Basic Algebras
- Introduction
- Basic Algebras
- Creation
BasicAlgebra(Q): SeqEnum[Tup] → AlgBasBasicAlgebra(F,R,s,P): AlgFr, SeqEnum, RngIntElt, SeqEnum → AlgBasBasicAlgebra(F,R): AlgFr, SeqEnum → AlgBasTensorProduct(A, B): AlgBas, AlgBas → AlgBasBasicAlgebra(G, k): GrpPerm, FldFin → AlgBasBasicAlgebra(G, k): GrpPC, FldFin → AlgBasBasicAlgebra(G): GrpPerm → AlgBass
- Special Basic Algebras
BasicAlgebra(A): AlgMat → AlgBasBasicAlgebraOfMatrixAlgebra(A): AlgMat → AlgBasBasicAlgebraOfEndomorphismAlgebra(M): ModRng → AlgBasBasicAlgebraOfHeckeAlgebra(G, H, F): GrpPerm, GrpPerm, FldFin → AlgBasBasicAlgebraOfHeckeAlgebra(G, H, F): GrpPC, GrpPC, FldFin → AlgBasBasicAlgebraOfHeckeAlgebra(G, H, F): GrpAb, GrpAb, FldFin → AlgBasBasicAlgebraOfSchurAlgebra(n, r, F): RngIntElt, RngIntElt, FldFin → AlgBasBasicAlgebraOfGroupAlgebra(G,F): GrpPerm, FldFin → AlgBasBasicAlgebraOfGroupAlgebra(G,F): GrpPC, FldFin → AlgBasBasicAlgebraOfGroupAlgebra(G,F): GrpAb, FldFin → AlgBasBasicAlgebra(S): SeqEnum → AlgBasBasicAlgebraOfBlockAlgebra(S): SeqEnum → AlgBasBasicAlgebraOfPrincipalBlock(G,k): GrpPerm, FldFin → AlgBasBasicAlgebraOfExtAlgebra(A): AlgBas → AlgBasBasicAlgebraOfExtAlgebra(A, n): AlgBas, RngIntElt → AlgBasBasicAlgebraOfExtAlgebra(A): Rec → AlgBasOppositeAlgebra(B): AlgBas → AlgBas- Example: Basic Algebra of an Alternating Group Algebra
- Example: Basic Algebra of a Schur Algebra
- A Database of Basic Algebras
- Access Functions
- Elementary Operations
- Boolean Functions
- Homomorphisms
hom<A -> B | S>: AlgBas, AlgBas, ModMatFldElt → MapKernel(phi): Map → ModTupFldImage(phi): Map → AlgBas, MapIsAlgebraHomomorphism(A, B, psi): AlgBas, AlgBas, Mtrx → BoolX * Y: Map, Map → MapIsAlgebraHomomorphism(A, B, psi): AlgBas, AlgBas, Map → BoolIsAlgebraHomomorphism(A, B, psi): AlgBasGrpP, AlgBasGrpP, Map → BoolIsAlgebraHomomorphism(A, B, psi): AlgBasGrpP, AlgBas, Map → BoolIsAlgebraHomomorphism(A, B, psi): AlgBas, AlgBasGrpP, Map → BoolIsAlgebraHomomorphism(psi): Map → Bool
- Subalgebras and Quotient Algebras
- Subalgebras and their Constructions
sub<A | S>: AlgBas, SeqEnum → AlgBas, MapSubalgebraFromBasis(A, V): AlgBas, SeqEnum → AlgBas, MapMaximalIdempotent(A, S): AlgBas, SeqEnum → AlgBasEltMinimalIdentity(A, S): AlgBas, SeqEnum[AlgBasElt] → AlgBasEltCentre(A): AlgBas → AlgBas, MapCentralizer(A,S): AlgBas, SeqEnum → AlgBas, MapMaximalCommutativeSubalgebra(A,S): AlgBas, SeqEnum → AlgBas, Map
- Ideals and their Construction
ideal< A | S>: AlgBas, SeqEnum[AlgBasElt] → ModTupFldideal< A | S>: AlgBasGrpP, SeqEnum[AlgBasElt] → ModTupFldLeftAnnihilator(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 → BoolIsLeftIdeal(A,S): AlgBas, ModTupFld → BoolIsRightIdeal(A, S): AlgBas, ModTupFld → BoolRandomIdealGeneratedBy(A, n): AlgBas, RngIntElt → ModTupFld
