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--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.
)abbrev domain ALGSC AlgebraGivenByStructuralConstants
++ Authors: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 22 January 1992
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: algebra, structural constants
++ Reference:
++ R.D. Schafer: An Introduction to Nonassociative Algebras
++ Academic Press, New York, 1966
++ Description:
++ AlgebraGivenByStructuralConstants implements finite rank algebras
++ over a commutative ring, given by the structural constants \spad{gamma}
++ with respect to a fixed basis \spad{[a1,..,an]}, where
++ \spad{gamma} is an \spad{n}-vector of n by n matrices
++ \spad{[(gammaijk) for k in 1..rank()]} defined by
++ \spad{ai * aj = gammaij1 * a1 + ... + gammaijn * an}.
++ The symbols for the fixed basis
++ have to be given as a list of symbols.
AlgebraGivenByStructuralConstants(R:Field, n : PositiveInteger,_
ls : List Symbol, gamma: Vector Matrix R ): public == private where
V ==> Vector
M ==> Matrix
I ==> Integer
NNI ==> NonNegativeInteger
REC ==> Record(particular: Union(V R,"failed"),basis: List V R)
LSMP ==> LinearSystemMatrixPackage(R,V R,V R, M R)
--public ==> FramedNonAssociativeAlgebra(R) with
public ==> Join(FramedNonAssociativeAlgebra(R), _
LeftModule(SquareMatrix(n,R)) ) with
coerce : Vector R -> %
++ coerce(v) converts a vector to a member of the algebra
++ by forming a linear combination with the basis element.
++ Note: the vector is assumed to have length equal to the
++ dimension of the algebra.
private ==> DirectProduct(n,R) add
Rep := DirectProduct(n,R)
x,y : %
dp : DirectProduct(n,R)
v : V R
recip(x) == recip(x)$FiniteRankNonAssociativeAlgebra_&(%,R)
(m:SquareMatrix(n,R))*(x:%) == apply((m :: Matrix R),x)
coerce v == directProduct(v) :: %
structuralConstants() == gamma
coordinates(x) == vector(entries(x :: Rep)$Rep)$Vector(R)
coordinates(x,b) ==
--not (maxIndex b = n) =>
-- error("coordinates: your 'basis' has not the right length")
m : NonNegativeInteger := (maxIndex b) :: NonNegativeInteger
transitionMatrix : Matrix R := new(n,m,0$R)$Matrix(R)
for i in 1..m repeat
setColumn!(transitionMatrix,i,coordinates(b.i))
res : REC := solve(transitionMatrix,coordinates(x))$LSMP
if (not every?(zero?$R,first res.basis)) then
error("coordinates: warning your 'basis' is linearly dependent")
(res.particular case "failed") =>
error("coordinates: first argument is not in linear span of second argument")
(res.particular) :: (Vector R)
basis() == [unitVector(i::PositiveInteger)::% for i in 1..n]
someBasis() == basis()$%
rank() == n
elt(x,i) == elt(x:Rep,i)$Rep
coerce(x:%):OutputForm ==
zero?(x::Rep)$Rep => (0$R) :: OutputForm
le : List OutputForm := nil
for i in 1..n repeat
coef : R := elt(x::Rep,i)
not zero?(coef)$R =>
one?(coef)$R =>
-- sy : OutputForm := elt(ls,i)$(List Symbol) :: OutputForm
le := cons(elt(ls,i)$(List Symbol) :: OutputForm, le)
le := cons(coef :: OutputForm * elt(ls,i)$(List Symbol)_
:: OutputForm, le)
reduce("+",le)
x * y ==
v : Vector R := new(n,0)
for k in 1..n repeat
h : R := 0
for i in 1..n repeat
for j in 1..n repeat
h := h +$R elt(x,i) *$R elt(y,j) *$R elt(gamma.k,i,j )
v.k := h
directProduct v
alternative?() ==
for i in 1..n repeat
-- expression for check of left alternative is symmetric in i and j:
