FiniteRankAlgebra(R, UP)¶

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

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (%, R) -> %

from RightModule R

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

*: (R, %) -> %

from LeftModule R

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

=: (%, %) -> Boolean

from BasicType

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %
associator: (%, %, %) -> %
characteristic: () -> NonNegativeInteger
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
coerce: % -> OutputForm
coerce: Integer -> %
coerce: R -> %

from Algebra R

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

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

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.

discriminant: Vector % -> R

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

hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

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

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)
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

AbelianGroup

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(%, %)

BiModule(R, R)

CancellationAbelianMonoid

Magma

MagmaWithUnit

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip

unitsKnown