Introduction

 
Basic Algebras

      Creation
            BasicAlgebra(Q) : SeqEnum[Tup] -> AlgBas
            BasicAlgebra(F,R,s,P): AlgFr, SeqEnum, RngIntElt, SeqEnum -> AlgBas
            BasicAlgebra(F,R) : AlgFr, SeqEnum -> AlgBas
            TensorProduct(A, B) : AlgBas, AlgBas-> AlgBas
            BasicAlgebra(G, k) : GrpPerm, FldFin -> AlgBas

      Special Basic Algebras
            BasicAlgebra(A): AlgMat -> AlgBas
            BasicAlgebraOfEndomorphismAlgebra(M): ModRng -> AlgBas
            BasicAlgebraOfHeckeAlgebra(G, H, F): GrpPerm, GrpPerm, FldFin) -> AlgBas
            BasicAlgebraOfSchurAlgebra(n, r, F): RngIntElt, RngIntElt, FldFin -> AlgBas
            BasicAlgebraOfGroupAlgebra(G,F): GrpPerm, FldFin -> AlgBas
            BasicAlgebra(S) : SeqEnum -> AlgBas
            BasicAlgebraOfBlockAlgebra(S) : SeqEnum -> AlgBas
            BasicAlgebraOfPrincipalBlock(G,k) : GrpPerm, FldFin -> AlgBas
            BasicAlgebraOfExtAlgebra(A) : AlgBas -> AlgBas
            BasicAlgebraOfExtAlgebra(A, n) : AlgBas, RngIntElt -> AlgBas
            BasicAlgebraOfExtAlgebra(A) : Rec -> AlgBas
            OppositeAlgebra(B) : AlgBas -> AlgBas
            Example AlgBas_GroupAlgebra (H94E1)
            Example AlgBas_SchurAlgebra (H94E2)

      A Database of Basic Algebras
            BasicAlgebraFromGroup(A, p, b) : MonStgElt, RngIntElt, RngIntElt -> AlgBas
            BasicAlgebraGroupNames() : -> SetIndx
            BasicAlgebraFromSchur(n, r, p) : RngIntElt, RngIntElt, RngIntElt -> AlgBas
            Example AlgBas_BasicDatabase (H94E3)

      Access Functions
            B . i : AlgBas, RngIntElt -> AlgBasElt
            BaseRing(B) : AlgBas -> Rng
            VectorSpace(B) : AlgBas -> ModTupFld
            Dimension(B) : AlgBas -> RngIntElt
            Basis(B) : AlgBas -> SeqEnum
            Generators(B) : AlgBas -> SeqEnum
            IdempotentGenerators(B) : AlgBas -> SeqEnum
            IdempotentPositions(B) : AlgBas -> SeqEnum
            NonIdempotentGenerators(B) : AlgBas -> SeqEnum
            Random(B) : AlgBas -> AlgBasElt
            NumberOfProjectives(B) : AlgBas -> RngIntElt
            NumberOfGenerators(B) : AlgBas -> RngIntElt
            DimensionsOfProjectiveModules(B) : AlgBas -> SeqEnum
            DimensionsOfInjectiveModules(B) : AlgBas -> SeqEnum

      Elementary Operations
            a + b : AlgBasElt, AlgBasElt -> AlgBasElt
            a * b : AlgBasElt, AlgBasElt -> AlgBasElt
            a ^ n : AlgBasElt, RngIntElt -> AlgBasElt
            Example AlgBas_BasicAlgebras (H94E4)
            Example AlgBas_BasicAlgebras-2 (H94E5)
            Example AlgBas_BasicAlgebras-3 (H94E6)

      Boolean Functions
            IsDimensionCompatible(B) : AlgBas -> Bool
            IsPathTree(B) : AlgBas -> Bool

            Commutativity
                  IsCommutative(A) : AlgBas -> Bool
                  IsCentral(A,x) : AlgBas, AlgBasElt -> BoolElt

            Other Properties
                  IsSymmetric(A) : AlgBas -> BoolElt
                  IsSelfInjective(A) : AlgBas -> BoolElt

 
Homomorphisms
      hom<A -> B | S> : AlgBas, AlgBas, ModMatFldElt -> Map
      Kernel(phi) : Map -> ModTupFld
      Image(phi) : Map -> AlgBas, Map
      IsAlgebraHomomorphism(A, B, psi) : AlgBas, AlgBas, Mtrx -> Bool
      X * Y : Map, Map -> Map
      IsAlgebraHomomorphism(A, B, psi) : AlgBas, AlgBas, Map -> Bool
      IsAlgebraHomomorphism(psi): Map -> Bool

