  
  
                                      [1XQPA[101X
  
  
                           [1XQuivers and Path Algebras[101X
  
  
                                  Version 1.34
  
  
                                  August 2022
  
  
                                  The QPA-team
  
  
  
  
  
  The QPA-team
      Email:    [7Xmailto:oyvind.solberg@ntnu.no[107X
      Homepage: [7Xhttps://folk.ntnu.no/oyvinso/QPA/[107X
      Address:  [33X[0;14YDepartment of Mathematical Sciences[133X
                [33X[0;14YNTNU[133X
                [33X[0;14YN-7491 Trondheim[133X
                [33X[0;14YNorway[133X
  
  
  
  -------------------------------------------------------
  [1XAbstract[101X
  [33X[0;0YThe   GAP4   deposited   package  [5XQPA[105X  extends  the  GAP  functionality  for
  computations  with  finite  dimensional  quotients of path algebras. [5XQPA[105X 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.[133X
  
  
  -------------------------------------------------------
  [1XCopyright[101X
  [33X[0;0Y© 2011-2021 by The QPA-team.[133X
  
  [33X[0;0Y[5XQPA[105X  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
  ([7Xhttps://www.gnu.org/licenses/gpl.html[107X).[133X
  
  [33X[0;0YIf  you  obtained [5XQPA[105X, 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 [5XQPA[105X, please cite it in the following form:[133X
  
  [33X[0;0YThe  QPA-team,  [5XQPA[105X  -  Quivers,  path algebras and representations, Version
  1.34; 2021 ([7Xhttps://folk.ntnu.no/oyvinso/QPA/[107X)[133X
  
  
  -------------------------------------------------------
  [1XAcknowledgements[101X
  [33X[0;0YThe system design of [5XQPA[105X 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:[133X
  
        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                                      
                                                                                                               
                                                                                                               
                                                                                                               
                                                                                                               
                                                                                                               
                                                                                                               
                                                                                                               
                                                                                                               
                                                                                                               
                                                                                                               
                                                                                                               
                                                                                                               
                                                                                                               
                                                                                                               
                                                                                                               
                                                                                                               
  
