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Acknowledgements
Introduction
Basic Algebras
Creation
Access Functions
Elementary Operations
Modules over Basic Algebras
Indecomposable Projective Modules
Creation
Access Functions
Predicates
Elementary Operations
Homomorphisms
Creation
Access Functions
Projective Covers
Opposite Algebras
Creation
Injective Modules
Cohomology
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(G, k) : GrpPerm, FldFin -> AlgBas
BasicAlgebra(F,R,s,P): AlgFr, SeqEnum, RngIntElt, SeqEnum -> AlgBas
BasicAlgebra(F,R) : AlgFr, SeqEnum -> AlgBas
TensorProduct(A, B) : AlgBas, AlgBas-> AlgBas
Access Functions
NumberOfProjectives(B) : 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 (H88E1)
Example AlgBas_BasicAlgebras-2 (H88E2)
Example AlgBas_BasicAlgebras-3 (H88E3)
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 (H88E4)
Example AlgBas_AModules-2 (H88E5)
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, 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
ProjectiveCover(M) : ModAlg -> ModAlg, ModMatFldElt, SeqEnum[ModMatFldElt], SeqEnum[ModMatFldElt], SeqEnum[RngIntElt]
ProjectiveResolution(M, n) : ModAlg, RngIntElt -> ModCpx, ModMatFldElt
CompactProjectiveResolution(M, n) : ModAlg, RngIntElt -> Rec
SyzygyModule(M, n) : ModAlg, RngIntElt -> ModAlg
SimpleHomologyDimensions(M) : ModAlg -> SeqEnum
Example AlgBas_Homomorphisms (H88E6)
Example AlgBas_Homomorphisms-2 (H88E7)
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 -> Rec
InjectiveSyzygyModule(M, n) : ModAlg, RngIntElt -> ModAlg
SimpleCohomologyDimensions(M) : ModAlg -> SeqEnum
Example AlgBas_Opposite (H88E8)
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 (H88E9)
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 (H88E10)
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 (H88E11)
Example AlgBas_A-infinity mod 3 (H88E12)
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
Bibliography
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