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\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra naalgc.spad}
\author{Johannes Grabmeier, Robert Wisbauer}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{category MONAD Monad}
<<category MONAD Monad>>=
)abbrev category MONAD Monad
++ Authors: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 11 June 1991
++ Basic Operations: *, **
++ Related Constructors: SemiGroup, Monoid, MonadWithUnit
++ Also See:
++ AMS Classifications:
++ Keywords: Monad, binary operation
++ Reference:
++ N. Jacobson: Structure and Representations of Jordan Algebras
++ AMS, Providence, 1968
++ Description:
++ Monad is the class of all multiplicative monads, i.e. sets
++ with a binary operation.
Monad(): Category == SetCategory with
--operations
*: (%,%) -> %
++ a*b is the product of \spad{a} and b in a set with
++ a binary operation.
rightPower: (%,PositiveInteger) -> %
++ rightPower(a,n) returns the \spad{n}-th right power of \spad{a},
++ i.e. \spad{rightPower(a,n) := rightPower(a,n-1) * a} and
++ \spad{rightPower(a,1) := a}.
leftPower: (%,PositiveInteger) -> %
++ leftPower(a,n) returns the \spad{n}-th left power of \spad{a},
++ i.e. \spad{leftPower(a,n) := a * leftPower(a,n-1)} and
++ \spad{leftPower(a,1) := a}.
**: (%,PositiveInteger) -> %
++ a**n returns the \spad{n}-th power of \spad{a},
++ defined by repeated squaring.
add
import RepeatedSquaring(%)
x:% ** n:PositiveInteger == expt(x,n)
rightPower(a,n) ==
one? n => a
res := a
for i in 1..(n-1) repeat res := res * a
res
leftPower(a,n) ==
one? n => a
res := a
for i in 1..(n-1) repeat res := a * res
res
@
\section{category MONADWU MonadWithUnit}
<<category MONADWU MonadWithUnit>>=
)abbrev category MONADWU MonadWithUnit
++ Authors: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 11 June 1991
++ Basic Operations: *, **, 1
++ Related Constructors: SemiGroup, Monoid, Monad
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Keywords: Monad with unit, binary operation
++ Reference:
++ N. Jacobson: Structure and Representations of Jordan Algebras
++ AMS, Providence, 1968
++ Description:
++ MonadWithUnit is the class of multiplicative monads with unit,
++ i.e. sets with a binary operation and a unit element.
++ Axioms
++ leftIdentity("*":(%,%)->%,1) \tab{30} 1*x=x
++ rightIdentity("*":(%,%)->%,1) \tab{30} x*1=x
++ Common Additional Axioms
++ unitsKnown---if "recip" says "failed", that PROVES input wasn't a unit
MonadWithUnit(): Category == Monad with
--constants
1: constant -> %
++ 1 returns the unit element, denoted by 1.
--operations
one?: % -> Boolean
++ one?(a) tests whether \spad{a} is the unit 1.
rightPower: (%,NonNegativeInteger) -> %
++ rightPower(a,n) returns the \spad{n}-th right power of \spad{a},
++ i.e. \spad{rightPower(a,n) := rightPower(a,n-1) * a} and
++ \spad{rightPower(a,0) := 1}.
leftPower: (%,NonNegativeInteger) -> %
++ leftPower(a,n) returns the \spad{n}-th left power of \spad{a},
++ i.e. \spad{leftPower(a,n) := a * leftPower(a,n-1)} and
++ \spad{leftPower(a,0) := 1}.
"**": (%,NonNegativeInteger) -> %
++ \spad{a**n} returns the \spad{n}-th power of \spad{a},
++ defined by repeated squaring.
recip: % -> Union(%,"failed")
++ recip(a) returns an element, which is both a left and a right
++ inverse of \spad{a},
++ or \spad{"failed"} if such an element doesn't exist or cannot
++ be determined (see unitsKnown).
leftRecip: % -> Union(%,"failed")
++ leftRecip(a) returns an element, which is a left inverse of \spad{a},
++ or \spad{"failed"} if such an element doesn't exist or cannot
++ be determined (see unitsKnown).
rightRecip: % -> Union(%,"failed")
++ rightRecip(a) returns an element, which is a right inverse of
++ \spad{a}, or \spad{"failed"} if such an element doesn't exist
++ or cannot be determined (see unitsKnown).
add
import RepeatedSquaring(%)
one? x == x = 1
x:% ** n:NonNegativeInteger ==
zero? n => 1
expt(x,n pretend PositiveInteger)
rightPower(a: %,n: NonNegativeInteger) ==
zero? n => 1
res := 1
for i in 1..n repeat res := res * a
res
leftPower(a: %,n: NonNegativeInteger) ==
zero? n => 1
res := 1
for i in 1..n repeat res := a * res
res
@
\section{category NARNG NonAssociativeRng}
<<category NARNG NonAssociativeRng>>=
)abbrev category NARNG NonAssociativeRng
++ Author: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 03 July 1991
++ Basic Operations: +, *, -, **
++ Related Constructors: Rng, Ring, NonAssociativeRing
++ Also See:
++ AMS Classifications:
++ Keywords: not associative ring
++ Reference:
++ R.D. Schafer: An Introduction to Nonassociative Algebras
++ Academic Press, New York, 1966
++ Description:
++ NonAssociativeRng is a basic ring-type structure, not necessarily
++ commutative or associative, and not necessarily with unit.
++ Axioms
++ x*(y+z) = x*y + x*z
++ (x+y)*z = x*z + y*z
++ Common Additional Axioms
++ noZeroDivisors ab = 0 => a=0 or b=0
NonAssociativeRng(): Category == Join(AbelianGroup,Monad) with
associator: (%,%,%) -> %
++ associator(a,b,c) returns \spad{(a*b)*c-a*(b*c)}.
commutator: (%,%) -> %
++ commutator(a,b) returns \spad{a*b-b*a}.
antiCommutator: (%,%) -> %
++ antiCommutator(a,b) returns \spad{a*b+b*a}.
add
associator(x,y,z) == (x*y)*z - x*(y*z)
commutator(x,y) == x*y - y*x
antiCommutator(x,y) == x*y + y*x
@
\section{category NASRING NonAssociativeRing}
<<category NASRING NonAssociativeRing>>=
)abbrev category NASRING NonAssociativeRing
++ Author: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 11 June 1991
++ Basic Operations: +, *, -, **
++ Related Constructors: NonAssociativeRng, Rng, Ring
++ Also See:
++ AMS Classifications:
++ Keywords: non-associative ring with unit
++ Reference:
++ R.D. Schafer: An Introduction to Nonassociative Algebras
++ Academic Press, New York, 1966
++ Description:
++ A NonAssociativeRing is a non associative rng which has a unit,
++ the multiplication is not necessarily commutative or associative.