  
  -------------------------------------------------------
  
  
  [1XContents (QPA)[101X
  
  1 [33X[0;0YIntroduction[133X
    1.1 [33X[0;0YGeneral aims[133X
    1.2 [33X[0;0YInstallation and system requirements[133X
  2 [33X[0;0YQuickstart[133X
    2.1 [33X[0;0YExample 1 -- quivers, path algebras and quotients of path algebras[133X
    2.2 [33X[0;0YExample 2 -- Introducing modules[133X
    2.3 [33X[0;0YExample 3 -- Constructing modules and module homomorphisms[133X
  3 [33X[0;0YQuivers[133X
    3.1 [33X[0;0YInformation class, Quivers[133X
      3.1-1 InfoQuiver
    3.2 [33X[0;0YConstructing Quivers[133X
      3.2-1 Quiver
      3.2-2 DynkinQuiver
      3.2-3 OrderedBy
    3.3 [33X[0;0YCategories and Properties of Quivers[133X
      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 [33X[0;0YOrderings of paths in a quiver[133X
    3.5 [33X[0;0YAttributes and Operations for Quivers[133X
      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 [33X[0;0YCategories and Properties of Paths[133X
      3.6-1 IsPath
      3.6-2 IsQuiverVertex
      3.6-3 IsArrow
      3.6-4 IsZeroPath
    3.7 [33X[0;0YAttributes and Operations of Paths[133X
      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 [33X[0;0YAttributes of Vertices[133X
      3.8-1 IncomingArrowsOfVertex
      3.8-2 OutgoingArrowsOfVertex
      3.8-3 InDegreeOfVertex
      3.8-4 OutDegreeOfVertex
      3.8-5 NeighborsOfVertex
    3.9 [33X[0;0YPosets[133X
      3.9-1 Poset
      3.9-2 Size
      3.9-3 UnderlyingSet
      3.9-4 PartialOrderOfPoset
  4 [33X[0;0YPath Algebras[133X
    4.1 [33X[0;0YIntroduction[133X
    4.2 [33X[0;0YConstructing Path Algebras[133X
      4.2-1 PathAlgebra
    4.3 [33X[0;0YCategories and Properties of Path Algebras[133X
      4.3-1 IsPathAlgebra
    4.4 [33X[0;0YAttributes and Operations for Path Algebras[133X
      4.4-1 AssociatedMonomialAlgebra
      4.4-2 QuiverOfPathAlgebra
      4.4-3 OrderingOfAlgebra
      4.4-4 .
    4.5 [33X[0;0YOperations on Path Algebra Elements[133X
      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 [33X[0;0YConstructing Quotients of Path Algebras[133X
      4.6-1 AssignGeneratorVariables
    4.7 [33X[0;0YIdeals and operations on ideals[133X
      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 [33X[0;0YCategories and properties of ideals[133X
      4.8-1 IsAdmissibleIdeal
      4.8-2 IsIdealInPathAlgebra
      4.8-3 IsMonomialIdeal
      4.8-4 IsQuadraticIdeal
    4.9 [33X[0;0YOperations on ideals[133X
      4.9-1 ProductOfIdeals
      4.9-2 QuadraticPerpOfPathAlgebraIdeal
    4.10 [33X[0;0YAttributes of ideals[133X
      4.10-1 GroebnerBasisOfIdeal
    4.11 [33X[0;0YCategories and Properties of Quotients of Path Algebras[133X
      4.11-1 IsAdmissibleQuotientOfPathAlgebra
      4.11-2 IsQuotientOfPathAlgebra
      4.11-3 IsFiniteDimensional
      4.11-4 IsCanonicalAlgebra
      4.11-5 IsDistributiveAlgebra
      4.11-6 IsFiniteGlobalDimensionAlgebra
      4.11-7 IsGentleAlgebra
      4.11-8 IsGorensteinAlgebra
      4.11-9 IsHereditaryAlgebra
      4.11-10 IsKroneckerAlgebra
      4.11-11 IsMonomialAlgebra
      4.11-12 IsNakayamaAlgebra
      4.11-13 IsQuiverAlgebra
      4.11-14 IsRadicalSquareZeroAlgebra
      4.11-15 IsSchurianAlgebra
      4.11-16 IsSelfinjectiveAlgebra
      4.11-17 IsSemicommutativeAlgebra
      4.11-18 IsSemisimpleAlgebra
      4.11-19 IsSpecialBiserialAlgebra
      4.11-20 IsStringAlgebra
      4.11-21 IsSymmetricAlgebra
      4.11-22 IsTriangularReduced
      4.11-23 IsWeaklySymmetricAlgebra
      4.11-24 BongartzTest
      4.11-25 IsFiniteTypeAlgebra
    4.12 [33X[0;0YOperations on String Algebras[133X
