# FramedAlgebra(R, UP)¶

A FramedAlgebra is a FiniteRankAlgebra together with a fixed R-module basis.

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: (%, %, %) -> %
basis: () -> Vector %

from FramedModule R

characteristic: () -> NonNegativeInteger
characteristicPolynomial: % -> UP

from FiniteRankAlgebra(R, UP)

charthRoot: % -> Union(%, failed) if R has CharacteristicNonZero
coerce: % -> OutputForm
coerce: Integer -> %
coerce: R -> %

from Algebra R

commutator: (%, %) -> %
convert: % -> InputForm if R has Finite
convert: % -> Vector R

from FramedModule R

convert: Vector R -> %

from FramedModule R

coordinates: % -> Vector R

from FramedModule R

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

from FiniteRankAlgebra(R, UP)

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

from FiniteRankAlgebra(R, UP)

coordinates: Vector % -> Matrix R

from FramedModule R

discriminant: () -> R

discriminant() = determinant(traceMatrix()).

discriminant: Vector % -> R

from FiniteRankAlgebra(R, UP)

enumerate: () -> List % if R has Finite

from Finite

hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

index: PositiveInteger -> % if R has Finite

from Finite

latex: % -> String

from SetCategory

leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftRecip: % -> Union(%, failed)

from MagmaWithUnit

lookup: % -> PositiveInteger if R has Finite

from Finite

minimalPolynomial: % -> UP if R has Field

from FiniteRankAlgebra(R, UP)

norm: % -> R

from FiniteRankAlgebra(R, UP)

one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

random: () -> % if R has Finite

from Finite

rank: () -> PositiveInteger

from FiniteRankAlgebra(R, UP)

recip: % -> Union(%, failed)

from MagmaWithUnit

regularRepresentation: % -> Matrix R

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

regularRepresentation: (%, Vector %) -> Matrix R

from FiniteRankAlgebra(R, UP)

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

from FiniteRankAlgebra(R, UP)

represents: Vector R -> %

from FramedModule R

rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightRecip: % -> Union(%, failed)

from MagmaWithUnit

sample: %

from AbelianMonoid

size: () -> NonNegativeInteger if R has Finite

from Finite

smaller?: (%, %) -> Boolean if R has Finite

from Comparable

subtractIfCan: (%, %) -> Union(%, failed)
trace: % -> R

from FiniteRankAlgebra(R, UP)

traceMatrix: () -> Matrix R

traceMatrix() is the n-by-n matrix ( Tr(vi * vj) ), where v1, …, vn are the elements of the fixed basis.

traceMatrix: Vector % -> Matrix R

from FiniteRankAlgebra(R, UP)

zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

BasicType

BiModule(%, %)

BiModule(R, R)

CancellationAbelianMonoid

Comparable if R has Finite

ConvertibleTo InputForm if R has Finite

Finite if R has Finite

FiniteRankAlgebra(R, UP)

Magma

MagmaWithUnit

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip

unitsKnown