-- expression for check of right alternative is symmetric in j and k:
for j in 1..i-1 repeat
for k in j..n repeat
-- right check
for r in 1..n repeat
res := 0$R
for l in 1..n repeat
res := res - _
(elt(gamma.l,j,k)+elt(gamma.l,k,j))*elt(gamma.r,i,l)+_
(elt(gamma.l,i,j)*elt(gamma.r,l,k) + elt(gamma.l,i,k)*_
elt(gamma.r,l,j) )
not zero? res =>
messagePrint("algebra is not right alternative")$OutputForm
return false
for j in i..n repeat
for k in 1..j-1 repeat
-- left check
for r in 1..n repeat
res := 0$R
for l in 1..n repeat
res := res + _
(elt(gamma.l,i,j)+elt(gamma.l,j,i))*elt(gamma.r,l,k)-_
(elt(gamma.l,j,k)*elt(gamma.r,i,l) + elt(gamma.l,i,k)*_
elt(gamma.r,j,l) )
not (zero? res) =>
messagePrint("algebra is not left alternative")$OutputForm
return false
for k in j..n repeat
-- left check
for r in 1..n repeat
res := 0$R
for l in 1..n repeat
res := res + _
(elt(gamma.l,i,j)+elt(gamma.l,j,i))*elt(gamma.r,l,k)-_
(elt(gamma.l,j,k)*elt(gamma.r,i,l) + elt(gamma.l,i,k)*_
elt(gamma.r,j,l) )
not (zero? res) =>
messagePrint("algebra is not left alternative")$OutputForm
return false
-- right check
for r in 1..n repeat
res := 0$R
for l in 1..n repeat
res := res - _
(elt(gamma.l,j,k)+elt(gamma.l,k,j))*elt(gamma.r,i,l)+_
(elt(gamma.l,i,j)*elt(gamma.r,l,k) + elt(gamma.l,i,k)*_
elt(gamma.r,l,j) )
not (zero? res) =>
messagePrint("algebra is not right alternative")$OutputForm
return false
messagePrint("algebra satisfies 2*associator(a,b,b) = 0 = 2*associator(a,a,b) = 0")$OutputForm
true
-- should be in the category, but is not exported
-- conditionsForIdempotents b ==
-- n := rank()
-- --gamma : Vector Matrix R := structuralConstants b
-- listOfNumbers : List String := [string(q)$String for q in 1..n]
-- symbolsForCoef : Vector Symbol :=
-- [concat("%", concat("x", i))::Symbol for i in listOfNumbers]
-- conditions : List Polynomial R := []
-- for k in 1..n repeat
-- xk := symbolsForCoef.k
-- p : Polynomial R := monomial( - 1$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
associative?() ==
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
for r in 1..n repeat
res := 0$R
for l in 1..n repeat
res := res + elt(gamma.l,i,j)*elt(gamma.r,l,k)-_
elt(gamma.l,j,k)*elt(gamma.r,i,l)
not (zero? res) =>
messagePrint("algebra is not associative")$OutputForm
return false
messagePrint("algebra is associative")$OutputForm
true
antiAssociative?() ==
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
for r in 1..n repeat
res := 0$R
for l in 1..n repeat
res := res + elt(gamma.l,i,j)*elt(gamma.r,l,k)+_
elt(gamma.l,j,k)*elt(gamma.r,i,l)
not (zero? res) =>
messagePrint("algebra is not anti-associative")$OutputForm
return false
messagePrint("algebra is anti-associative")$OutputForm
true
commutative?() ==
for i in 1..n repeat
for j in (i+1)..n repeat
for k in 1..n repeat
not ( elt(gamma.k,i,j)=elt(gamma.k,j,i) ) =>
messagePrint("algebra is not commutative")$OutputForm
return false
messagePrint("algebra is commutative")$OutputForm
true
antiCommutative?() ==
for i in 1..n repeat
for j in i..n repeat
for k in 1..n repeat
not zero? (i=j => elt(gamma.k,i,i); elt(gamma.k,i,j)+elt(gamma.k,j,i) ) =>
messagePrint("algebra is not anti-commutative")$OutputForm
return false
messagePrint("algebra is anti-commutative")$OutputForm
true
leftAlternative?() ==
