[Next][Prev] [Right] [Left] [Up] [Index] [Root]

ASSOCIATIVE ALGEBRAS

 
Acknowledgements
 
Introduction
 
Construction of Associative Algebras
      Construction of an Associative Structure Constant Algebra
      Associative Structure Constant Algebras from other Algebras
 
Operations on Algebras and their Elements
      Operations on Algebras
      Operations on Elements
      Representations
      Decomposition of an Algebra
 
Orders
      Creation of Orders
      Attributes
      Bases of Orders
      Predicates
      Operations with Orders
 
Elements of Orders
      Creation of Elements
      Arithmetic of Elements
      Predicates on Elements
      Other Operations with Elements
 
Ideals of Orders
      Creation of Ideals
      Attributes of Ideals
      Arithmetic for Ideals
      Predicates on Ideals
      Other Operations on Ideals
 
Quaternionic Orders
 
Bibliography







DETAILS

 
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 (H81E1)

      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 (H81E2)
            DirectSumDecomposition(A) : AlgAssV -> [ AlgAssV ], [ AlgAssVElt ]
            CentralIdempotents(A) : AlgAssV -> SeqEnum, SeqEnum
            Example AlgAss_id_pots (H81E3)

 
Orders

      Creation 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 (H81E4)
            Example AlgAss_ord_creat_cyc (H81E5)
            MaximalOrder(A) : AlgAssV[FldRat] -> AlgAssVOrd
            Example AlgAss_max_ord (H81E6)

      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
            Basis(O) : AlgAssVOrd -> SeqEnum
            PseudoBasis(O) : AlgAssVOrd[RngOrd] -> SeqEnum
            PseudoMatrix(O) : AlgAssVOrd[RngOrd]> -> PMat
            ZBasis(O) : AlgAssVOrd[RngOrd] -> [AlgAssVElt]
            Generators(O) : AlgAssVOrd -> [AlgAssVElt]
            Example AlgAss_bases (H81E7)

      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 (H81E8)

 
Elements of Orders

      Creation of Elements
            O ! 0 : AlgAssVOrd, RngIntElt -> AlgAssVOrdElt
            O ! 1 : AlgAssVOrd, RngIntElt -> AlgAssVOrdElt
            O . i : AlgAssVOrd, RngIntElt -> AlgAssVElt
            O ! x : AlgAssVOrd, Any -> AlgAssVOrdElt
            Random(O) : AlgAssVOrd -> 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

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

      Attributes of Ideals
            Algebra(I) : AlgAssVOrdIdl -> AlgAssV
            Order(I) : AlgAssVOrdIdl -> AlgAssVOrd
            LeftOrder(I) : AlgAssVOrdIdl[RngOrd] -> AlgAssVOrd
            Basis(I) : AlgAssVOrdIdl -> SeqEnum
            BasisMatrix(I) : AlgAssVOrdIdl -> AlgMatElt
            PseudoBasis(I) : AlgAssVOrdIdl[RngOrd] -> SeqEnum
            PseudoMatrix(I) : AlgAssVOrdIdl[RngOrd] -> PMat
            ZBasis(I) : AlgAssVOrdIdl[RngOrd] -> [AlgAssVOrdElt]
            Generators(I) : AlgAssVOrdIdl[RngOrd] -> [AlgAssVOrdElt]
            Denominator(I) : AlgAssVOrdIdl -> RngElt

      Arithmetic for Ideals
            I + J : AlgAssVOrdIdl, AlgAssVOrdIdl -> AlgAssVOrdIdl
            I * J: AlgAssVOrdIdl, AlgAssVOrdIdl -> AlgAssVOrdIdl, AlgAssVOrdIdl
            a * I: RngElt, AlgAssVOrdIdl -> 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 (H81E9)

 
Quaternionic Orders

 
Bibliography

[Next][Prev] [Right] [____] [Up] [Index] [Root]
Version: V2.19 of Wed Apr 24 15:09:57 EST 2013