Introduction

 
Construction of Associative Algebras

      Construction of an Associative Structure Constant Algebra
            AssociativeAlgebra< R, n | Q : parameters > : Rng, RngIntElt, SeqEnum -> AlgAss
            AssociativeAlgebra< R, n | T : parameters > : Rng, RngIntElt, SeqEnum -> AlgAss
            AssociativeAlgebra(A) : AlgGen -> AlgAss
            ChangeBasis(A, B) : AlgAss, [AlgAssElt] -> AlgAss

      Associative Structure Constant Algebras from other Algebras
            Algebra(A) : AlgGrp -> AlgAss, Map
            Algebra(F, E) : FldFin, FldFin -> AlgAss, Map;
            AlgebraOverCenter(A) : Alg -> AlgAss, Map;

 
Operations on Algebras and their Elements

      Operations on Algebras
            Centre(A) : AlgAss -> AlgAss
            Centralizer(A, S) : AlgAss, AlgAss -> AlgAss
            Idealizer(A, B: parameters) : AlgAss, AlgAss -> AlgAss
            LieAlgebra(A) : AlgAss -> AlgGen, Map
            CommutatorModule(A, B) : AlgAss, AlgAss -> ModTupRng
            CommutatorIdeal(A, B) : AlgAss, AlgAss -> AlgAss
            LeftAnnihilator(A, B) : AlgAss, AlgAss -> AlgAss, AlgAss
            RightAnnihilator(A, B) : AlgAss, AlgAss -> AlgAss, AlgAss
            Example AlgAss_liealg (H90E1)
            RestrictionOfScalars(A) : AlgAss[FldAlg] -> AlgAss, Map
            Example AlgAss_restrict (H90E2)

      Operations on Elements
            Centralizer(A, s) : AlgAss, AlgAssElt -> AlgAss
            LieBracket(a, b) : AlgAssElt, AlgAssElt -> AlgAssElt
            IsScalar(a) : AlgAssElt -> BoolElt, RngElt
            RepresentationMatrix(a, M : parameters) : AlgAssElt, AlgAss -> AlgMatElt

      Representations
            MatrixAlgebra(A) : AlgAss -> AlgMat
            MatrixAlgebra(A, M : parameters) : AlgAss, AlgAss -> AlgMat, Map
            RegularRepresentation(A : parameters) : AlgAss -> AlgMat, Map

      Decomposition of an Algebra
            JacobsonRadical(A) : AlgAssV -> AlgAssV
            Example AlgAss_jac_rad (H90E3)
            DirectSumDecomposition(A) : AlgAssV -> [ AlgAssV ], [ AlgAssVElt ]
            CentralIdempotents(A) : AlgAssV -> SeqEnum, SeqEnum
            Example AlgAss_id_pots (H90E4)

 
Orders

      Construction of Orders
            Order(R, S) : Rng, SeqEnum[AlgAssVElt] -> AlgAssVOrd
            Order(S) : SeqEnum[AlgAssVElt[FldAlg]] -> AlgAssVOrd
            Order(S, I) : SeqEnum[AlgAssVElt[FldAlg]], SeqEnum[RngOrdFracIdl] -> AlgAssVOrd
            Order(A, m, I) : AlgAssV[FldOrd], AlgMatElt[FldOrd], SeqEnum[RngOrdFracIdl] -> AlgAssVOrd
            Order(A, pm) : AlgAssV[FldOrd], PMat -> AlgAssVOrd
            Example AlgAss_ord_creat_cyc (H90E5)
            Example AlgAss_ord_creat_cyc (H90E6)
            MaximalOrder(A) : AlgAssV[FldRat] -> AlgAssVOrd
            MaximalOrder(A) : AlgAssV[FldAlg] -> AlgAssVOrd
            MaximalOrder(O) : AlgAssVOrd[RngInt] -> AlgAssVOrd
            MaximalOrderFinite(A) : AlgAssV[FldFunRat] -> AlgAssVOrd
            MaximalOrderInfinite(A) : AlgAssV[FldFunRat] -> AlgAssVOrd
            MaximalOrderFinite(A) : AlgAssV[FldFun] -> AlgAssVOrd
            MaximalOrder(O) : AlgAssVOrd[RngUPol] -> AlgAssVOrd
            Example AlgAss_max_ord (H90E7)
            Example AlgAss_max_ord_alg (H90E8)
            RestrictionOfScalars(O) : AlgAssVOrd[RngOrd] -> AlgAssVOrd, Map
            Example AlgAss_restrict (H90E9)