for i in 1..n repeat
-- expression is symmetric in i and j:
for j in i..n repeat
for k in 1..n repeat
for r in 1..n repeat
res := 0$R
for l in 1..n repeat
res := res + (elt(gamma.l,i,j)+elt(gamma.l,j,i))*elt(gamma.r,l,k)-_
(elt(gamma.l,j,k)*elt(gamma.r,i,l) + elt(gamma.l,i,k)*elt(gamma.r,j,l) )
not (zero? res) =>
messagePrint("algebra is not left alternative")$OutputForm
return false
messagePrint("algebra is left alternative")$OutputForm
true
rightAlternative?() ==
for i in 1..n repeat
for j in 1..n repeat
-- expression is symmetric in j and k:
for k in j..n repeat
for r in 1..n repeat
res := 0$R
for l in 1..n repeat
res := res - (elt(gamma.l,j,k)+elt(gamma.l,k,j))*elt(gamma.r,i,l)+_
(elt(gamma.l,i,j)*elt(gamma.r,l,k) + elt(gamma.l,i,k)*elt(gamma.r,l,j) )
not (zero? res) =>
messagePrint("algebra is not right alternative")$OutputForm
return false
messagePrint("algebra is right alternative")$OutputForm
true
flexible?() ==
for i in 1..n repeat
for j in 1..n repeat
-- expression is symmetric in i and k:
for k in i..n repeat
for r in 1..n repeat
res := 0$R
for l in 1..n repeat
res := res + elt(gamma.l,i,j)*elt(gamma.r,l,k)-_
elt(gamma.l,j,k)*elt(gamma.r,i,l)+_
elt(gamma.l,k,j)*elt(gamma.r,l,i)-_
elt(gamma.l,j,i)*elt(gamma.r,k,l)
not (zero? res) =>
messagePrint("algebra is not flexible")$OutputForm
return false
messagePrint("algebra is flexible")$OutputForm
true
lieAdmissible?() ==
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
for r in 1..n repeat
res := 0$R
for l in 1..n repeat
res := res_
+ (elt(gamma.l,i,j)-elt(gamma.l,j,i))*(elt(gamma.r,l,k)-elt(gamma.r,k,l)) _
+ (elt(gamma.l,j,k)-elt(gamma.l,k,j))*(elt(gamma.r,l,i)-elt(gamma.r,i,l)) _
+ (elt(gamma.l,k,i)-elt(gamma.l,i,k))*(elt(gamma.r,l,j)-elt(gamma.r,j,l))
not (zero? res) =>
messagePrint("algebra is not Lie admissible")$OutputForm
return false
messagePrint("algebra is Lie admissible")$OutputForm
true
jordanAdmissible?() ==
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 w in 1..n repeat
for t in 1..n repeat
res := 0$R
for l in 1..n repeat
for r in 1..n repeat
res := res_
+ (elt(gamma.l,i,j)+elt(gamma.l,j,i))_
* (elt(gamma.r,w,k)+elt(gamma.r,k,w))_
* (elt(gamma.t,l,r)+elt(gamma.t,r,l))_
- (elt(gamma.r,w,k)+elt(gamma.r,k,w))_
* (elt(gamma.l,j,r)+elt(gamma.l,r,j))_
* (elt(gamma.t,i,l)+elt(gamma.t,l,i))_
+ (elt(gamma.l,w,j)+elt(gamma.l,j,w))_
* (elt(gamma.r,k,i)+elt(gamma.r,i,k))_
* (elt(gamma.t,l,r)+elt(gamma.t,r,l))_
- (elt(gamma.r,k,i)+elt(gamma.r,k,i))_
* (elt(gamma.l,j,r)+elt(gamma.l,r,j))_
* (elt(gamma.t,w,l)+elt(gamma.t,l,w))_
+ (elt(gamma.l,k,j)+elt(gamma.l,j,k))_
* (elt(gamma.r,i,w)+elt(gamma.r,w,i))_
* (elt(gamma.t,l,r)+elt(gamma.t,r,l))_
- (elt(gamma.r,i,w)+elt(gamma.r,w,i))_
* (elt(gamma.l,j,r)+elt(gamma.l,r,j))_
* (elt(gamma.t,k,l)+elt(gamma.t,l,k))
not (zero? res) =>
messagePrint("algebra is not Jordan admissible")$OutputForm
return false
messagePrint("algebra is Jordan admissible")$OutputForm
true
jordanAlgebra?() ==
recip(2 * 1$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
for t in 1..n repeat
res := 0$R
for r in 1..n repeat
for s in 1..n repeat
res := res + _
elt(gamma.r,i,j)*elt(gamma.s,l,k)*elt(gamma.t,r,s) - _
elt(gamma.r,l,k)*elt(gamma.s,j,r)*elt(gamma.t,i,s) + _
elt(gamma.r,l,j)*elt(gamma.s,k,k)*elt(gamma.t,r,s) - _
elt(gamma.r,k,i)*elt(gamma.s,j,r)*elt(gamma.t,l,s) + _