      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 [33X[0;0YAttributes and Operations (for Quotients) of Path Algebras[133X
      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 [33X[0;0YAttributes and Operations on Elements of Quotients of Path Algebra[133X
      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 [33X[0;0YPredefined classes and classes of (quotients of) path algebras[133X
      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 [33X[0;0YOpposite algebras[133X
      4.16-1 OppositePath
      4.16-2 OppositePathAlgebra
      4.16-3 OppositePathAlgebraElement
      4.16-4 OppositeAlgebraHomomorphism
    4.17 [33X[0;0YTensor products of path algebras[133X
      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 [33X[0;0YOperations on quiver algebras[133X
      4.18-1 QuiverAlgebraOfAmodAeA
      4.18-2 QuiverAlgebraOfeAe
    4.19 [33X[0;0YFinite dimensional algebras over finite fields[133X
      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 [33X[0;0YAlgebras[133X
      4.20-1 LiftingCompleteSetOfOrthogonalIdempotents
      4.20-2 LiftingIdempotent
    4.21 [33X[0;0YSaving and reading quotients of path algebras to and from a file[133X
      4.21-1 ReadAlgebra
      4.21-2 SaveAlgebra
  5 [33X[0;0YGroebner Basis[133X
    5.1 [33X[0;0YConstructing a Groebner Basis[133X
      5.1-1 InfoGroebnerBasis
      5.1-2 GroebnerBasis
    5.2 [33X[0;0YCategories and Properties of Groebner Basis[133X
      5.2-1 IsCompletelyReducedGroebnerBasis
      5.2-2 IsCompleteGroebnerBasis
      5.2-3 IsGroebnerBasis
      5.2-4 IsHomogeneousGroebnerBasis
      5.2-5 IsTipReducedGroebnerBasis
    5.3 [33X[0;0YAttributes and Operations for Groebner Basis[133X
      5.3-1 AdmitsFinitelyManyNontips
      5.3-2 CompletelyReduce
      5.3-3 CompletelyReduceGroebnerBasis
      5.3-4 Enumerator
      5.3-5 IsPrefixOfTipInTipIdeal
      5.3-6 Iterator
      5.3-7 Nontips
      5.3-8 NontipSize
      5.3-9 TipReduce
      5.3-10 TipReduceGroebnerBasis
    5.4 [33X[0;0YRight Groebner Basis[133X
      5.4-1 IsRightGroebnerBasis
      5.4-2 RightGroebnerBasis
      5.4-3 RightGroebnerBasisOfIdeal
  6 [33X[0;0YRight Modules over Path Algebras[133X
    6.1 [33X[0;0YModules of matrix type[133X
      6.1-1 RightModuleOverPathAlgebra
      6.1-2 RightAlgebraModuleToPathAlgebraMatModule
      6.1-3 \=
    6.2 [33X[0;0YCategories Of Matrix Modules[133X
      6.2-1 IsPathAlgebraMatModule
    6.3 [33X[0;0YActing on Module Elements[133X
      6.3-1 ^
    6.4 [33X[0;0YOperations on representations[133X
      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 [33X[0;0YSpecial representations[133X
      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 [33X[0;0YFunctors on representations[133X
      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
    6.7 [33X[0;0YVertex projective modules and submodules thereof[133X
      6.7-1 RightProjectiveModule
      6.7-2 CompletelyReduceGroebnerBasisForModule
      6.7-3 IsLeftDivisible
      6.7-4 IsPathAlgebraModule
      6.7-5 IsPathAlgebraVector
      6.7-6 LeadingCoefficient (of PathAlgebraVector)
      6.7-7 LeadingComponent
      6.7-8 LeadingPosition
      6.7-9 LeadingTerm (of PathAlgebraVector)
      6.7-10 LeftDivision
      6.7-11 ^
      6.7-12 <
      6.7-13 /
      6.7-14 PathAlgebraVector
      6.7-15 ProjectivePathAlgebraPresentation
      6.7-16 RightGroebnerBasisOfModule
      6.7-17 TargetVertex
      6.7-18 UniformGeneratorsOfModule
      6.7-19 Vectorize
  7 [33X[0;0YHomomorphisms of Right Modules over Path Algebras[133X
    7.1 [33X[0;0YCategories and representation of homomorphisms[133X
      7.1-1 IsPathAlgebraModuleHomomorphism
      7.1-2 RightModuleHomOverAlgebra