NonAssociativeRing(): Category == Join(NonAssociativeRng,MonadWithUnit) with
--operations
characteristic: NonNegativeInteger
++ characteristic() returns the characteristic of the ring.
--we can not make this a constant, since some domains are mutable
coerce: Integer -> %
++ coerce(n) coerces the integer n to an element of the ring.
add
n:Integer
coerce(n) == n * 1$%
@
\section{category NAALG NonAssociativeAlgebra}
<<category NAALG NonAssociativeAlgebra>>=
)abbrev category NAALG NonAssociativeAlgebra
++ Author: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 11 June 1991
++ Basic Operations: +, -, *, **
++ Related Constructors: Algebra
++ Also See:
++ AMS Classifications:
++ Keywords: nonassociative algebra
++ Reference:
++ R.D. Schafer: An Introduction to Nonassociative Algebras
++ Academic Press, New York, 1966
++ Description:
++ NonAssociativeAlgebra is the category of non associative algebras
++ (modules which are themselves non associative rngs).
++ Axioms
++ r*(a*b) = (r*a)*b = a*(r*b)
NonAssociativeAlgebra(R:CommutativeRing): Category == _
Join(NonAssociativeRng, Module R) with
--operations
plenaryPower : (%,PositiveInteger) -> %
++ plenaryPower(a,n) is recursively defined to be
++ \spad{plenaryPower(a,n-1)*plenaryPower(a,n-1)} for \spad{n>1}
++ and \spad{a} for \spad{n=1}.
add
plenaryPower(a,n) ==
one? n => a
n1 : PositiveInteger := (n-1)::NonNegativeInteger::PositiveInteger
plenaryPower(a,n1) * plenaryPower(a,n1)
@
\section{category FINAALG FiniteRankNonAssociativeAlgebra}
<<category FINAALG FiniteRankNonAssociativeAlgebra>>=
)abbrev category FINAALG FiniteRankNonAssociativeAlgebra
++ Author: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 12 June 1991
++ Basic Operations: +,-,*,**, someBasis
++ Related Constructors: FramedNonAssociativeAlgebra, FramedAlgebra,
++ FiniteRankAssociativeAlgebra
++ Also See:
++ AMS Classifications:
++ Keywords: nonassociative algebra, basis
++ References:
++ R.D. Schafer: An Introduction to Nonassociative Algebras
++ Academic Press, New York, 1966
++
++ R. Wisbauer: Bimodule Structure of Algebra
++ Lecture Notes Univ. Duesseldorf 1991
++ Description:
++ A FiniteRankNonAssociativeAlgebra is a non associative algebra over
++ a commutative ring R which is a free \spad{R}-module of finite rank.
FiniteRankNonAssociativeAlgebra(R:CommutativeRing):
Category == NonAssociativeAlgebra R with
someBasis : () -> Vector %
++ someBasis() returns some \spad{R}-module basis.
rank : () -> PositiveInteger
++ rank() returns the rank of the algebra as \spad{R}-module.
conditionsForIdempotents: Vector % -> List Polynomial R
++ conditionsForIdempotents([v1,...,vn]) determines a complete list
++ of polynomial equations for the coefficients of idempotents
++ with respect to the \spad{R}-module basis \spad{v1},...,\spad{vn}.
structuralConstants: Vector % -> Vector Matrix R
++ structuralConstants([v1,v2,...,vm]) calculates the structural
++ constants \spad{[(gammaijk) for k in 1..m]} defined by
++ \spad{vi * vj = gammaij1 * v1 + ... + gammaijm * vm},
++ where \spad{[v1,...,vm]} is an \spad{R}-module basis
++ of a subalgebra.
leftRegularRepresentation: (% , Vector %) -> Matrix R
++ leftRegularRepresentation(a,[v1,...,vn]) returns the matrix of
++ the linear map defined by left multiplication by \spad{a}
++ with respect to the \spad{R}-module basis \spad{[v1,...,vn]}.
rightRegularRepresentation: (% , Vector %) -> Matrix R
++ rightRegularRepresentation(a,[v1,...,vn]) returns the matrix of
++ the linear map defined by right multiplication by \spad{a}
++ with respect to the \spad{R}-module basis \spad{[v1,...,vn]}.
leftTrace: % -> R
++ leftTrace(a) returns the trace of the left regular representation
++ of \spad{a}.
rightTrace: % -> R
++ rightTrace(a) returns the trace of the right regular representation
++ of \spad{a}.
leftNorm: % -> R
++ leftNorm(a) returns the determinant of the left regular representation
++ of \spad{a}.
rightNorm: % -> R
++ rightNorm(a) returns the determinant of the right regular
++ representation of \spad{a}.
coordinates: (%, Vector %) -> Vector R
++ coordinates(a,[v1,...,vn]) returns the coordinates of \spad{a}
++ with respect to the \spad{R}-module basis \spad{v1},...,\spad{vn}.
coordinates: (Vector %, Vector %) -> Matrix R
++ coordinates([a1,...,am],[v1,...,vn]) returns a matrix whose
++ i-th row is formed by the coordinates of \spad{ai}
++ with respect to the \spad{R}-module basis \spad{v1},...,\spad{vn}.
represents: (Vector R, Vector %) -> %
++ represents([a1,...,am],[v1,...,vm]) returns the linear
++ combination \spad{a1*vm + ... + an*vm}.
leftDiscriminant: Vector % -> R
++ leftDiscriminant([v1,...,vn]) returns the determinant of the
++ \spad{n}-by-\spad{n} matrix whose element at the \spad{i}-th row
++ and \spad{j}-th column is given by the left trace of the product
++ \spad{vi*vj}.