elt(gamma.r,k,j)*elt(gamma.s,i,k)*elt(gamma.t,r,s) - _
elt(gamma.r,i,l)*elt(gamma.s,j,r)*elt(gamma.t,k,s)
not zero? res =>
messagePrint("this is not a Jordan algebra")$OutputForm
return false
messagePrint("this is a Jordan algebra")$OutputForm
true
jacobiIdentity?() ==
for i in 1..n repeat
for j in 1..n repeat
for k in 1..n repeat
for r in 1..n repeat
res := 0$R
for s in 1..n repeat
res := res + elt(gamma.r,i,j)*elt(gamma.s,j,k) +_
elt(gamma.r,j,k)*elt(gamma.s,k,i) +_
elt(gamma.r,k,i)*elt(gamma.s,i,j)
not zero? res =>
messagePrint("Jacobi identity does not hold")$OutputForm
return false
messagePrint("Jacobi identity holds")$OutputForm
true
)abbrev package ALGPKG AlgebraPackage
++ Authors: J. Grabmeier, R. Wisbauer
++ Date Created: 04 March 1991
++ Date Last Updated: 04 April 1992
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: rank, nucleus, nucloid, structural constants
++ Reference:
++ R.S. Pierce: Associative Algebras
++ Graduate Texts in Mathematics 88
++ Springer-Verlag, Heidelberg, 1982, ISBN 0-387-90693-2
++
++ R.D. Schafer: An Introduction to Nonassociative Algebras
++ Academic Press, New York, 1966
++
++ A. Woerz-Busekros: Algebra in Genetics
++ Lectures Notes in Biomathematics 36,
++ Springer-Verlag, Heidelberg, 1980
++ Description:
++ AlgebraPackage assembles a variety of useful functions for
++ general algebras.
AlgebraPackage(R:IntegralDomain, A: FramedNonAssociativeAlgebra(R)): _
public == private where
V ==> Vector
M ==> Matrix
I ==> Integer
NNI ==> NonNegativeInteger
REC ==> Record(particular: Union(V R,"failed"),basis: List V R)
LSMP ==> LinearSystemMatrixPackage(R,V R,V R, M R)
public ==> with
leftRank: A -> NonNegativeInteger
++ leftRank(x) determines the number of linearly independent elements
++ in \spad{x*b1},...,\spad{x*bn},
++ where \spad{b=[b1,...,bn]} is a basis.
rightRank: A -> NonNegativeInteger
++ rightRank(x) determines the number of linearly independent elements
++ in \spad{b1*x},...,\spad{bn*x},
++ where \spad{b=[b1,...,bn]} is a basis.
doubleRank: A -> NonNegativeInteger
++ doubleRank(x) determines the number of linearly
++ independent elements
++ in \spad{b1*x},...,\spad{x*bn},
++ where \spad{b=[b1,...,bn]} is a basis.
weakBiRank: A -> NonNegativeInteger
++ weakBiRank(x) determines the number of
++ linearly independent elements
++ in the \spad{bi*x*bj}, \spad{i,j=1,...,n},
++ where \spad{b=[b1,...,bn]} is a basis.
biRank: A -> NonNegativeInteger
++ biRank(x) determines the number of linearly independent elements
++ in \spad{x}, \spad{x*bi}, \spad{bi*x}, \spad{bi*x*bj},
++ \spad{i,j=1,...,n},
++ where \spad{b=[b1,...,bn]} is a basis.
++ Note: if \spad{A} has a unit,
++ then \spadfunFrom{doubleRank}{AlgebraPackage},
++ \spadfunFrom{weakBiRank}{AlgebraPackage}
++ and \spadfunFrom{biRank}{AlgebraPackage} coincide.
basisOfCommutingElements: () -> List A
++ basisOfCommutingElements() returns a basis of the space of
++ all x of \spad{A} satisfying \spad{0 = commutator(x,a)} for all
++ \spad{a} in \spad{A}.
basisOfLeftAnnihilator: A -> List A
++ basisOfLeftAnnihilator(a) returns a basis of the space of
++ all x of \spad{A} satisfying \spad{0 = x*a}.
basisOfRightAnnihilator: A -> List A
++ basisOfRightAnnihilator(a) returns a basis of the space of
++ all x of \spad{A} satisfying \spad{0 = a*x}.
