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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
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;
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 (H84E1)
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 (H84E2)
DirectSumDecomposition(A) : AlgAssV -> [ AlgAssV ], [ AlgAssVElt ]
CentralIdempotents(A) : AlgAssV -> SeqEnum, SeqEnum
Example AlgAss_id_pots (H84E3)
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 (H84E4)
Example AlgAss_ord_creat_cyc (H84E5)
MaximalOrder(A) : AlgAssV[FldRat] -> AlgAssVOrd
Example AlgAss_max_ord (H84E6)
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 (H84E7)
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 (H84E8)
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 (H84E9)
Bibliography
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