 
Subalgebras and Quotient Algebras

      Subalgebras and their Constructions
            sub<A | S> : AlgBas, SeqEnum -> AlgBas, Map
            SubalgebraFromBasis(A, V) : AlgBas, SeqEnum -> AlgBas, Map
            MaximalIdempotent(A, S) : AlgBas, SeqEnum -> AlgBasElt
            MinimalIdentity(A, S) : AlgBas, SeqEnum[AlgBasElt] -> AlgBasElt
            Centre(A) : AlgBas -> AlgBas, Map
            Centralizer(A,S) : AlgBas, SeqEnum -> AlgBas, Map
            MaximalCommutativeSubalgebra(A,S) : AlgBas, SeqEnum -> AlgBas, Map

      Ideals and their Construction
            ideal< A | S> : AlgBas, SeqEnum[AlgBasElt] -> ModTupFld
            LeftAnnihilator(A, S) : AlgBas, SeqEnum[AlgBasElt] -> SeqEnum[AlgBasElt]
            RightAnnihilator(A, S) : AlgBas, SeqEnum[AlgBasElt] -> SeqEnum[AlgBaselt]
            Annihilator(A,S) : AlgBas, SeqEnum[AlgBasElt] -> SeqEnum[AlgBasElt]
            IsIdeal(A, S) : AlgBas, ModTupFld -> Bool
            IsLeftIdeal(A,S) : AlgBas, ModTupFld -> Bool
            IsRightIdeal(A, S) : AlgBas, ModTupFld -> Bool
            RandomIdealGeneratedBy(A, n) : AlgBas, RngIntElt -> ModTupFld

      Quotient Algebras
            quo<A | S> : AlgBas, ModTupFld -> AlgBas, Map
            CoverAlgebra(A) : AlgBas -> AlgBas, ModMatFldElt
            GradedCoverAlgebra(A) : AlgBas -> AlgBas, ModMatFldElt
            TruncatedAlgebra(A, n) : AlgBas, RngIntElt -> AlgBas, ModMatFldElt

      Units
            GeneratorsOfGroupOfUnits(A) : AlgBas -> SeqEnum, SeqEnum
            NoncentralGeneratorsOfGroupOfUnits(A) : AlgBas -> SeqEnum, SeqEnum

 
Minimal Forms and Gradings
      MinimalGeneratorForm(A) : AlgBas -> Rec
      MinimalGeneratorFormAlgebra(A) : AlgBas -> AlgBas
      AssociatedGradedAlgebra(A) : AlgBas -> AlgBas
      GradedCapHomomorphism(A) : AlgBas -> ModMatFldElt
      GradedCapHomomorphism(A, B, mu) : AlgBas, AlgBas, ModMatFldElt -> ModMatFldElt
      BuildHomomorphismFromGradedCap(A, B, phi) : AlgBas, AlgBas, ModMatFldElt -> ModMatFldElt
      ChangeIdempotents(A, S) : AlgBas, SeqEnum -> AlgBas, Map
      Example AlgBas_GradedHomomorphism (H94E7)
      Example AlgBas_GradedHomomorphisms-2 (H94E8)

 
Automorphisms and Isomorphisms
      GradedAutomorphismGroupMatchingIdempotents(A) : AlgBas -> GrpMat, SeqEnum, SecEnum
      GradedAutomorphismGroup(A) : AlgBas -> GrpMat, SeqEnum[ModMatFldElt], SeqEnum[ModMatFldElt], SeqEnum[ModMatFldElt]
      IsGradedIsomorphic(A, B) : AlgBas, AlgBas -> Bool, ModMatFldElt
      AutomorphismGroupMatchingIdempotents(A) : AlgBas -> AlgBas, ModMatFldElt
      AutomorphismGroup(A) : AlgBas -> GrpMat, SeqEnum, SeqEnum, SeqEnum
      InnerAutomorphismGroup(A) : AlgBas -> GrpMat
      IsIsomorphic(A, B) : AlgBas, AlgBas -> Bool, Map
      Example AlgBas_Automorphism group (H94E9)
      Example AlgBas_modify presentation (H94E10)
      Example AlgBas_GradedGroupAlgebras (H94E11)

 
Quiver and Relations
      Quiver(A) : AlgBas -> SeqEnum
      QuiverAndRelations(A) : AlgBas -> SeqEnum, SeqEnum, SeqEnum
      Example AlgBas_Quiver (H94E12)

 
Modules over Basic Algebras

      Indecomposable Projective Modules
            ProjectiveModule(B, i) : AlgBas, RngIntElt -> ModRng
            PathTree(B, i) : AlgBas, RngIntElt -> ModRng
            ActionGenerator(B, i) : AlgBas, RngIntElt -> SeqEnum
            IdempotentActionGenerators(B, i) : AlgBas, RngIntElt -> SeqEnum
            NonIdempotentActionGenerators(B, i) : AlgBas, RngIntElt -> SeqEnum
            Injection(B, i, v) : AlgBas, RngIntElt, ModRngElt -> AlgBasElt

