- 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 (H92E1)
- Example AlgBas_SchurAlgebra (H92E2)
- A Database of Basic Algebras
- Access Functions
- Elementary Operations
- Boolean Functions
- 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
- Units
- 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 (H92E7)
- Example AlgBas_GradedHomomorphisms-2 (H92E8)
- 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 (H92E9)
- Example AlgBas_modify presentation (H92E10)
- Example AlgBas_GradedGroupAlgebras (H92E11)
- Quiver and Relations
- 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
- Predicates
- Elementary Operations
- 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
- 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 (H92E18)
- Example AlgBas_Homomorphisms-2 (H92E19)
- 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 (H92E20)
- 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 (H92E21)
- Ext-Algebras
- Group Algebras of p-groups
- Access Functions
- Projective Resolutions
- Cohomology Generators
- Cohomology Rings
- 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 (H92E23)
- 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 (H92E24)
- Example AlgBas_A-infinity mod 3 (H92E25)
- 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 (H92E26)
- Bibliography
V2.28, 13 July 2023