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)
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)
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
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)
Bibliography
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Version: V2.19 of
Mon Dec 17 14:40:36 EST 2012