++ Note: the same as \spad{determinant(leftTraceMatrix([v1,...,vn]))}.
rightDiscriminant: Vector % -> R
++ rightDiscriminant([v1,...,vn]) returns the determinant of the
++ \spad{n}-by-\spad{n} matrix whose element at the \spad{i}-th row
++ and \spad{j}-th column is given by the right trace of the product
++ \spad{vi*vj}.
++ Note: the same as \spad{determinant(rightTraceMatrix([v1,...,vn]))}.
leftTraceMatrix: Vector % -> Matrix R
++ leftTraceMatrix([v1,...,vn]) is the \spad{n}-by-\spad{n} matrix
++ whose element at the \spad{i}-th row and \spad{j}-th column is given
++ by the left trace of the product \spad{vi*vj}.
rightTraceMatrix: Vector % -> Matrix R
++ rightTraceMatrix([v1,...,vn]) is the \spad{n}-by-\spad{n} matrix
++ whose element at the \spad{i}-th row and \spad{j}-th column is given
++ by the right trace of the product \spad{vi*vj}.
leftCharacteristicPolynomial: % -> SparseUnivariatePolynomial R
++ leftCharacteristicPolynomial(a) returns the characteristic
++ polynomial of the left regular representation of \spad{a}
++ with respect to any basis.
rightCharacteristicPolynomial: % -> SparseUnivariatePolynomial R
++ rightCharacteristicPolynomial(a) returns the characteristic
++ polynomial of the right regular representation of \spad{a}
++ with respect to any basis.
--we not necessarily have a unit
--if R has CharacteristicZero then CharacteristicZero
--if R has CharacteristicNonZero then CharacteristicNonZero
commutative?:()-> Boolean
++ commutative?() tests if multiplication in the algebra
++ is commutative.
antiCommutative?:()-> Boolean
++ antiCommutative?() tests if \spad{a*a = 0}
++ for all \spad{a} in the algebra.
++ Note: this implies \spad{a*b + b*a = 0} for all \spad{a} and \spad{b}.
associative?:()-> Boolean
++ associative?() tests if multiplication in algebra
++ is associative.
antiAssociative?:()-> Boolean
++ antiAssociative?() tests if multiplication in algebra
++ is anti-associative, i.e. \spad{(a*b)*c + a*(b*c) = 0}
++ for all \spad{a},b,c in the algebra.
leftAlternative?: ()-> Boolean
++ leftAlternative?() tests if \spad{2*associator(a,a,b) = 0}
++ for all \spad{a}, b in the algebra.
++ Note: we only can test this; in general we don't know
++ whether \spad{2*a=0} implies \spad{a=0}.
rightAlternative?: ()-> Boolean
++ rightAlternative?() tests if \spad{2*associator(a,b,b) = 0}
++ for all \spad{a}, b in the algebra.
++ Note: we only can test this; in general we don't know
++ whether \spad{2*a=0} implies \spad{a=0}.
flexible?: ()-> Boolean
++ flexible?() tests if \spad{2*associator(a,b,a) = 0}
++ for all \spad{a}, b in the algebra.
++ Note: we only can test this; in general we don't know
++ whether \spad{2*a=0} implies \spad{a=0}.
alternative?: ()-> Boolean
++ alternative?() tests if
++ \spad{2*associator(a,a,b) = 0 = 2*associator(a,b,b)}
++ for all \spad{a}, b in the algebra.
++ Note: we only can test this; in general we don't know
++ whether \spad{2*a=0} implies \spad{a=0}.
powerAssociative?:()-> Boolean
++ powerAssociative?() tests if all subalgebras
++ generated by a single element are associative.
jacobiIdentity?:() -> Boolean
++ jacobiIdentity?() tests if \spad{(a*b)*c + (b*c)*a + (c*a)*b = 0}
++ for all \spad{a},b,c in the algebra. For example, this holds
++ for crossed products of 3-dimensional vectors.
lieAdmissible?: () -> Boolean
++ lieAdmissible?() tests if the algebra defined by the commutators
++ is a Lie algebra, i.e. satisfies the Jacobi identity.
++ The property of anticommutativity follows from definition.
jordanAdmissible?: () -> Boolean
++ jordanAdmissible?() tests if 2 is invertible in the
++ coefficient domain and the multiplication defined by
++ \spad{(1/2)(a*b+b*a)} determines a
++ Jordan algebra, i.e. satisfies the Jordan identity.
++ The property of \spadatt{commutative("*")}
++ follows from by definition.
noncommutativeJordanAlgebra?: () -> Boolean
++ noncommutativeJordanAlgebra?() tests if the algebra
++ is flexible and Jordan admissible.
jordanAlgebra?:() -> Boolean
++ jordanAlgebra?() tests if the algebra is commutative,
++ characteristic is not 2, and \spad{(a*b)*a**2 - a*(b*a**2) = 0}
++ for all \spad{a},b,c in the algebra (Jordan identity).
++ Example:
++ for every associative algebra \spad{(A,+,@)} we can construct a
++ Jordan algebra \spad{(A,+,*)}, where \spad{a*b := (a@b+b@a)/2}.
lieAlgebra?:() -> Boolean
++ lieAlgebra?() tests if the algebra is anticommutative
++ and \spad{(a*b)*c + (b*c)*a + (c*a)*b = 0}
++ for all \spad{a},b,c in the algebra (Jacobi identity).
++ Example:
++ for every associative algebra \spad{(A,+,@)} we can construct a
++ Lie algebra \spad{(A,+,*)}, where \spad{a*b := a@b-b@a}.
if R has IntegralDomain then
-- we not neccessarily have a unit, hence we don't inherit
-- the next 3 functions anc hence copy them from MonadWithUnit:
recip: % -> Union(%,"failed")
++ recip(a) returns an element, which is both a left and a right
++ inverse of \spad{a},
++ or \spad{"failed"} if there is no unit element, if such an
++ element doesn't exist or cannot be determined (see unitsKnown).
leftRecip: % -> Union(%,"failed")
++ leftRecip(a) returns an element, which is a left inverse of \spad{a},
++ or \spad{"failed"} if there is no unit element, if such an
++ element doesn't exist or cannot be determined (see unitsKnown).
rightRecip: % -> Union(%,"failed")
++ rightRecip(a) returns an element, which is a right inverse of
++ \spad{a},
++ or \spad{"failed"} if there is no unit element, if such an
++ element doesn't exist or cannot be determined (see unitsKnown).
associatorDependence:() -> List Vector R
++ associatorDependence() looks for the associator identities, i.e.