    7.2 [33X[0;0YGeneralities of homomorphisms[133X
      7.2-1 \= (maps)
      7.2-2 \+ (maps)
      7.2-3 \* (maps)
      7.2-4 CoKernelOfWhat
      7.2-5 IdentityMapping
      7.2-6 ImageElm
      7.2-7 ImagesSet
      7.2-8 ImageOfWhat
      7.2-9 IsInjective
      7.2-10 IsIsomorphism
      7.2-11 IsLeftMinimal
      7.2-12 IsRightMinimal
      7.2-13 IsSplitEpimorphism
      7.2-14 IsSplitMonomorphism
      7.2-15 IsSurjective
      7.2-16 IsZero
      7.2-17 KernelOfWhat
      7.2-18 LeftInverseOfHomomorphism
      7.2-19 MatricesOfPathAlgebraMatModuleHomomorphism
      7.2-20 PathAlgebraOfMatModuleMap
      7.2-21 PreImagesRepresentative
      7.2-22 Range
      7.2-23 RightInverseOfHomomorphism
      7.2-24 Source
      7.2-25 Zero
      7.2-26 ZeroMapping
      7.2-27 HomomorphismFromImages
    7.3 [33X[0;0YHomomorphisms and modules constructed from homomorphisms and modules[133X
      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 [33X[0;0YHomological algebra[133X
    8.1 [33X[0;0YHomological algebra[133X
      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 [33X[0;0YAuslander-Reiten theory[133X
    9.1 [33X[0;0YAlmost split sequences and AR-quivers[133X
      9.1-1 AlmostSplitSequence
      9.1-2 AlmostSplitSequenceInPerpT
      9.1-3 IrreducibleMorphismsEndingIn
      9.1-4 IsTauPeriodic
      9.1-5 PredecessorOfModule
  10 [33X[0;0YChain complexes[133X
    10.1 [33X[0;0YIntroduction[133X
    10.2 [33X[0;0YInfinite lists[133X
      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 [33X[0;0YRepresentation of categories[133X
      10.3-1 IsCat
      10.3-2 CatOfRightAlgebraModules
    10.4 [33X[0;0YMaking a complex[133X
      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 [33X[0;0YInformation about a complex[133X
      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 [33X[0;0YTransforming and combining complexes[133X
      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 [33X[0;0YChain maps[133X
      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 [33X[0;0YProjective resolutions and the bounded derived category[133X
    11.1 [33X[0;0YProjective and injective complexes[133X
      11.1-1 InjectiveResolution
      11.1-2 IsProjectiveComplex
      11.1-3 IsInjectiveComplex
      11.1-4 ProjectiveResolution
    11.2 [33X[0;0YThe bounded derived category[133X
      11.2-1 ProjectiveResolutionOfComplex
      11.2-2 ProjectiveToInjectiveComplex
      11.2-3 TauOfComplex
      11.2-4 [33X[0;0YExample[133X
      11.2-5 StarOfMapBetweenProjectives
  12 [33X[0;0YCombinatorial representation theory[133X
    12.1 [33X[0;0YIntroduction[133X
    12.2 [33X[0;0YDifferent unit forms[133X
      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
  13 [33X[0;0YDegeneration order for modules in finite type[133X
    13.1 [33X[0;0YIntroduction[133X
    13.2 [33X[0;0YBasic definitions[133X
    13.3 [33X[0;0YDefining Auslander-Reiten quiver in finite type[133X
      13.3-1 ARQuiverNumerical
      13.3-2 IsARQuiverNumerical
      13.3-3 NumberOfIndecomposables
      13.3-4 NumberOfProjectives
    13.4 [33X[0;0YElementary operations[133X
      13.4-1 DimensionVector
      13.4-2 DimHom
      13.4-3 DimEnd
      13.4-4 OrbitDim
      13.4-5 OrbitCodim
      13.4-6 DegOrderLEQ
      13.4-7 DegOrderLEQNC
      13.4-8 PrintMultiplicityVector
      13.4-9 PrintMultiplicityVectors
    13.5 [33X[0;0YOperations returning families of modules[133X
      13.5-1 ModulesOfDimVect
      13.5-2 DegOrderPredecessors
      13.5-3 DegOrderDirectPredecessors
      13.5-4 DegOrderPredecessorsWithDirect
      13.5-5 DegOrderSuccessors
      13.5-6 DegOrderDirectSuccessors
      13.5-7 DegOrderSuccessorsWithDirect
  
  
  [32X