      Attributes
            BaseRing(O) : AlgAssVOrd -> Rng
            Algebra(O) : AlgAssVOrd -> AlgAssV
            Degree(O) : AlgAssVOrd -> RngIntElt
            Discriminant(O) : AlgAssVOrd[RngOrd] -> RngOrdIdl
            FactoredDiscriminant(O) : AlgAssVOrd[RngOrd] -> [Tup]
            MultiplicationTable(O) : AlgAssVOrd -> SeqEnum
            Module(O) : AlgAssVOrd[RngOrd] -> PMat
            TraceZeroSubspace(O) : AlgAssVOrd -> SeqEnum

      Bases of Orders
            PseudoBasis(O) : AlgAssVOrd[RngOrd] -> SeqEnum
            PseudoMatrix(O) : AlgAssVOrd[RngOrd]> -> PMat
            Basis(O) : AlgAssVOrd -> SeqEnum
            ZBasis(O) : AlgAssVOrd[RngOrd] -> [AlgAssVElt]
            Generators(O) : AlgAssVOrd -> [AlgAssVElt]
            LocalBasis(O, p) : AlgAssVOrd, RngOrdIdl -> [AlgAssElt]
            Example AlgAss_bases (H90E10)

      Predicates
            O1 eq O2 : AlgAssVOrd, AlgAssVOrd -> BoolElt
            x in O : AlgAssVElt, AlgAssVOrd -> BoolElt

      Operations with Orders
            Adjoin(O, x) : AlgAssVOrd, AlgAssVElt -> AlgAssVOrd
            O1 + O2 : AlgAssVOrd[RngOrd], AlgAssVOrd[RngOrd] -> AlgAssVOrd
            O1 meet O2 : AlgAssVOrd[RngOrd], AlgAssVOrd[RngOrd] -> AlgAssVOrd
            O ^ x : AlgAssVOrd, AlgAssElt -> AlgAssvOrd
            Example AlgAss_sumandadjoin (H90E11)

 
Elements of Orders

      Construction of Elements
            O ! 0 : AlgAssVOrd, RngIntElt -> AlgAssVOrdElt
            O ! 1 : AlgAssVOrd, RngIntElt -> AlgAssVOrdElt
            O . i : AlgAssVOrd, RngIntElt -> AlgAssVElt
            O ! x : AlgAssVOrd, Any -> AlgAssVOrdElt

      Arithmetic of Elements
            x + y : AlgAssVOrdElt, AlgAssVOrdElt -> AlgAssVOrdElt
            x - y : AlgAssVOrdElt, AlgAssVOrdElt -> AlgAssVOrdElt
            - x : AlgAssVOrdElt -> AlgAssVOrdElt
            x * y : AlgAssVOrdElt, AlgAssVOrdElt -> AlgAssVOrdElt
            u * c : AlgAssVOrdElt, RngElt -> AlgAssVOrdElt
            x / y : AlgAssVOrdElt, AlgAssVOrdElt -> AlgAssVElt
            x div y : AlgAssVOrdElt, AlgAssVOrdElt -> AlgAssVOrdElt
            x ^ n : AlgAssVOrdElt, RngIntElt -> AlgAssVOrdElt

      Predicates on Elements
            x eq y : AlgAssVOrdElt, AlgAssVOrdElt -> BoolElt
            x ne y : AlgAssVOrdElt, AlgAssVOrdElt -> BoolElt
            IsZero(x) : AlgAssVOrdElt -> BoolElt
            IsUnit(a) : AlgAssVOrdElt -> BoolElt
            IsScalar(x) : AlgAssVOrdElt -> BoolElt, RngElt