++ finds a basis of the solutions of the linear combinations of the
++ six permutations of \spad{associator(a,b,c)} which yield 0,
++ for all \spad{a},b,c in the algebra.
++ The order of the permutations is \spad{123 231 312 132 321 213}.
leftMinimalPolynomial : % -> SparseUnivariatePolynomial R
++ leftMinimalPolynomial(a) returns the polynomial determined by the
++ smallest non-trivial linear combination of left powers of \spad{a}.
++ Note: the polynomial never has a constant term as in general
++ the algebra has no unit.
rightMinimalPolynomial : % -> SparseUnivariatePolynomial R
++ rightMinimalPolynomial(a) returns the polynomial determined by the
++ smallest non-trivial linear
++ combination of right powers of \spad{a}.
++ Note: the polynomial never has a constant term as in general
++ the algebra has no unit.
leftUnits:() -> Union(Record(particular: %, basis: List %), "failed")
++ leftUnits() returns the affine space of all left units of the
++ algebra, or \spad{"failed"} if there is none.
rightUnits:() -> Union(Record(particular: %, basis: List %), "failed")
++ rightUnits() returns the affine space of all right units of the
++ algebra, or \spad{"failed"} if there is none.
leftUnit:() -> Union(%, "failed")
++ leftUnit() returns a left unit of the algebra
++ (not necessarily unique), or \spad{"failed"} if there is none.
rightUnit:() -> Union(%, "failed")
++ rightUnit() returns a right unit of the algebra
++ (not necessarily unique), or \spad{"failed"} if there is none.
unit:() -> Union(%, "failed")
++ unit() returns a unit of the algebra (necessarily unique),
++ or \spad{"failed"} if there is none.
-- we not necessarily have a unit, hence we can't say anything
-- about characteristic
-- if R has CharacteristicZero then CharacteristicZero
-- if R has CharacteristicNonZero then CharacteristicNonZero
unitsKnown
++ unitsKnown means that \spadfun{recip} truly yields reciprocal
++ or \spad{"failed"} if not a unit,
++ similarly for \spadfun{leftRecip} and
++ \spadfun{rightRecip}. The reason is that we use left, respectively
++ right, minimal polynomials to decide this question.
add
--n := rank()
--b := someBasis()
--gamma : Vector Matrix R := structuralConstants b
-- here is a problem: there seems to be a problem having local
-- variables in the capsule of a category, furthermore
-- see the commented code of conditionsForIdempotents, where
-- we call structuralConstants, which also doesn't work
-- at runtime, i.e. is not properly inherited, hence for
-- the moment we put the code for
-- conditionsForIdempotents, structuralConstants, unit, leftUnit,
-- rightUnit into the domain constructor ALGSC
V ==> Vector
M ==> Matrix
REC ==> Record(particular: Union(V R,"failed"),basis: List V R)
LSMP ==> LinearSystemMatrixPackage(R,V R,V R, M R)
SUP ==> SparseUnivariatePolynomial
NNI ==> NonNegativeInteger
-- next 2 functions: use a general characteristicPolynomial
leftCharacteristicPolynomial a ==
n := rank()$%
ma : Matrix R := leftRegularRepresentation(a,someBasis()$%)
mb : Matrix SUP R := zero(n,n)
for i in 1..n repeat
for j in 1..n repeat
mb(i,j):=
i=j => monomial(ma(i,j),0)$SUP(R) - monomial(1,1)$SUP(R)
monomial(ma(i,j),1)$SUP(R)
determinant mb
rightCharacteristicPolynomial a ==
n := rank()$%
ma : Matrix R := rightRegularRepresentation(a,someBasis()$%)
mb : Matrix SUP R := zero(n,n)
for i in 1..n repeat
for j in 1..n repeat
mb(i,j):=
i=j => monomial(ma(i,j),0)$SUP(R) - monomial(1,1)$SUP(R)
monomial(ma(i,j),1)$SUP(R)
determinant mb
leftTrace a ==
t : R := 0
ma : Matrix R := leftRegularRepresentation(a,someBasis()$%)
for i in 1..rank()$% repeat
t := t + elt(ma,i,i)
t
rightTrace a ==
t : R := 0
ma : Matrix R := rightRegularRepresentation(a,someBasis()$%)
for i in 1..rank()$% repeat
t := t + elt(ma,i,i)
t
leftNorm a == determinant leftRegularRepresentation(a,someBasis()$%)
rightNorm a == determinant rightRegularRepresentation(a,someBasis()$%)
antiAssociative?() ==
b := someBasis()
n := rank()
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
not zero? ( (b.i*b.j)*b.k + b.i*(b.j*b.k) ) =>
messagePrint("algebra is not anti-associative")$OutputForm
return false
messagePrint("algebra is anti-associative")$OutputForm
true
jordanAdmissible?() ==
b := someBasis()
n := rank()
recip(2 * 1$R) case "failed" =>
messagePrint("this algebra is not Jordan admissible, as 2 is not invertible in the ground ring")$OutputForm
false
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
for l in 1..n repeat
not zero? ( _
antiCommutator(antiCommutator(b.i,b.j),antiCommutator(b.l,b.k)) + _
antiCommutator(antiCommutator(b.l,b.j),antiCommutator(b.k,b.i)) + _
antiCommutator(antiCommutator(b.k,b.j),antiCommutator(b.i,b.l)) _
) =>
messagePrint("this algebra is not Jordan admissible")$OutputForm
return false
messagePrint("this algebra is Jordan admissible")$OutputForm
true
lieAdmissible?() ==
n := rank()
b := someBasis()
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
not zero? (commutator(commutator(b.i,b.j),b.k) _