      Creation
            AModule(B, Q) : AlgBas, SeqEnum[AlgMatElt] -> ModRng
            ProjectiveModule(B, S) : AlgBas, SeqEnum[RngIntElt] -> ModAlg, SeqEnum, SeqEnum
            IrreducibleModule(B, i) : AlgBas, RngIntElt -> ModAlg
            ZeroModule(B) : AlgBas -> ModAlg
            RightRegularModule(B) : AlgBas -> ModAlg
            RegularRepresentation(v) : AlgBasElt -> AlgMatElt
            Restriction(M, B, xi) : ModAlgBas, AlgBas, ModMatFldElt -> ModAlgBas
            ChangeAlgebra(M, B, xi) : ModAlgBas , AlgBas, Map -> ModAlgBas
            JacobsonRadical(M) : ModAlg -> ModAlg
            Socle(M) : ModAlg -> ModAlg

      Access Functions
            Algebra(M) : ModAlg -> AlgBas
            Dimension(M) : ModAlg -> RngIntElt
            Action(M) : ModAlg -> AlgMat
            IsomorphismTypesOfRadicalLayers(M) : ModAlgBas -> SeqEnum
            IsomorphismTypesOfSocleLayers(M) : ModAlgBas -> SeqEnum
            IsomorphismTypesOfBasicAlgebraSequence(S) : SeqEnum -> SeqEnum
            Example AlgBas_restriction-to-center (H94E13)
            Example AlgBas_ChangeAlgebras-2 (H94E14)
            Example AlgBas_RadicalLayers (H94E15)

      Predicates
            IsSemisimple(M) : ModAlg -> BoolElt, SeqEnum
            IsProjective(M) : ModAlg -> BoolElt, SeqEnum
            IsInjective(M) : ModAlg -> BoolElt, SeqEnum

      Elementary Operations
            m * b : ModAlgElt, AlgBasElt -> ModAlgElt
            Example AlgBas_AModules (H94E16)
            Example AlgBas_AModules-2 (H94E17)

 
Homomorphisms of Modules

      Creation
            AHom(M, N) : ModAlg, ModAlg -> ModMatFld
            PHom(M,N) : ModAlg, ModAlg -> ModMatFld
            ZeroMap(M, N) : ModAlg, ModAlg -> ModMatFld
            LiftHomomorphism(x, n) : ModAlgElt, RngIntElt -> ModMatFldElt
            LiftHomomorphism(X, N) : SeqEnum[ModAlgElt], SeqEnum[RngIntElt] -> ModMatFldElt
            Pushout(M, f1, N1, f2, N2) : ModAlg, ModMatFldElt, ModAlg, ModMatFldElt, ModAlg -> ModAlg, ModMatFldElt, ModMatFldElt
            Pullback(f1, M1, f2, M2, N) : ModAlg, ModMatFldElt, ModAlg, ModMatFldElt, ModAlg -> ModAlg, ModMatFldElt, ModMatFldElt

      Access Functions
            IsModuleHomomorphism(f) : ModMatFldElt -> BoolElt
            Domain(f) : ModMatFldElt -> ModAlg
            Codomain(f) : ModMatFldElt -> ModAlg
            Kernel(f) : ModMatFldElt -> ModAlg,ModMatFldElt
            Cokernel(f) : ModMatFldElt -> ModAlg,ModMatFldElt

      Projective Covers and Resolutions
            ProjectiveCover(M) : ModAlg -> ModAlg, ModMatFldElt, SeqEnum[ModMatFldElt], SeqEnum[ModMatFldElt], SeqEnum[RngIntElt]
            ProjectiveResolution(M, n) : ModAlg, RngIntElt -> ModCpx, ModMatFldElt
            CompactProjectiveResolution(M, n) : ModAlg, RngIntElt -> Rec
            CompactProjectiveResolutionsOfSimpleModules(A,n) : AlgBas, RngIntElt -> SeqEnum
            SyzygyModule(M, n) : ModAlg, RngIntElt -> ModAlg
            SimpleHomologyDimensions(M) : ModAlg -> SeqEnum
            Example AlgBas_Homomorphisms (H94E18)
            Example AlgBas_Homomorphisms-2 (H94E19)

 
Duals and Injectives
      Dual(M) : ModAlg -> ModAlg
      BaseChangeMatrix(A) : AlgBas -> ModAlg

      Injective Modules
            InjectiveModule(B, i) : AlgBas, RngIntElt -> ModAlg
            InjectiveHull(M) : ModAlg -> ModAlg, ModMatFldElt, SeqEnum[ModMatFldElt], SeqEnum[ModMatFldElt], SeqEnum[RngIntElt]
            InjectiveResolution(M, n) : ModAlg, RngIntElt -> ModCpx, ModMatFldElt
            CompactInjectiveResolution(M, n) : ModAlg, RngIntElt -> Rec
            InjectiveSyzygyModule(M, n) : ModAlg, RngIntElt -> ModAlg
            SimpleCohomologyDimensions(M) : ModAlg -> SeqEnum
            Example AlgBas_Opposite (H94E20)