basisOfLeftNucleus: () -> List A
++ basisOfLeftNucleus() returns a basis of the space of
++ all x of \spad{A} satisfying \spad{0 = associator(x,a,b)}
++ for all \spad{a},b in \spad{A}.
basisOfRightNucleus: () -> List A
++ basisOfRightNucleus() returns a basis of the space of
++ all x of \spad{A} satisfying \spad{0 = associator(a,b,x)}
++ for all \spad{a},b in \spad{A}.
basisOfMiddleNucleus: () -> List A
++ basisOfMiddleNucleus() returns a basis of the space of
++ all x of \spad{A} satisfying \spad{0 = associator(a,x,b)}
++ for all \spad{a},b in \spad{A}.
basisOfNucleus: () -> List A
++ basisOfNucleus() returns a basis of the space of all x of \spad{A} satisfying
++ \spad{associator(x,a,b) = associator(a,x,b) = associator(a,b,x) = 0}
++ for all \spad{a},b in \spad{A}.
basisOfCenter: () -> List A
++ basisOfCenter() returns a basis of the space of
++ all x of \spad{A} satisfying \spad{commutator(x,a) = 0} and
++ \spad{associator(x,a,b) = associator(a,x,b) = associator(a,b,x) = 0}
++ for all \spad{a},b in \spad{A}.
basisOfLeftNucloid:()-> List Matrix R
++ basisOfLeftNucloid() returns a basis of the space of
++ endomorphisms of \spad{A} as right module.
++ Note: left nucloid coincides with left nucleus if \spad{A} has a unit.
basisOfRightNucloid:()-> List Matrix R
++ basisOfRightNucloid() returns a basis of the space of
++ endomorphisms of \spad{A} as left module.
++ Note: right nucloid coincides with right nucleus if \spad{A} has a unit.
basisOfCentroid:()-> List Matrix R
++ basisOfCentroid() returns a basis of the centroid, i.e. the
++ endomorphism ring of \spad{A} considered as \spad{(A,A)}-bimodule.
radicalOfLeftTraceForm: () -> List A
++ radicalOfLeftTraceForm() returns basis for null space of
++ \spad{leftTraceMatrix()}, if the algebra is
++ associative, alternative or a Jordan algebra, then this
++ space equals the radical (maximal nil ideal) of the algebra.
if R has EuclideanDomain then
basis : V A -> V A
++ basis(va) selects a basis from the elements of va.
private ==> add
-- constants
n : PositiveInteger := rank()$A
n2 : PositiveInteger := n*n
n3 : PositiveInteger := n*n2
gamma : Vector Matrix R := structuralConstants()$A
-- local functions
convVM : Vector R -> Matrix R
-- converts n2-vector to (n,n)-matrix row by row
convMV : Matrix R -> Vector R
-- converts n-square matrix to n2-vector row by row
convVM v ==
cond : Matrix(R) := new(n,n,0$R)$M(R)
z : Integer := 0
for i in 1..n repeat
for j in 1..n repeat
z := z+1
setelt(cond,i,j,v.z)
cond
-- convMV m ==
-- vec : Vector(R) := new(n*n,0$R)
-- z : Integer := 0
-- for i in 1..n repeat
-- for j in 1..n repeat
-- z := z+1
-- setelt(vec,z,elt(m,i,j))
-- vec
radicalOfLeftTraceForm() ==
ma : M R := leftTraceMatrix()$A
map(represents, nullSpace ma)$ListFunctions2(Vector R, A)
basisOfLeftAnnihilator a ==
ca : M R := transpose (coordinates(a) :: M R)
cond : M R := reduce(vertConcat$(M R),
[ca*transpose(gamma.i) for i in 1..#gamma])
map(represents, nullSpace cond)$ListFunctions2(Vector R, A)
basisOfRightAnnihilator a ==
ca : M R := transpose (coordinates(a) :: M R)