+ commutator(commutator(b.j,b.k),b.i) _
+ commutator(commutator(b.k,b.i),b.j)) =>
messagePrint("this algebra is not Lie admissible")$OutputForm
return false
messagePrint("this algebra is Lie admissible")$OutputForm
true
-- conditionsForIdempotents b ==
-- n := rank()
-- gamma : Vector Matrix R := structuralConstants b
-- listOfNumbers : List String := [STRINGIMAGE(q)$Lisp for q in 1..n]
-- symbolsForCoef : Vector Symbol :=
-- [concat("%", concat("x", i))::Symbol for i in listOfNumbers]
-- conditions : List Polynomial R := []
-- for k in 1..n repeat
-- xk := symbolsForCoef.k
-- p : Polynomial R := monomial( - 1$Polynomial(R), [xk], [1] )
-- for i in 1..n repeat
-- for j in 1..n repeat
-- xi := symbolsForCoef.i
-- xj := symbolsForCoef.j
-- p := p + monomial(_
-- elt((gamma.k),i,j) :: Polynomial(R), [xi,xj], [1,1])
-- conditions := cons(p,conditions)
-- conditions
structuralConstants b ==
--n := rank()
-- be careful with the possibility that b is not a basis
m : NonNegativeInteger := (maxIndex b) :: NonNegativeInteger
sC : Vector Matrix R := [new(m,m,0$R) for k in 1..m]
for i in 1..m repeat
for j in 1..m repeat
covec : Vector R := coordinates(b.i * b.j, b)
for k in 1..m repeat
setelt( sC.k, i, j, covec.k )
sC
if R has IntegralDomain then
leftRecip x ==
zero? x => "failed"
lu := leftUnit()
lu case "failed" => "failed"
b := someBasis()
xx : % := (lu :: %)
k : PositiveInteger := 1
cond : Matrix R := coordinates(xx,b) :: Matrix(R)
listOfPowers : List % := [xx]
while rank(cond) = k repeat
k := k+1
xx := xx*x
listOfPowers := cons(xx,listOfPowers)
cond := horizConcat(cond, coordinates(xx,b) :: Matrix(R) )
vectorOfCoef : Vector R := (nullSpace(cond)$Matrix(R)).first
invC := recip vectorOfCoef.1
invC case "failed" => "failed"
invCR : R := - (invC :: R)
reduce(_+,[(invCR*vectorOfCoef.i)*power for i in _
2..maxIndex vectorOfCoef for power in reverse listOfPowers])
rightRecip x ==
zero? x => "failed"
ru := rightUnit()
ru case "failed" => "failed"
b := someBasis()
xx : % := (ru :: %)
k : PositiveInteger := 1
cond : Matrix R := coordinates(xx,b) :: Matrix(R)
listOfPowers : List % := [xx]
while rank(cond) = k repeat
k := k+1
xx := x*xx
listOfPowers := cons(xx,listOfPowers)
cond := horizConcat(cond, coordinates(xx,b) :: Matrix(R) )
vectorOfCoef : Vector R := (nullSpace(cond)$Matrix(R)).first
invC := recip vectorOfCoef.1
invC case "failed" => "failed"
invCR : R := - (invC :: R)
reduce(_+,[(invCR*vectorOfCoef.i)*power for i in _
2..maxIndex vectorOfCoef for power in reverse listOfPowers])
recip x ==
lrx := leftRecip x
lrx case "failed" => "failed"
rrx := rightRecip x
rrx case "failed" => "failed"
(lrx :: %) ~= (rrx :: %) => "failed"
lrx :: %
leftMinimalPolynomial x ==
zero? x => monomial(1$R,1)$(SparseUnivariatePolynomial R)
b := someBasis()
xx : % := x
k : PositiveInteger := 1
cond : Matrix R := coordinates(xx,b) :: Matrix(R)
while rank(cond) = k repeat
k := k+1
xx := x*xx
cond := horizConcat(cond, coordinates(xx,b) :: Matrix(R) )
vectorOfCoef : Vector R := (nullSpace(cond)$Matrix(R)).first
res : SparseUnivariatePolynomial R := 0
for i in 1..k repeat
res := res+monomial(vectorOfCoef.i,i)$(SparseUnivariatePolynomial R)
res
rightMinimalPolynomial x ==
zero? x => monomial(1$R,1)$(SparseUnivariatePolynomial R)
b := someBasis()
xx : % := x
k : PositiveInteger := 1
cond : Matrix R := coordinates(xx,b) :: Matrix(R)
while rank(cond) = k repeat
k := k+1
xx := xx*x
cond := horizConcat(cond, coordinates(xx,b) :: Matrix(R) )
vectorOfCoef : Vector R := (nullSpace(cond)$Matrix(R)).first
res : SparseUnivariatePolynomial R := 0
for i in 1..k repeat
res := res+monomial(vectorOfCoef.i,i)$(SparseUnivariatePolynomial R)
res
associatorDependence() ==
n := rank()
b := someBasis()
cond : Matrix(R) := new(n**4,6,0$R)$Matrix(R)
z : Integer := 0
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
a123 : Vector R := coordinates(associator(b.i,b.j,b.k),b)
a231 : Vector R := coordinates(associator(b.j,b.k,b.i),b)
a312 : Vector R := coordinates(associator(b.k,b.i,b.j),b)
a132 : Vector R := coordinates(associator(b.i,b.k,b.j),b)
a321 : Vector R := coordinates(associator(b.k,b.j,b.i),b)
a213 : Vector R := coordinates(associator(b.j,b.i,b.k),b)
for r in 1..n repeat
z:= z+1
setelt(cond,z,1,elt(a123,r))
setelt(cond,z,2,elt(a231,r))
setelt(cond,z,3,elt(a312,r))
setelt(cond,z,4,elt(a132,r))
setelt(cond,z,5,elt(a321,r))
setelt(cond,z,6,elt(a213,r))
nullSpace(cond)
jacobiIdentity?() ==
n := rank()
b := someBasis()
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
not zero? ((b.i*b.j)*b.k + (b.j*b.k)*b.i + (b.k*b.i)*b.j) =>
messagePrint("Jacobi identity does not hold")$OutputForm
return false
messagePrint("Jacobi identity holds")$OutputForm
true
lieAlgebra?() ==
not antiCommutative?() =>
messagePrint("this is not a Lie algebra")$OutputForm
false
not jacobiIdentity?() =>
messagePrint("this is not a Lie algebra")$OutputForm
false
messagePrint("this is a Lie algebra")$OutputForm
true
jordanAlgebra?() ==
b := someBasis()
n := rank()
