Construction of General Algebras and their Elements
Construction of a General Algebra
Algebra< R, n | Q > : Rng, RngIntElt, SeqEnum -> AlgGen
AssociativeAlgebra< R, n | Q > : Rng, RngIntElt, SeqEnum -> AlgAss
QuaternionAlgebra< K | a, b > : Rng, RngElt, RngElt -> AlgQuat
LieAlgebra< R, n | Q > : Rng, RngIntElt, SeqEnum -> AlgLie
LieAlgebra(A) : AlgAss -> AlgLie
GroupAlgebra(R, G) : Rng, Grp -> AlgGrp
MatrixAlgebra(R, n) : Rng, RngInt -> AlgMat
Construction of an Element of a General Algebra
Zero(A) : AlgGen -> AlgGenElt
One(A) : AlgGen -> AlgGenElt
Random(A) : AlgGen -> AlgGenElt
Construction of Subalgebras, Ideals and Quotient Algebras
Subalgebras and Ideals
sub< A | L > : AlgGen, List -> AlgGen, Map
lideal< A | L > : AlgGen, List -> AlgGen, Map
rideal< A | L > : AlgGen, List -> AlgGen, Map
ideal< A | L > : AlgGen, List -> AlgGen, Map
Quotient Algebras
quo< A | L > : AlgGen, List -> AlgGen, Map
A / S : AlgGen, AlgGen -> AlgGen
Operations on Algebras and Subalgebras
Invariants of an Algebra
CoefficientRing(A) : AlgGen -> Rng
Dimension(A) : AlgGen -> RngIntElt
# A : AlgGen -> RngIntElt
Changing Rings
ChangeRing(A, S) : AlgGen, Rng -> AlgGen, Map
ChangeRing(A, S, f) : AlgGen, Rng, Map -> AlgGen, Map
Bases
BasisElement(A, i) : AlgGen, RngIntElt -> AlgGenElt
Basis(A) : AlgGen -> [ AlgGenElt ]
IsIndependent(Q) : [ AlgGen ] -> BoolElt
ExtendBasis(S, A) : AlgGen, AlgGen -> [ AlgElt ]
Decomposition of an Algebra
CompositionSeries(A) : AlgGen -> [ AlgGen ], [ AlgGen ], AlgMatElt
CompositionFactors(A) : AlgGen -> [ AlgGen ]
MinimalLeftIdeals(A : parameters) : AlgGen -> [ AlgGen ], BoolElt
MaximalLeftIdeals(A : parameters) : AlgGen -> [ AlgGen ], BoolElt
JacobsonRadical(A) : AlgGen -> AlgGen
IsSemisimple(A) : AlgGen -> BoolElt
IsSimple(A) : AlgGen -> BoolElt
Example AlgGen_quaternions (H79E1)
Operations on Subalgebras
IsZero(A) : AlgGen -> BoolElt
A eq B : AlgGen, AlgGen -> BoolElt
A ne B : AlgGen, AlgGen -> BoolElt
A subset B : AlgGen, AlgGen -> BoolElt
A notsubset B : AlgGen, AlgGen -> BoolElt
A meet B : AlgGen, AlgGen -> AlgGen
A * B : AlgGen, AlgGen -> AlgGen
A ^ n : AlgGen, RngIntElt -> AlgGen
Morphism(A, B) : AlgGen, AlgGen -> Map
Operations on Elements of an Algebra
Operations on Elements
a + b : AlgGenElt, AlgGenElt -> AlgGenElt
- a : AlgGenElt -> AlgGenElt
a - b : AlgGenElt, AlgGenElt -> AlgGenElt
a * b : AlgGenElt, AlgGenElt -> AlgGenElt
a * r : AlgGenElt, RngElt -> AlgGenElt
a / r : AlgGenElt, RngElt -> AlgGenElt
a ^ n : AlgGenElt, RngIntElt -> AlgGenElt
MinimalPolynomial(a) : AlgGenElt -> RngUPolElt
Parent(a) : AlgGenElt -> AlgGen
Comparisons and Membership
a eq b : AlgGenElt, AlgGenElt -> BoolElt
a ne b : AlgGenElt, AlgGenElt -> BoolElt
a in A : AlgGenElt, AlgGen -> BoolElt
a notin A : AlgGenElt, AlgGen -> BoolElt
Predicates on Elements
IsZero(a) : AlgGenElt -> BoolElt
IsOne(a) : AlgGenElt -> BoolElt
IsMinusOne(a) : AlgGenElt -> BoolElt
IsUnit(a) : AlgGenElt -> BoolElt, AlgGenElt
IsRegular(a) : AlgGenElt -> BoolElt
IsZeroDivisor(a) : AlgGenElt -> BoolElt
IsIdempotent(a) : AlgGenElt -> BoolElt
IsNilpotent(a) : AlgGenElt -> BoolElt, RngIntElt
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Version: V2.19 of
Mon Dec 17 14:40:36 EST 2012