- Quotient Algebras
- Units
- Minimal Forms and Gradings
MinimalGeneratorForm(A): AlgBas → RecMinimalGeneratorFormAlgebra(A): AlgBas → AlgBasAssociatedGradedAlgebra(A): AlgBas → AlgBasGradedCapHomomorphism(A): AlgBas → ModMatFldEltGradedCapHomomorphism(A, B, mu): AlgBas, AlgBas, ModMatFldElt → ModMatFldEltBuildHomomorphismFromGradedCap(A, B, phi): AlgBas, AlgBas, ModMatFldElt → ModMatFldEltChangeIdempotents(A, S): AlgBas, SeqEnum → AlgBas, MapChangeIdempotents(A, S): AlgBas, GrpPermElt → AlgBas, Map- Example: Graded Properties of a Schur Algebra
- Example: Reordering Primitive Idempotents
- Automorphisms and Isomorphisms
GradedAutomorphismGroupMatchingIdempotents(A): AlgBas → GrpMat, SeqEnum, SecEnumGradedAutomorphismGroup(A): AlgBas → GrpMat, SeqEnum[ModMatFldElt], SeqEnum[ModMatFldElt], SeqEnum[ModMatFldElt]IsGradedIsomorphic(A, B): AlgBas, AlgBas → Bool, ModMatFldEltAutomorphismGroupMatchingIdempotents(A): AlgBas → AlgBas, ModMatFldEltAutomorphismGroup(A): AlgBas → GrpMat, SeqEnum, SeqEnum, SeqEnumInnerAutomorphismGroup(A): AlgBas → GrpMatIsIsomorphic(A, B): AlgBas, AlgBas → Bool, Map- Example: Automorphism Group of a Group Algebra
- Example: Changing a Presentation
- Example: Graded Group Algebras of Order 32
- Quiver and Relations
- Modules over Basic Algebras
- Indecomposable Projective Modules
ProjectiveModule(B, i): AlgBas, RngIntElt → ModRngPathTree(B, i): AlgBas, RngIntElt → ModRngActionGenerator(B, i): AlgBas, RngIntElt → SeqEnumIdempotentActionGenerators(B, i): AlgBas, RngIntElt → SeqEnumNonIdempotentActionGenerators(B, i): AlgBas, RngIntElt → SeqEnumInjection(B, i, v): AlgBas, RngIntElt, ModRngElt → AlgBasElt
- Creation
AModule(B, Q): AlgBas, SeqEnum[AlgMatElt] → ModRngProjectiveModule(B, S): AlgBas, SeqEnum[RngIntElt] → ModAlg, SeqEnum, SeqEnumIrreducibleModule(B, i): AlgBas, RngIntElt → ModAlgSimpleModule(B, i): AlgBas, RngIntElt → ModAlgZeroModule(B): AlgBas → ModAlgRightRegularModule(B): AlgBas → ModAlgRegularRepresentation(v): AlgBasElt → AlgMatEltRestriction(M, B, xi): ModAlgBas, AlgBas, ModMatFldElt → ModAlgBasChangeAlgebra(M, B, xi): ModAlgBas, AlgBas, Map → ModAlgBasChangeAlgebra(M, B, xi): ModAlgBas, AlgBas, ModMatFldElt → ModAlgBasJacobsonRadical(M): ModAlg → ModAlgSocle(M): ModAlg → ModAlg
- Access Functions
- Predicates
- Elementary Operations
- Homomorphisms of Modules
- Creation