      Other Operations with Elements
            ElementToSequence(x) : AlgAssVOrdElt -> SeqEnum
            Norm(x) : AlgAssVOrdElt -> RngElt
            Trace(x) : AlgAssVOrdElt -> RngElt
            LeftRepresentationMatrix(e) : AlgAssVOrdElt -> AlgMatElt
            RepresentationMatrix(a) : AlgAssVOrdElt -> AlgMatElt
            CharacteristicPolynomial(x) : AlgAssVOrdElt -> RngUPolElt
            MinimalPolynomial(x) : AlgAssVOrdElt -> RngUPolElt

 
Ideals of Orders

      Construction of Ideals
            lideal<O | E> : AlgAssVOrd, [AlgAssVOrdElt] -> AlgAssVOrdIdl
            lideal<O | M> : AlgAssVOrd, PMat -> AlgAssVOrdIdl
            O * e : AlgAssVOrd, RngElt -> AlgAssVOrdIdl

      Attributes of Ideals
            Order(I) : AlgAssVOrdIdl -> AlgAssVOrd
            Algebra(I) : AlgAssVOrdIdl -> AlgAssV
            LeftOrder(I) : AlgAssVOrdIdl -> AlgAssVOrd
            ArithmeticRadical(O, p) : AlgAssVOrd[RngOrd], RngOrdIdl -> AlgAssVOrdIdl
            RadicalIdealizer(O, p) : AlgAssVOrd[RngOrd], RngOrdIdl -> AlgAssVOrd
            Neighbors(I, p) : AlgAssVOrdIdl, RngOrdIdl -> SeqEnum[AlgAssVOrdIdl]
            ElementsOfNorm(I, n) : AlgAssVOrdIdl, RngElt -> SeqEnum

      Bases of Ideals
            Denominator(I) : AlgAssVOrdIdl -> RngElt
            PseudoBasis(I) : AlgAssVOrdIdl[RngOrd] -> SeqEnum
            PseudoMatrix(I) : AlgAssVOrdIdl[RngOrd] -> PMat
            Basis(I) : AlgAssVOrdIdl -> SeqEnum
            BasisMatrix(I) : AlgAssVOrdIdl -> AlgMatElt
            ZBasis(I) : AlgAssVOrdIdl[RngOrd] -> [AlgAssVOrdElt]
            Generators(I) : AlgAssVOrdIdl[RngOrd] -> [AlgAssVOrdElt]
            LocalBasis(I, p) : AlgAssVOrdIdl, RngOrdIdl -> [AlgAssElt]
            PseudoBasis(I, R) : AlgAssVOrdIdl[RngOrd], Str -> SeqEnum

      Arithmetic for Ideals
            I + J : AlgAssVOrdIdl, AlgAssVOrdIdl -> AlgAssVOrdIdl
            I * J: AlgAssVOrdIdl, AlgAssVOrdIdl -> AlgAssVOrdIdl, AlgAssVOrdIdl
            a * I: RngElt, AlgAssVOrdIdl -> AlgAssVOrdIdl
            A * I : RngOrdFracIdl, AlgAssVOrdIdl[RngOrd]) -> AlgAssVOrdIdl
            Colon(J, I): AlgAssVOrdIdl[RngOrd], AlgAssVOrdIdl[RngOrd] -> PMat
            MultiplicatorRing(I): AlgAssVOrdIdl -> AlgAssVOrd

      Predicates on Ideals
            IsLeftIdeal(I) : AlgAssVOrdIdl -> BoolElt
            I eq J : AlgAssVOrdIdl, AlgAssVOrdIdl -> BoolElt
            I subset J : AlgAssVOrdIdl, AlgAssVOrdIdl -> BoolElt
            a in I : AlgAssVElt, AlgAssVOrdIdl -> BoolElt

      Other Operations on Ideals
            Norm(I) : AlgAssVOrdIdl[RngOrd] -> RngOrdIdl
            Example AlgAss_sumandadjoin (H90E12)

 
Bibliography

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