recip(2 * 1$R) case "failed" =>
messagePrint("this is not a Jordan algebra, as 2 is not invertible in the ground ring")$OutputForm
false
not commutative?() =>
messagePrint("this is not a Jordan algebra")$OutputForm
false
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
for l in 1..n repeat
not zero? (associator(b.i,b.j,b.l*b.k)+_
associator(b.l,b.j,b.k*b.i)+associator(b.k,b.j,b.i*b.l)) =>
messagePrint("not a Jordan algebra")$OutputForm
return false
messagePrint("this is a Jordan algebra")$OutputForm
true
noncommutativeJordanAlgebra?() ==
b := someBasis()
n := rank()
recip(2 * 1$R) case "failed" =>
messagePrint("this is not a noncommutative Jordan algebra, as 2 is not invertible in the ground ring")$OutputForm
false
not flexible?()$% =>
messagePrint("this is not a noncommutative Jordan algebra, as it is not flexible")$OutputForm
false
not jordanAdmissible?()$% =>
messagePrint("this is not a noncommutative Jordan algebra, as it is not Jordan admissible")$OutputForm
false
messagePrint("this is a noncommutative Jordan algebra")$OutputForm
true
antiCommutative?() ==
b := someBasis()
n := rank()
for i in 1..n repeat
for j in i..n repeat
not zero? (i=j => b.i*b.i; b.i*b.j + b.j*b.i) =>
messagePrint("algebra is not anti-commutative")$OutputForm
return false
messagePrint("algebra is anti-commutative")$OutputForm
true
commutative?() ==
b := someBasis()
n := rank()
for i in 1..n repeat
for j in i+1..n repeat
not zero? commutator(b.i,b.j) =>
messagePrint("algebra is not commutative")$OutputForm
return false
messagePrint("algebra is commutative")$OutputForm
true
associative?() ==
b := someBasis()
n := rank()
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
not zero? associator(b.i,b.j,b.k) =>
messagePrint("algebra is not associative")$OutputForm
return false
messagePrint("algebra is associative")$OutputForm
true
leftAlternative?() ==
b := someBasis()
n := rank()
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
not zero? (associator(b.i,b.j,b.k) + associator(b.j,b.i,b.k)) =>
messagePrint("algebra is not left alternative")$OutputForm
return false
messagePrint("algebra satisfies 2*associator(a,a,b) = 0")$OutputForm
true
rightAlternative?() ==
b := someBasis()
n := rank()
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
not zero? (associator(b.i,b.j,b.k) + associator(b.i,b.k,b.j)) =>
messagePrint("algebra is not right alternative")$OutputForm
return false
messagePrint("algebra satisfies 2*associator(a,b,b) = 0")$OutputForm
true
flexible?() ==
b := someBasis()
n := rank()
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
not zero? (associator(b.i,b.j,b.k) + associator(b.k,b.j,b.i)) =>
messagePrint("algebra is not flexible")$OutputForm
return false
messagePrint("algebra satisfies 2*associator(a,b,a) = 0")$OutputForm
true
alternative?() ==
b := someBasis()
n := rank()
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
not zero? (associator(b.i,b.j,b.k) + associator(b.j,b.i,b.k)) =>
messagePrint("algebra is not alternative")$OutputForm
return false
not zero? (associator(b.i,b.j,b.k) + associator(b.i,b.k,b.j)) =>
messagePrint("algebra is not alternative")$OutputForm
return false
messagePrint("algebra satisfies 2*associator(a,b,b) = 0 = 2*associator(a,a,b) = 0")$OutputForm
true
leftDiscriminant v == determinant leftTraceMatrix v
rightDiscriminant v == determinant rightTraceMatrix v
coordinates(v:Vector %, b:Vector %) ==
m := new(#v, #b, 0)$Matrix(R)
for i in minIndex v .. maxIndex v for j in minRowIndex m .. repeat
setRow_!(m, j, coordinates(qelt(v, i), b))
m
represents(v, b) ==
m := minIndex v - 1
reduce(_+,[v(i+m) * b(i+m) for i in 1..maxIndex b])
leftTraceMatrix v ==
matrix [[leftTrace(v.i*v.j) for j in minIndex v..maxIndex v]$List(R)
for i in minIndex v .. maxIndex v]$List(List R)
rightTraceMatrix v ==
matrix [[rightTrace(v.i*v.j) for j in minIndex v..maxIndex v]$List(R)
for i in minIndex v .. maxIndex v]$List(List R)
leftRegularRepresentation(x, b) ==
m := minIndex b - 1
matrix
[parts coordinates(x*b(i+m),b) for i in 1..rank()]$List(List R)
rightRegularRepresentation(x, b) ==
m := minIndex b - 1
matrix
[parts coordinates(b(i+m)*x,b) for i in 1..rank()]$List(List R)
@
\section{category FRNAALG FramedNonAssociativeAlgebra}
<<category FRNAALG FramedNonAssociativeAlgebra>>=
)abbrev category FRNAALG FramedNonAssociativeAlgebra
++ Author: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 11 June 1991
++ Basic Operations: +,-,*,**,basis
++ Related Constructors: FiniteRankNonAssociativeAlgebra, FramedAlgebra,
++ FiniteRankAssociativeAlgebra
++ Also See:
++ AMS Classifications:
++ Keywords: nonassociative algebra, basis
++ Reference:
++ R.D. Schafer: An Introduction to Nonassociative Algebras
++ Academic Press, New York, 1966
++ Description:
++ FramedNonAssociativeAlgebra(R) is a
++ \spadtype{FiniteRankNonAssociativeAlgebra} (i.e. a non associative
++ algebra over R which is a free \spad{R}-module of finite rank)
++ over a commutative ring R together with a fixed \spad{R}-module basis.