AHom(M, N): ModAlg, ModAlg → ModMatFldPHom(M,N): ModAlg, ModAlg → ModMatFldZeroMap(M, N): ModAlg, ModAlg → ModMatFldLiftHomomorphism(x, n): ModAlgElt, RngIntElt → ModMatFldEltLiftHomomorphism(X, N): SeqEnum[ModAlgElt], SeqEnum[RngIntElt] → ModMatFldEltPushout(M, f1, N1, f2, N2): ModAlg, ModMatFldElt, ModAlg, ModMatFldElt, ModAlg → ModAlg, ModMatFldElt, ModMatFldEltPullback(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, ModMatFldEltCompactProjectiveResolution(M, n): ModAlg, RngIntElt → RecCompactProjectiveResolutionsOfSimpleModules(A,n): AlgBas, RngIntElt → SeqEnumSyzygyModule(M, n): ModAlg, RngIntElt → ModAlgSimpleHomologyDimensions(M): ModAlg → SeqEnum- Example: Projective Resolution for a Quiver Algebra
- Example: Quotient of an Extraspecial Group Algebra
- Duals and Injectives
Dual(M): ModAlg → ModAlgBaseChangeMatrix(A): AlgBas → ModAlg- Injective Modules
InjectiveModule(B, i): AlgBas, RngIntElt → ModAlgInjectiveHull(M): ModAlg → ModAlg, ModMatFldElt, SeqEnum[ModMatFldElt], SeqEnum[ModMatFldElt], SeqEnum[RngIntElt]InjectiveResolution(M, n): ModAlg, RngIntElt → ModCpx, ModMatFldEltCompactInjectiveResolution(M, n): ModAlg, RngIntElt → RecInjectiveSyzygyModule(M, n): ModAlg, RngIntElt → ModAlgSimpleCohomologyDimensions(M): ModAlg → SeqEnum- Example: Injective Modules of a Quiver Algebra
- Cohomology
CohomologyRingGenerators(P): Rec → RecCohomologyRightModuleGenerators(P, Q, CQ): Rec, Rec, Rec → RecCohomologyLeftModuleGenerators(P, CP, Q): Tup, Tup, Tup → TupDegreesOfCohomologyGenerators(C): Rec → SeqEnumCohomologyGeneratorToChainMap(P,Q,C,n): ModCpx, ModCpx, Rec, RngIntElt → MapChnCohomologyGeneratorToChainMap(P, C, n): ModCpx, Tup, RngIntElt → MapChn- Example: Cohomology of the Principal Block of M11
- 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, SeqEnumRestrictResolution(PR, RD): Rec, Rec → ModCpxRestrictionChainMap(P1,P2): Rec, Rec → MapChnRestrictionOfGenerators(PR1, PR2, AC1, AC2, REL2): Rec, Rec, Rec, Rec, Rec → SeqEnumInflationMap(PR2, PR1, AC2, AC1, REL1, theta): Rec, Rec, Rec, Rec, Rec → SeqEnum- Example: Restriction and Inflation of Cohomology
- A-infinity Algebra Structures on Group Cohomology
AInfinityRecord(G,n): Grp, RngIntElt → RecMasseyProduct(Aoo,terms): Rec, SeqEnum[RngElt] → RngEltHighProduct(Aoo,terms): Rec, SeqEnum[RngElt] → RngEltHighMap(Aoo,terms): Rec, SeqEnum[RngElt] → MapChn- Example: A-infinity Structure in Characteristic 2
- Example: A-infinity Structure in Characteristic 3
- Homological Algebra Toolkit
ActionMatrix(A,x): AlgBas, Mtrx → ModMatFldEltCohomologyRingQuotient(CR): Rec → Rng, MapLiftToChainmap(P,f,d): ModCpx, Mtrx, RngIntElt → MapChnNullHomotopy(f): MapChn → MapChnIsNullHomotopy(f,H): MapChn, MapChn → BoolEltChainmapToCohomology(f,CR): MapChn, Rec → RngEltCohomologyToChainmap(xi,CR,P): RngElt, Rec, ModCpx → MapChn- Example: Null Homotopies for a Cyclic Group
- Bibliography