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)
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)
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)
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
Mon Dec 17 14:40:36 EST 2012