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QPA

Quivers and Path Algebras

2.0-dev

09/04/2015

This manual documents the GAP package QPA, version 2.0-dev, revision 5db4193, 2015-09-15 15:51:01 +0200.

Øystein Skartsæterhagen
Email: oysteini@math.ntnu.no
Homepage: http://www.math.ntnu.no/~oysteini/

Contents

1.1 Categories for paths and quivers

  1.1-1 IsPath
  1.1-2 IsPrimitivePath
  1.1-3 IsNontrivialPath
  1.1-4 IsCompositePath
  1.1-5 IsVertex
  1.1-6 IsArrow
  1.1-7 IsQuiver
  1.1-8 IsLeftQuiver
  1.1-9 IsRightQuiver

1.2 Constructing quivers

  1.2-1 LeftQuiver
  1.2-2 RightQuiver
  1.2-3 Quiver
  1.2-4 MakeLabelsFromPattern

1.3 Information about a quiver

  1.3-1 Label
  1.3-2 LabelAsString
  1.3-3 QuiverCategory

1.4 Accessing paths in a quiver

  1.4-1 Vertices
  1.4-2 Arrows
  1.4-3 VertexLabels
  1.4-4 ArrowLabels
  1.4-5 NumberOfVertices
  1.4-6 NumberOfArrows
  1.4-7 ArrowSourceIndices
  1.4-8 ArrowTargetIndices
  1.4-9 PrimitivePaths
  1.4-10 Vertex
  1.4-11 Arrow
  1.4-12 PrimitivePathByLabel
  1.4-13 PathFromString
  1.4-14 \.

1.5 Information about a path

  1.5-1 QuiverOfPath
  1.5-2 Source
  1.5-3 Length
  1.5-4 Label
  1.5-5 LabelAsString
  1.5-6 VertexNumber
  1.5-7 ArrowNumber

1.6 Composition of paths

  1.6-1 ComposePaths
  1.6-2 Composable
  1.6-3 ComposableLR
  1.6-4 PathFromArrowList
  1.6-5 \*

1.7 Decomposition of paths

1.7-1 ArrowList
1.7-2 AsList

1.8 Path manipulation

1.8-1 TranslatePath

1.9 Subpaths and path division

  1.9-1 Subpath
  1.9-2 SubpathIndex
  1.9-3 ExtractSubpath
  1.9-4 \/
  1.9-5 PathOverlaps

1.10 Quiver constructions

  1.10-1 OppositeQuiver
  1.10-2 OppositePath
  1.10-3 QuiverProduct
  1.10-4 QuiverProduct
  1.10-5 ProjectPathFromProductQuiver
  1.10-6 IncludePathInProductQuiver
  1.10-7 PathInProductQuiver
  1.10-8 PathInProductQuiver

2 Quiver algebras

2.1 Categories for algebras, elements and ideals

  2.1-1 IsQuiverAlgebraElement
  2.1-2 IsQuiverAlgebra
  2.1-3 IsPathAlgebraElement
  2.1-4 IsPathAlgebra
  2.1-5 IsPathIdeal
  2.1-6 IsQuotientOfPathAlgebraElement
  2.1-7 IsQuotientOfPathAlgebra

2.2 Constructing algebras

2.2-1 PathAlgebra
2.2-2 QuotientOfPathAlgebra

2.3 Information about an algebra

  2.3-1 QuiverOfAlgebra
  2.3-2 RelationsOfAlgebra
  2.3-3 IdealOfQuotient
  2.3-4 LeftActingDomain
  2.3-5 PathAlgebra

2.4 Accessing algebra elements

  2.4-1 PathAlgebraElement
  2.4-2 QuotientOfPathAlgebraElement
  2.4-3 PathAsAlgebraElement
  2.4-4 AlgebraElementByLabel
  2.4-5 AlgebraElementFromString

2.5 Information about an element

  2.5-1 AlgebraOfElement
  2.5-2 Coefficients
  2.5-3 Paths
  2.5-4 IsUniform
  2.5-5 Representative

