BASIC ALGEBRAS

DETAILS

 
Introduction

 
Basic Algebras

      Creation
            BasicAlgebra(Q) : SeqEnum[Tup] -> AlgBas
            BasicAlgebra(G, k) : GrpPerm, FldFin -> AlgBas
            BasicAlgebra(FA, N, LR, R) : AlgFP, RngIntElt, SeqEnum, SeqEnum -> AlgBas
            TensorProduct(A, B) : AlgBas, AlgBas-> AlgBas

      Access Functions
            NumberOfProjectives(A) : AlgBas -> RngIntElt
            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
            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 (H81E1)

 
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
            JacobsonRadical(M) : ModAlg -> ModAlg
            Socle(M) : ModAlg -> ModAlg

      Access Functions
            Algebra(M) : ModAlg -> AlgBas
            Dimension(M) : ModAlg -> RngIntElt
            Action(M) : ModAlg -> AlgMat

      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 (H81E2)

 
Homomorphisms

      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, S) : SeqEnum[ModAlgElt], SeqEnum[RngIntElt] -> ModMatFldElt
            Pushout(M, f1, N1, f2, N2) : ModAlg, ModMatFldElt, ModAlg, ModMatFldElt, ModAlg -> ModAlg, ModMatFldElt, ModMatFldElt
            Pullback(N, f1, M1, f2, M2) : 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
            ProjectiveCover(M) : ModAlg -> ModAlg, ModMatFldElt, SeqEnum[ModMatFldElt], SeqEnum[ModMatFldElt], SeqEnum[RngIntElt]
            ProjectiveResolution(M, n) : ModAlg, RngIntElt -> ModCpx, ModMatFldElt
            CompactProjectiveResolution(M, n) : ModAlg, RngIntElt -> Tup
            SyzygyModule(M, n) : ModAlg, RngIntElt -> ModAlg
            SimpleHomologyDimensions(M) : ModAlg -> SeqEnum
            Example AlgBas_Homomorphisms (H81E3)

 
Opposite Algebras

      Creation
            OppositeAlgebra(B) : AlgBas -> AlgBas
            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 -> List, ModMatFldElt
            InjectiveSyzygyModule(M, n) : ModAlg, RngIntElt -> ModAlg
            SimpleCohomologyDimensions(M) : ModAlg -> SeqEnum
            Example AlgBas_Opposite (H81E4)

 
Cohomology
      CohomologyRingGenerators(P) : Tup -> Tup
      CohomologyRightModuleGenerators(P, Q, CQ) : Tup, Tup, Tup -> Tup
      CohomologyLeftModuleGenerators(P, CP, Q) : Tup, Tup, Tup -> Tup
      DegreesOfCohomologyGenerators(C) : Tup -> SeqEnum
      CohomologyGeneratorToChainMap(P,Q,C,n) : ModCpx, ModCpx, Tup, RngIntElt -> MapChn
      CohomologyGeneratorToChainMap(P,C,n) : ModCpx, Tup, RngIntElt -> MapChn
      Example AlgBas_Cohomology (H81E5)