FramedNonAssociativeAlgebra(R:CommutativeRing):
Category == FiniteRankNonAssociativeAlgebra(R) with
--operations
basis: () -> Vector %
++ basis() returns the fixed \spad{R}-module basis.
coordinates: % -> Vector R
++ coordinates(a) returns the coordinates of \spad{a}
++ with respect to the
++ fixed \spad{R}-module basis.
coordinates: Vector % -> Matrix R
++ coordinates([a1,...,am]) returns a matrix whose i-th row
++ is formed by the coordinates of \spad{ai} with respect to the
++ fixed \spad{R}-module basis.
elt : (%,Integer) -> R
++ elt(a,i) returns the i-th coefficient of \spad{a} with respect to the
++ fixed \spad{R}-module basis.
structuralConstants:() -> Vector Matrix R
++ structuralConstants() calculates the structural constants
++ \spad{[(gammaijk) for k in 1..rank()]} defined by
++ \spad{vi * vj = gammaij1 * v1 + ... + gammaijn * vn},
++ where \spad{v1},...,\spad{vn} is the fixed \spad{R}-module basis.
conditionsForIdempotents: () -> List Polynomial R
++ conditionsForIdempotents() determines a complete list
++ of polynomial equations for the coefficients of idempotents
++ with respect to the fixed \spad{R}-module basis.
represents: Vector R -> %
++ represents([a1,...,an]) returns \spad{a1*v1 + ... + an*vn},
++ where \spad{v1}, ..., \spad{vn} are the elements of the
++ fixed \spad{R}-module basis.
convert: % -> Vector R
++ convert(a) returns the coordinates of \spad{a} with respect to the
++ fixed \spad{R}-module basis.
convert: Vector R -> %
++ convert([a1,...,an]) returns \spad{a1*v1 + ... + an*vn},
++ where \spad{v1}, ..., \spad{vn} are the elements of the
++ fixed \spad{R}-module basis.
leftDiscriminant : () -> R
++ leftDiscriminant() returns the
++ determinant of the \spad{n}-by-\spad{n}
++ matrix whose element at the \spad{i}-th row and \spad{j}-th column is
++ given by the left trace of the product \spad{vi*vj}, where
++ \spad{v1},...,\spad{vn} are the
++ elements of the fixed \spad{R}-module basis.
++ Note: the same as \spad{determinant(leftTraceMatrix())}.
rightDiscriminant : () -> R
++ rightDiscriminant() returns the determinant of the \spad{n}-by-\spad{n}
++ matrix whose element at the \spad{i}-th row and \spad{j}-th column is
++ given by the right trace of the product \spad{vi*vj}, where
++ \spad{v1},...,\spad{vn} are the elements of
++ the fixed \spad{R}-module basis.
++ Note: the same as \spad{determinant(rightTraceMatrix())}.
leftTraceMatrix : () -> Matrix R
++ leftTraceMatrix() is the \spad{n}-by-\spad{n}
++ matrix whose element at the \spad{i}-th row and \spad{j}-th column is
++ given by left trace of the product \spad{vi*vj},
++ where \spad{v1},...,\spad{vn} are the
++ elements of the fixed \spad{R}-module
++ basis.
rightTraceMatrix : () -> Matrix R
++ rightTraceMatrix() is the \spad{n}-by-\spad{n}
++ matrix whose element at the \spad{i}-th row and \spad{j}-th column is
++ given by the right trace of the product \spad{vi*vj}, where
++ \spad{v1},...,\spad{vn} are the elements
++ of the fixed \spad{R}-module basis.
leftRegularRepresentation : % -> Matrix R
++ leftRegularRepresentation(a) returns the matrix of the linear
++ map defined by left multiplication by \spad{a} with respect
++ to the fixed \spad{R}-module basis.
rightRegularRepresentation : % -> Matrix R
++ rightRegularRepresentation(a) returns the matrix of the linear
++ map defined by right multiplication by \spad{a} with respect
++ to the fixed \spad{R}-module basis.
if R has Field then
leftRankPolynomial : () -> SparseUnivariatePolynomial Polynomial R
++ leftRankPolynomial() calculates the left minimal polynomial
++ of the generic element in the algebra,
++ defined by the same structural
++ constants over the polynomial ring in symbolic coefficients with
++ respect to the fixed basis.
rightRankPolynomial : () -> SparseUnivariatePolynomial Polynomial R
++ rightRankPolynomial() calculates the right minimal polynomial
++ of the generic element in the algebra,
++ defined by the same structural
++ constants over the polynomial ring in symbolic coefficients with
++ respect to the fixed basis.
apply: (Matrix R, %) -> %
++ apply(m,a) defines a left operation of n by n matrices
++ where n is the rank of the algebra in terms of matrix-vector
++ multiplication, this is a substitute for a left module structure.
++ Error: if shape of matrix doesn't fit.
--attributes
--attributes
--separable <=> discriminant() ~= 0
add
V ==> Vector
M ==> Matrix
P ==> Polynomial
F ==> Fraction
REC ==> Record(particular: Union(V R,"failed"),basis: List V R)
LSMP ==> LinearSystemMatrixPackage(R,V R,V R, M R)
CVMP ==> CoerceVectorMatrixPackage(R)
--GA ==> GenericNonAssociativeAlgebra(R,rank()$%,_
-- [random()$Character :: String :: Symbol for i in 1..rank()$%], _
-- structuralConstants()$%)
--y : GA := generic()
if R has Field then
leftRankPolynomial() ==
n := rank()
b := basis()
gamma : Vector Matrix R := structuralConstants b
listOfNumbers : List String := [STRINGIMAGE(q)$Lisp for q in 1..n]
symbolsForCoef : Vector Symbol :=
[concat("%", concat("x", i))::Symbol for i in listOfNumbers]
xx : M P R
mo : P R
x : M P R := new(1,n,0)
for i in 1..n repeat
mo := monomial(1, [symbolsForCoef.i], [1])$(P R)
qsetelt_!(x,1,i,mo)
y : M P R := copy x
k : PositiveInteger := 1
cond : M P R := copy x
-- multiplication in the generic algebra means using
-- the structural matrices as bilinear forms.