2.6 Manipulation of elements

2.6-1 TranslateAlgebraElement

2.7 Operations related to Groebner basis theory

  2.7-1 LeadingPath
  2.7-2 LeadingCoefficient
  2.7-3 LeadingTerm
  2.7-4 NonLeadingTerms
  2.7-5 DivideByList
  2.7-6 Reduce
  2.7-7 OverlapRelation
  2.7-8 OverlapRelations
  2.7-9 TipReduce
  2.7-10 ComputeGroebnerBasis
  2.7-11 GroebnerBasis

2.8 Algebra constructions

  2.8-1 OppositeAlgebra
  2.8-2 OppositeAlgebraElement
  2.8-3 TensorProductOfAlgebras
  2.8-4 IsTensorProductOfAlgebras

3 Representations

3.1 Categories for representations and their elements

3.1-1 IsQuiverRepresentationElement
3.1-2 IsQuiverRepresentation

3.2 Constructing elements

3.2-1 QuiverRepresentationElement
3.2-2 QuiverRepresentationElementByVertices

3.3 Information about an element

  3.3-1 RepresentationOfElement
  3.3-2 ElementVectors
  3.3-3 ElementVector
  3.3-4 ElementVector

3.4 Acting on elements

3.4-1 PathAction
3.4-2 QuiverAlgebraAction

3.5 Constructing representations

  3.5-1 QuiverRepresentation
  3.5-2 QuiverRepresentationByArrows
  3.5-3 AsRepresentationOfQuotientAlgebra
  3.5-4 ZeroRepresentation
  3.5-5 AlgebraOfRepresentation
  3.5-6 QuiverOfRepresentation
  3.5-7 FieldOfRepresentation
  3.5-8 VertexDimensions
  3.5-9 VertexDimension
  3.5-10 VertexDimension
  3.5-11 MatricesOfRepresentation
  3.5-12 MatrixForArrow
  3.5-13 MatrixForArrow
  3.5-14 MatrixForPath
  3.5-15 MatrixForAlgebraElement

4.1 Categories for modules and module elements

  4.1-1 IsQuiverModuleElement
  4.1-2 IsQuiverModule
  4.1-3 IsLeftQuiverModuleElement
  4.1-4 IsLeftQuiverModule
  4.1-5 IsRightQuiverModuleElement
  4.1-6 IsRightQuiverModule
  4.1-7 IsQuiverBimoduleElement
  4.1-8 IsQuiverBimodule

4.2 Constructing modules

  4.2-1 LeftQuiverModule
  4.2-2 LeftQuiverModuleByArrows
  4.2-3 LeftZeroModule
  4.2-4 RightQuiverModule
  4.2-5 RightQuiverModuleByArrows
  4.2-6 RightZeroModule
  4.2-7 AlgebraForLeftModules
  4.2-8 AlgebraForRightModules
  4.2-9 AsModule
  4.2-10 AsLeftModule
  4.2-11 AsRightModule
  4.2-12 AsBimodule

4.3 Information about a module

  4.3-1 UnderlyingRepresentation
  4.3-2 QuiverOfModule
  4.3-3 FieldOfModule
  4.3-4 VertexDimensions
  4.3-5 VertexDimension
  4.3-6 VertexDimension

4.4 Constructing elements

  4.4-1 QuiverModuleElement
  4.4-2 QuiverModuleElementByVertices
  4.4-3 AsModuleElement

4.5 Information about an element

  4.5-1 UnderlyingRepresentationElement
  4.5-2 ModuleOfElement
  4.5-3 ElementVectors
  4.5-4 ElementVector
  4.5-5 ElementVector

4.6 Acting on elements

4.6-1 \^
4.6-2 \^

5.1 Empty matrices

  5.1-1 IsEmptyMatrix
  5.1-2 MakeEmptyMatrix
  5.1-3 RingOfEmptyMatrix
  5.1-4 MakeZeroMatrix

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