorize

. Vectorize(M, components) (function)

Arguments: M – a module over a path algebra, components – a list of elements of M .
Returns: a vector in M from a list of path algebra elements components , which defines the

components in the resulting vector.
The returned vector is normalized, so the vector’s components may not match the input compo-


Page 79

Chapter 7

Homomorphisms of Right Modules over
Path Algebras

This chapter describes the categories, representations, attributes, and operations on homomorphisms
between representations of quivers.

Given two homorphisms f : L→M and g : M→N, then the composition is written f ∗g. The ele-
ments in the modules or the representations of a quiver are row vectors. Therefore the homomorphisms
between two modules are acting on these row vectors, that is, if mi is in M[i] and gi : M[i]→ N[i] rep-
resents the linear map, then the value of g applied to mi is the matrix product mi ∗gi.

The example used throughout this chapter is the following.

gap> Q := Quiver(3,[[1,2,"a"],[1,2,"b"],[2,2,"c"],[2,3,"d"],[3,1,"e"]]);;
gap> KQ := PathAlgebra(Rationals, Q);;
gap> AssignGeneratorVariables(KQ);;
gap> rels := [d*e,c^2,a*c*d-b*d,e*a];;
gap> A := KQ/rels;;
gap> mat :=[["a",[[1,2],[0,3],[1,5]]],["b",[[2,0],[3,0],[5,0]]],
> ["c",[[0,0],[1,0]]],["d",[[1,2],[0,1]]],["e",[[0,0,0],[0,0,0]]]];;
gap> N := RightModuleOverPathAlgebra(A,mat);;

7.1 Categories and representation of homomorphisms

7.1.1 IsPathAlgebraModuleHomomorphism

. IsPathAlgebraModuleHomomorphism(f ) (filter)

Arguments: f - any object in GAP.
Returns: true or false depending on if f belongs to the categories

This defines the category IsPathAlgebraModuleHomomorphism.

7.1.2 RightModuleHomOverAlgebra

. RightModuleHomOverAlgebra(M, N, mats) (operation)


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QPA 155

MorphismOnKernel, 99
MorphismsOfChainMap, 133

NakayamaAlgebra, 43
NakayamaAutomorphism, 40
NakayamaFunctorOfModule, 72
NakayamaFunctorOfModuleHomomorphism, 72
NakayamaPermutation, 41
NegativeInfinity, 109
NegativePart, 116
NegativePartFrom, 117
NeighborsOfVertex, 22
NewValueCallback, 112
Nontips, 56
NontipSize, 56
N_RigidModule, 103
NthPowerOfArrowIdeal, 31
NthSyzygy, 100
NthSyzygyNC, 101
NumberOfArrows, 18

CotiltingModule, 101

TiltingModule, 101
NumberOfIndecomposables, 144
NumberOfNonIsoDirSummands, 68
NumberOfProjectives, 145
NumberOfVertices, 18

ObjectOfComplex, 123
OppositePath, 45
OppositePathAlgebra, 45
OppositePathAlgebraElement, 45
OppositeQuiver, 18
OrbitCodim, 146
OrbitDim, 145
OrderedBy, 15
OrderingOfAlgebra, 25
OrderingOfQuiver, 18
OrderOfNakayamaAutomorphism, 41
OriginalPathAlgebra, 42
OutDegreeOfVertex, 22
OutgoingArrowsOfVertex, 22

PartialOrderOfPoset, 23
PathAlgebra, 24
PathAlgebraOfMatModuleMap, 85
PathAlgebraVector, 77

PathsOfLengthTwo, 31

for a list P and a set of relations rel, 23
PosetAlgebra, 43
PosetOfPosetAlgebra, 43
PositiveInfinity, 109
PositivePart, 116
PositivePartFrom, 117
PositiveRootsOfUnitForm, 140
PredecessorOfModule, 106
PreImagesRepresentative, 85
PrimitiveIdempotents, 50
PrintMultiplicityVector, 147
PrintMultiplicityVectors, 147
ProductOfIdeals, 33
ProjDimension, 101
ProjDimensionOfModule, 101
ProjectFromProductQuiver, 47
ProjectiveCover, 101
ProjectivePathAlgebraPresentation, 77
ProjectiveResolution, 136
ProjectiveResolutionOfComplex, 136

Module, 102
ProjectiveToInjectiveComplex, 137
ProjectiveToInjectiveFiniteComplex, 137
PullBack, 102
PushOut, 102

QuadraticFormOfUnitForm, 140
QuadraticPerpOfPathAlgebraIdeal, 33

adjacenymatrix, 13
lists of vertices and arrows, 13
no. of vertices, list of arrows, 13

QuiverOfPathAlgebra, 25
QuiverProduct, 46
QuiverProductDecomposition, 46

RadicalOfModule, 68
RadicalOfModuleInclusion, 91
RadicalSeries, 68
RadicalSeriesOfAlgebra, 41
Range, 86
ReadAlgebra, 51
RejectOfModule, 91
RelationsOfAlgebra, 28

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QPA 156

RepeatingList, 112
RestrictionViaAlgebraHomomorphism, 72

Module, 59
RightApproximationByPerpT, 103
RightFacMApproximation, 103
RightGroebnerBasis, 57
RightGroebnerBasisOfIdeal, 57
RightGroebnerBasisOfModule, 77
RightInverseOfHomomorphism, 86
RightMinimalVersion, 91
RightModuleHomOverAlgebra, 79

no dimension vector, 58
with dimension vector, 58

no dimension vector, 58

Complement, 103

Complement, 103

RightProjectiveModule, 73
RightSubMApproximation, 103

SaveAlgebra, 51
SeparatedQuiver, 19
Shift, 113, 118, 127
ShiftUnsigned, 127
ShortExactSequence, 123
SimpleModules, 71
SimpleTensor, 48
Size, 23
SocleOfModule, 69
SocleOfModuleInclusion, 92
SocleSeries, 69
Source, 86
SourceOfPath, 20
Splice, 118
StalkComplex, 123
StarOfMapBetweenDecompProjectives, 138
StarOfMapBetweenIndecProjectives, 138
StarOfMapBetweenProjectives, 138
StarOfModule, 72
StarOfModuleHomomorphism, 72
StartPosition, 111
SubRepresentation, 69
SubRepresentationInclusion, 92

SumOfSubmodules, 69
SupportModuleElement, 69
SymmetricMatrixOfUnitForm, 140
SyzygyCosyzygyTruncation, 129
SyzygyTruncation, 129

TargetOfPath, 20
TargetVertex, 78
TauOfComplex, 137
TensorAlgebrasInclusion, 48
TensorProductDecomposition, 48
TensorProductOfAlgebras, 48
TensorProductOfModules, 73
TiltingModule, 104
Tip, 27
TipCoefficient, 27
TipMonomial, 27
TipReduce, 56
TipReduceGroebnerBasis, 56
TitsUnitFormOfAlgebra, 140
TopOfModule, 70
TopOfModuleProjection, 92
TraceOfModule, 92
TransposeOfDual, 73
TransposeOfModule, 73
TrD, 73
TrivialExtensionOfQuiverAlgebra, 50
TruncatedPathAlgebra, 44

UnderlyingSet, 23
UniformGeneratorsOfModule, 78
UnitForm, 141
UpperBound, 117, 125

Vectorize, 78
VertexPosition, 28
VerticesOfQuiver, 17

WalkOfPath, 21

YonedaProduct, 128

Zero, 86
ZeroChainMap, 131
ZeroComplex, 122
ZeroMapping, 86
ZeroModule, 71

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