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BASIC ALGEBRAS

 
Acknowledgements
 
Introduction
 
Basic Algebras
      Creation
      Special Basic Algebras
      Access Functions
      Elementary Operations
      Boolean Functions
 
Homomorphisms
 
Subalgebras and Quotient Algebras
      Subalgebras and their Constructions
      Ideals and their Construction
      Quotient Algebras
 
Minimal Forms and Gradings
 
Automorphisms and Isomorphisms
 
Modules over Basic Algebras
      Indecomposable Projective Modules
      Creation
      Access Functions
      Predicates
      Elementary Operations
 
Homomorphisms of Modules
      Creation
      Access Functions
      Projective Covers and Resolutions
 
Duals and Injectives
      Injective Modules
 
Cohomology
      Ext-Algebras
 
Group Algebras of p-groups
      Access Functions
      Projective Resolutions
      Cohomology Generators
      Cohomology Rings
      Restrictions and Inflations
 
A-infinity Algebra Structures on Group Cohomology
      Homological Algebra Toolkit
 
Bibliography







DETAILS

 
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) : AlgBas, RngIntElt -> AlgBas
            BasicAlgebraOfExtAlgebra(A) : Rec -> AlgBas
            OppositeAlgebra(B) : AlgBas -> AlgBas
            Example AlgBas_GroupAlgebra (H85E1)
            Example AlgBas_SchurAlgebra (H85E2)

      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 (H85E3)
            Example AlgBas_BasicAlgebras-2 (H85E4)
            Example AlgBas_BasicAlgebras-3 (H85E5)

      Boolean Functions
            IsDimensionCompatible(B) : AlgBas -> Bool
            IsPathTree(B) : AlgBas -> Bool
            IsCommutative(A) : AlgBas -> Bool
            IsCentral(A,x) : AlgBas, AlgBasElt -> BoolElt

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

 
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 (H85E6)
      Example AlgBas_GradedHomomorphisms-2 (H85E7)

 
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
      IsIsomorphic(A, B) : AlgBas, AlgBas -> Bool, Map
      Example AlgBas_Automorphism group (H85E8)
      Example AlgBas_modify presentation (H85E9)
      Example AlgBas_GradedGroupAlgebras (H85E10)

 
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 (H85E11)
            Example AlgBas_ChangeAlgebras-2 (H85E12)
            Example AlgBas_RadicalLayers (H85E13)

      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 (H85E14)
            Example AlgBas_AModules-2 (H85E15)

 
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 (H85E16)
            Example AlgBas_Homomorphisms-2 (H85E17)

 
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 (H85E18)

 
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 (H85E19)

      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 (H85E20)

 
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 (H85E21)

 
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 (H85E22)
      Example AlgBas_A-infinity mod 3 (H85E23)

      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 (H85E24)

 
Bibliography

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013