 
Cohomology
      CohomologyRingGenerators(P) : Rec -> Rec
      CohomologyRightModuleGenerators(P, Q, CQ) : Rec, Rec, Rec -> Rec
      CohomologyLeftModuleGenerators(P, CP, Q) : Tup, Tup, Tup -> Tup
      DegreesOfCohomologyGenerators(C) : Rec -> SeqEnum
      CohomologyGeneratorToChainMap(P,Q,C,n) : ModCpx, ModCpx, Rec, RngIntElt -> MapChn
      CohomologyGeneratorToChainMap(P, C, n) : ModCpx, Tup, RngIntElt -> MapChn
      Example AlgBas_Cohomology-2 (H94E21)

      Ext-Algebras
            ExtAlgebra(A, n): AlgBas,RngIntElt -> Rec
            BasicAlgebraOfExtAlgebra(ext) : Rec -> AlgBas
            BasicAlgebraOfExtAlgebra(A): AlgBas -> AlgBas
            BasicAlgebraOfExtAlgebra(A, n): AlgBas, RngIntElt -> AlgBas
            SumOfBettiNumbersOfSimpleModules(A, n) : AlgBas, RngIntElt -> RngIntElt
            Example AlgBas_ExtAlgebra (H94E22)

 
Group Algebras of p-groups

      Access Functions
            Group(A) : AlgBasGrpP -> Grp
            PCGroup(A) : AlgBasGrpP -> Grp
            PCMap(A) : AlgBasGrpP -> Map
            AModule(M) : ModGrp -> ModAlg
            GModule(M) : AlgBasGrpP -> ModGrp, ModGrp
            GModule(M) : ModAlgBas -> ModGrp

      Projective Resolutions
            ResolutionData(A) : AlgBasGrpP -> Rec
            CompactProjectiveResolutionPGroup(M, n) : ModAlgBas, RngIntElt -> Rec
            ProjectiveResolutionPGroup(PR) : Rec -> ModCpx
            ProjectiveResolution(M, n) : ModAlgBas, RngIntElt -> ModCpx, ModMatFldElt
            ProjectiveResolution(PR) : Rec -> ModCpx, ModMatFldElt

      Cohomology Generators
            AllCompactChainMaps(PR) : Rec -> Rec
            CohomologyElementToChainMap(P, d, n) : ModCpx ,RngIntElt, RngIntElt -> MapChn
            CohomologyElementToCompactChainMap(PR, d, n): Rec, RngIntElt, RngIntElt -> ModMatFldElt

      Cohomology Rings
            CohomologyRing(k, n) : ModAlgBas, RngIntElt -> Rec
            MinimalRelations(R) : Rec -> SeqEnum

      Restrictions and Inflations
            RestrictionData(A,B) : AlgBasGrpP, AlgBasGrpP -> ModMatFldElt, ModMatFldElt, SeqEnum
            RestrictResolution(PR, RD) : Rec, Rec -> ModCpx
            RestrictionChainMap(P1,P2) : Rec, Rec -> MapChn
            RestrictionOfGenerators(PR1, PR2, AC1, AC2, REL2) : Rec, Rec, Rec, Rec, Rec -> SeqEnum
            InflationMap(PR2, PR1, AC2, AC1, REL1, theta) : Rec, Rec, Rec, Rec, Rec -> SeqEnum
            Example AlgBas_CohomologyRing (H94E23)

 
A-infinity Algebra Structures on Group Cohomology
      AInfinityRecord(G,n) : Grp, RngIntElt -> Rec
      MasseyProduct(Aoo,terms) : Rec, SeqEnum[RngElt] -> RngElt
      HighMap(Aoo,terms) : Rec, SeqEnum[RngElt] -> MapChn
      Example AlgBas_A-infinity mod 2 (H94E24)
      Example AlgBas_A-infinity mod 3 (H94E25)

      Homological Algebra Toolkit
            ActionMatrix(A,x) : AlgBas, Mtrx -> ModMatFldElt
            CohomologyRingQuotient(CR) : Rec -> Rng,Map
            LiftToChainmap(P,f,d) : ModCpx, Mtrx, RngIntElt -> MapChn
            NullHomotopy(f) : MapChn -> MapChn
            IsNullHomotopy(f,H) : MapChn, MapChn -> BoolElt
            ChainmapToCohomology(f,CR) : MapChn, Rec -> RngElt
            CohomologyToChainmap(xi,CR,P) : RngElt, Rec, ModCpx -> MapChn
            Example AlgBas_Nullhomotopy (H94E26)

 
Bibliography

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