cond : M R := reduce(vertConcat$(M R),
[ca*(gamma.i) for i in 1..#gamma])
map(represents, nullSpace cond)$ListFunctions2(Vector R, A)
basisOfLeftNucloid() ==
cond : Matrix(R) := new(n3,n2,0$R)$M(R)
condo: Matrix(R) := new(n3,n2,0$R)$M(R)
z : Integer := 0
for i in 1..n repeat
for j in 1..n repeat
r1 : Integer := 0
for k in 1..n repeat
z := z + 1
-- z equals (i-1)*n*n+(j-1)*n+k (loop-invariant)
r2 : Integer := i
for r in 1..n repeat
r1 := r1 + 1
-- here r1 equals (k-1)*n+r (loop-invariant)
setelt(cond,z,r1,elt(gamma.r,i,j))
-- here r2 equals (r-1)*n+i (loop-invariant)
setelt(condo,z,r2,-elt(gamma.k,r,j))
r2 := r2 + n
[convVM(sol) for sol in nullSpace(cond+condo)]
basisOfCommutingElements() ==
--gamma1 := first gamma
--gamma1 := gamma1 - transpose gamma1
--cond : Matrix(R) := gamma1 :: Matrix(R)
--for i in 2..n repeat
-- gammak := gamma.i
-- gammak := gammak - transpose gammak
-- cond := vertConcat(cond, gammak :: Matrix(R))$Matrix(R)
--map(represents, nullSpace cond)$ListFunctions2(Vector R, A)
cond : M R := reduce(vertConcat$(M R),
[(gam := gamma.i) - transpose gam for i in 1..#gamma])
map(represents, nullSpace cond)$ListFunctions2(Vector R, A)
basisOfLeftNucleus() ==
condi: Matrix(R) := new(n3,n,0$R)$Matrix(R)
z : Integer := 0
for k in 1..n repeat
for j in 1..n repeat
for s in 1..n repeat
z := z+1
for i in 1..n repeat
entry : R := 0
for l in 1..n repeat
entry := entry+elt(gamma.l,j,k)*elt(gamma.s,i,l)_
-elt(gamma.l,i,j)*elt(gamma.s,l,k)
setelt(condi,z,i,entry)$Matrix(R)
map(represents, nullSpace condi)$ListFunctions2(Vector R,A)
basisOfRightNucleus() ==
condo : Matrix(R) := new(n3,n,0$R)$Matrix(R)
z : Integer := 0
for k in 1..n repeat
for j in 1..n repeat
for s in 1..n repeat
z := z+1
for i in 1..n repeat
entry : R := 0
for l in 1..n repeat
entry := entry+elt(gamma.l,k,i)*elt(gamma.s,j,l) _
-elt(gamma.l,j,k)*elt(gamma.s,l,i)
setelt(condo,z,i,entry)$Matrix(R)
map(represents, nullSpace condo)$ListFunctions2(Vector R,A)
basisOfMiddleNucleus() ==
conda : Matrix(R) := new(n3,n,0$R)$Matrix(R)
z : Integer := 0
for k in 1..n repeat
for j in 1..n repeat
for s in 1..n repeat
z := z+1
for i in 1..n repeat
entry : R := 0
for l in 1..n repeat
entry := entry+elt(gamma.l,j,i)*elt(gamma.s,l,k)
-elt(gamma.l,i,k)*elt(gamma.s,j,l)
setelt(conda,z,i,entry)$Matrix(R)
map(represents, nullSpace conda)$ListFunctions2(Vector R,A)
basisOfNucleus() ==
condi: Matrix(R) := new(3*n3,n,0$R)$Matrix(R)
z : Integer := 0
u : Integer := n3
w : Integer := 2*n3
for k in 1..n repeat
for j in 1..n repeat
for s in 1..n repeat
z := z+1
u := u+1
w := w+1
for i in 1..n repeat
entry : R := 0
enter : R := 0
ent : R := 0
for l in 1..n repeat
entry := entry + elt(gamma.l,j,k)*elt(gamma.s,i,l) _
- elt(gamma.l,i,j)*elt(gamma.s,l,k)
enter := enter + elt(gamma.l,k,i)*elt(gamma.s,j,l) _
- elt(gamma.l,j,k)*elt(gamma.s,l,i)
ent := ent + elt(gamma.l,j,k)*elt(gamma.s,i,l) _
- elt(gamma.l,j,i)*elt(gamma.s,l,k)
setelt(condi,z,i,entry)$Matrix(R)
setelt(condi,u,i,enter)$Matrix(R)
setelt(condi,w,i,ent)$Matrix(R)
map(represents, nullSpace condi)$ListFunctions2(Vector R,A)