-- left multiplication by x, we prepare for that:
genGamma : V M P R := coerceP$CVMP gamma
x := reduce(horizConcat,[x*genGamma(i) for i in 1..#genGamma])
while rank(cond) = k repeat
k := k+1
for i in 1..n repeat
setelt(xx,[1],[i],x*transpose y)
y := copy xx
cond := horizConcat(cond, xx)
vectorOfCoef : Vector P R := (nullSpace(cond)$Matrix(P R)).first
res : SparseUnivariatePolynomial P R := 0
for i in 1..k repeat
res := res+monomial(vectorOfCoef.i,i)$(SparseUnivariatePolynomial P R)
res
rightRankPolynomial() ==
n := rank()
b := basis()
gamma : Vector Matrix R := structuralConstants b
listOfNumbers : List String := [STRINGIMAGE(q)$Lisp for q in 1..n]
symbolsForCoef : Vector Symbol :=
[concat("%", concat("x", i))::Symbol for i in listOfNumbers]
xx : M P R
mo : P R
x : M P R := new(1,n,0)
for i in 1..n repeat
mo := monomial(1, [symbolsForCoef.i], [1])$(P R)
qsetelt_!(x,1,i,mo)
y : M P R := copy x
k : PositiveInteger := 1
cond : M P R := copy x
-- multiplication in the generic algebra means using
-- the structural matrices as bilinear forms.
-- left multiplication by x, we prepare for that:
genGamma : V M P R := coerceP$CVMP gamma
x := reduce(horizConcat,[genGamma(i)*transpose x for i in 1..#genGamma])
while rank(cond) = k repeat
k := k+1
for i in 1..n repeat
setelt(xx,[1],[i],y * transpose x)
y := copy xx
cond := horizConcat(cond, xx)
vectorOfCoef : Vector P R := (nullSpace(cond)$Matrix(P R)).first
res : SparseUnivariatePolynomial P R := 0
for i in 1..k repeat
res := res+monomial(vectorOfCoef.i,i)$(SparseUnivariatePolynomial P R)
res
leftUnitsInternal : () -> REC
leftUnitsInternal() ==
n := rank()
b := basis()
gamma : Vector Matrix R := structuralConstants b
cond : Matrix(R) := new(n**2,n,0$R)$Matrix(R)
rhs : Vector(R) := new(n**2,0$R)$Vector(R)
z : Integer := 0
addOn : R := 0
for k in 1..n repeat
for i in 1..n repeat
z := z+1 -- index for the rows
addOn :=
k=i => 1
0
setelt(rhs,z,addOn)$Vector(R)
for j in 1..n repeat -- index for the columns
setelt(cond,z,j,elt(gamma.k,j,i))$Matrix(R)
solve(cond,rhs)$LSMP
leftUnit() ==
res : REC := leftUnitsInternal()
res.particular case "failed" =>
messagePrint("this algebra has no left unit")$OutputForm
"failed"
represents (res.particular :: V R)
leftUnits() ==
res : REC := leftUnitsInternal()
res.particular case "failed" =>
messagePrint("this algebra has no left unit")$OutputForm
"failed"
[represents(res.particular :: V R)$%, _
map(represents, res.basis)$ListFunctions2(Vector R, %) ]
rightUnitsInternal : () -> REC
rightUnitsInternal() ==
n := rank()
b := basis()
gamma : Vector Matrix R := structuralConstants b
condo : Matrix(R) := new(n**2,n,0$R)$Matrix(R)
rhs : Vector(R) := new(n**2,0$R)$Vector(R)
z : Integer := 0
addOn : R := 0
for k in 1..n repeat
for i in 1..n repeat
z := z+1 -- index for the rows
addOn :=
k=i => 1
0
setelt(rhs,z,addOn)$Vector(R)
for j in 1..n repeat -- index for the columns
setelt(condo,z,j,elt(gamma.k,i,j))$Matrix(R)
solve(condo,rhs)$LSMP
rightUnit() ==
res : REC := rightUnitsInternal()
res.particular case "failed" =>
messagePrint("this algebra has no right unit")$OutputForm
"failed"
represents (res.particular :: V R)
rightUnits() ==
res : REC := rightUnitsInternal()
res.particular case "failed" =>
messagePrint("this algebra has no right unit")$OutputForm
"failed"
[represents(res.particular :: V R)$%, _
map(represents, res.basis)$ListFunctions2(Vector R, %) ]
unit() ==
n := rank()
b := basis()
gamma : Vector Matrix R := structuralConstants b
cond : Matrix(R) := new(2*n**2,n,0$R)$Matrix(R)
rhs : Vector(R) := new(2*n**2,0$R)$Vector(R)
z : Integer := 0
u : Integer := n*n
addOn : R := 0
for k in 1..n repeat
for i in 1..n repeat
z := z+1 -- index for the rows
addOn :=
k=i => 1
0
setelt(rhs,z,addOn)$Vector(R)
setelt(rhs,u,addOn)$Vector(R)
for j in 1..n repeat -- index for the columns
setelt(cond,z,j,elt(gamma.k,j,i))$Matrix(R)
setelt(cond,u,j,elt(gamma.k,i,j))$Matrix(R)
res : REC := solve(cond,rhs)$LSMP
res.particular case "failed" =>
messagePrint("this algebra has no unit")$OutputForm
"failed"
represents (res.particular :: V R)
apply(m:Matrix(R),a:%) ==
v : Vector R := coordinates(a)
v := m *$Matrix(R) v
convert v
structuralConstants() == structuralConstants basis()
conditionsForIdempotents() == conditionsForIdempotents basis()
convert(x:%):Vector(R) == coordinates(x, basis())
convert(v:Vector R):% == represents(v, basis())
leftTraceMatrix() == leftTraceMatrix basis()
rightTraceMatrix() == rightTraceMatrix basis()
leftDiscriminant() == leftDiscriminant basis()
rightDiscriminant() == rightDiscriminant basis()
leftRegularRepresentation x == leftRegularRepresentation(x, basis())
rightRegularRepresentation x == rightRegularRepresentation(x, basis())
coordinates(x: %) == coordinates(x, basis())
represents(v:Vector R):%== represents(v, basis())
coordinates(v:Vector %) ==
m := new(#v, rank(), 0)$Matrix(R)
for i in minIndex v .. maxIndex v for j in minRowIndex m .. repeat
setRow_!(m, j, coordinates qelt(v, i))
m
@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>
<<category MONAD Monad>>
<<category MONADWU MonadWithUnit>>
<<category NARNG NonAssociativeRng>>
<<category NASRING NonAssociativeRing>>
<<category NAALG NonAssociativeAlgebra>>
<<category FINAALG FiniteRankNonAssociativeAlgebra>>
<<category FRNAALG FramedNonAssociativeAlgebra>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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