FiniteRankAlgebra(R, UP)

algcat.spad line 70 [edit on github]

A FiniteRankAlgebra is an algebra over a commutative ring R which is a free R-module of finite rank.

* : (%, %) -> %
from Magma
* : (%, R) -> %
from RightModule(R)
* : (R, %) -> %
from LeftModule(R)
* : (Integer, %) -> %
from AbelianGroup
* : (NonNegativeInteger, %) -> %
from AbelianMonoid
* : (PositiveInteger, %) -> %
from AbelianSemiGroup
+ : (%, %) -> %
from AbelianSemiGroup
- : % -> %
from AbelianGroup
- : (%, %) -> %
from AbelianGroup
0 : () -> %
from AbelianMonoid
1 : () -> %
from MagmaWithUnit
= : (%, %) -> Boolean
from BasicType
^ : (%, NonNegativeInteger) -> %
from MagmaWithUnit
^ : (%, PositiveInteger) -> %
from Magma
annihilate? : (%, %) -> Boolean
from Rng
antiCommutator : (%, %) -> %
from NonAssociativeSemiRng
associator : (%, %, %) -> %
from NonAssociativeRng
characteristic : () -> NonNegativeInteger
from NonAssociativeRing
characteristicPolynomial : % -> UP

characteristicPolynomial(a) returns the characteristic polynomial of the regular representation of a with respect to any basis.

charthRoot : % -> Union(%, "failed") if R has CharacteristicNonZero
from CharacteristicNonZero
coerce : R -> %
from Algebra(R)
coerce : Integer -> %
from NonAssociativeRing
coerce : % -> OutputForm
from CoercibleTo(OutputForm)
commutator : (%, %) -> %
from NonAssociativeRng
coordinates : (Vector(%), Vector(%)) -> Matrix(R)

coordinates([v1, ..., vm], basis) returns the coordinates of the vi's with to the basis basis. The coordinates of vi are contained in the ith row of the matrix returned by this function.

coordinates : (%, Vector(%)) -> Vector(R)

coordinates(a, basis) returns the coordinates of a with respect to the basis basis.

discriminant : Vector(%) -> R

discriminant([v1, .., vn]) returns determinant(traceMatrix([v1, .., vn])).

latex : % -> String
from SetCategory
leftPower : (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower : (%, PositiveInteger) -> %
from Magma
leftRecip : % -> Union(%, "failed")
from MagmaWithUnit
minimalPolynomial : % -> UP if R has Field

minimalPolynomial(a) returns the minimal polynomial of a.

norm : % -> R

norm(a) returns the determinant of the regular representation of a with respect to any basis.

one? : % -> Boolean
from MagmaWithUnit
opposite? : (%, %) -> Boolean
from AbelianMonoid
plenaryPower : (%, PositiveInteger) -> %
from NonAssociativeAlgebra(R)
rank : () -> PositiveInteger

rank() returns the rank of the algebra.

recip : % -> Union(%, "failed")
from MagmaWithUnit
regularRepresentation : (%, Vector(%)) -> Matrix(R)

regularRepresentation(a, basis) returns the matrix m of the linear map defined by left multiplication by a with respect to the basis basis. That is for all x we have coordinates(a*x, basis) = m*coordinates(x, basis).

represents : (Vector(R), Vector(%)) -> %

represents([a1, .., an], [v1, .., vn]) returns a1*v1 + ... + an*vn.

rightPower : (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower : (%, PositiveInteger) -> %
from Magma
rightRecip : % -> Union(%, "failed")
from MagmaWithUnit
sample : () -> %
from AbelianMonoid
subtractIfCan : (%, %) -> Union(%, "failed")
from CancellationAbelianMonoid
trace : % -> R

trace(a) returns the trace of the regular representation of a with respect to any basis.

traceMatrix : Vector(%) -> Matrix(R)

traceMatrix([v1, .., vn]) is the n-by-n matrix ( Tr(vi * vj) )

zero? : % -> Boolean
from AbelianMonoid
~= : (%, %) -> Boolean
from BasicType

CharacteristicNonZero

Algebra(R)

RightModule(%)

Monoid

AbelianMonoid

BiModule(R, R)

NonAssociativeAlgebra(R)

CancellationAbelianMonoid

MagmaWithUnit

NonAssociativeRing

AbelianGroup

LeftModule(%)

LeftModule(R)

SetCategory

Rng

TwoSidedRecip

Magma

SemiGroup

BiModule(%, %)

unitsKnown

CoercibleTo(OutputForm)

AbelianSemiGroup

NonAssociativeSemiRing

Module(R)

RightModule(R)

NonAssociativeRng

Ring

SemiRng

NonAssociativeSemiRng

CharacteristicZero

BasicType

SemiRing