basisOfCenter() ==
gamma1 := first gamma
gamma1 := gamma1 - transpose gamma1
cond : Matrix(R) := gamma1 :: Matrix(R)
for i in 2..n repeat
gammak := gamma.i
gammak := gammak - transpose gammak
cond := vertConcat(cond, gammak :: Matrix(R))$Matrix(R)
B := cond :: Matrix(R)
condi: Matrix(R) := new(2*n3,n,0$R)$Matrix(R)
z : Integer := 0
u : Integer := n3
for k in 1..n repeat
for j in 1..n repeat
for s in 1..n repeat
z := z+1
u := u+1
for i in 1..n repeat
entry : R := 0
enter : R := 0
for l in 1..n repeat
entry := entry + elt(gamma.l,j,k)*elt(gamma.s,i,l) _
- elt(gamma.l,i,j)*elt(gamma.s,l,k)
enter := enter + elt(gamma.l,k,i)*elt(gamma.s,j,l) _
- elt(gamma.l,j,k)*elt(gamma.s,l,i)
setelt(condi,z,i,entry)$Matrix(R)
setelt(condi,u,i,enter)$Matrix(R)
D := vertConcat(condi,B)$Matrix(R)
map(represents, nullSpace D)$ListFunctions2(Vector R, A)
basisOfRightNucloid() ==
cond : Matrix(R) := new(n3,n2,0$R)$M(R)
condo: Matrix(R) := new(n3,n2,0$R)$M(R)
z : Integer := 0
for i in 1..n repeat
for j in 1..n repeat
r1 : Integer := 0
for k in 1..n repeat
z := z + 1
-- z equals (i-1)*n*n+(j-1)*n+k (loop-invariant)
r2 : Integer := i
for r in 1..n repeat
r1 := r1 + 1
-- here r1 equals (k-1)*n+r (loop-invariant)
setelt(cond,z,r1,elt(gamma.r,j,i))
-- here r2 equals (r-1)*n+i (loop-invariant)
setelt(condo,z,r2,-elt(gamma.k,j,r))
r2 := r2 + n
[convVM(sol) for sol in nullSpace(cond+condo)]
basisOfCentroid() ==
cond : Matrix(R) := new(2*n3,n2,0$R)$M(R)
condo: Matrix(R) := new(2*n3,n2,0$R)$M(R)
z : Integer := 0
u : Integer := n3
for i in 1..n repeat
for j in 1..n repeat
r1 : Integer := 0
for k in 1..n repeat
z := z + 1
u := u + 1
-- z equals (i-1)*n*n+(j-1)*n+k (loop-invariant)
-- u equals n**3 + (i-1)*n*n+(j-1)*n+k (loop-invariant)
r2 : Integer := i
for r in 1..n repeat
r1 := r1 + 1
-- here r1 equals (k-1)*n+r (loop-invariant)
setelt(cond,z,r1,elt(gamma.r,i,j))
setelt(cond,u,r1,elt(gamma.r,j,i))
-- here r2 equals (r-1)*n+i (loop-invariant)
setelt(condo,z,r2,-elt(gamma.k,r,j))
setelt(condo,u,r2,-elt(gamma.k,j,r))
r2 := r2 + n
[convVM(sol) for sol in nullSpace(cond+condo)]
doubleRank x ==
cond : Matrix(R) := new(2*n,n,0$R)
for k in 1..n repeat
z : Integer := 0
u : Integer := n
for j in 1..n repeat
z := z+1
u := u+1
entry : R := 0
enter : R := 0
for i in 1..n repeat
entry := entry + elt(x,i)*elt(gamma.k,j,i)
enter := enter + elt(x,i)*elt(gamma.k,i,j)
setelt(cond,z,k,entry)$Matrix(R)
setelt(cond,u,k,enter)$Matrix(R)
rank(cond)$(M R)
weakBiRank(x) ==
cond : Matrix(R) := new(n2,n,0$R)$Matrix(R)
z : Integer := 0
for i in 1..n repeat
for j in 1..n repeat
z := z+1
for k in 1..n repeat
entry : R := 0
for l in 1..n repeat
for s in 1..n repeat
entry:=entry+elt(x,l)*elt(gamma.s,i,l)*elt(gamma.k,s,j)
setelt(cond,z,k,entry)$Matrix(R)
rank(cond)$(M R)
biRank(x) ==
cond : Matrix(R) := new(n2+2*n+1,n,0$R)$Matrix(R)
z : Integer := 0
for j in 1..n repeat
for i in 1..n repeat
z := z+1
for k in 1..n repeat
entry : R := 0
for l in 1..n repeat
for s in 1..n repeat
entry:=entry+elt(x,l)*elt(gamma.s,i,l)*elt(gamma.k,s,j)
setelt(cond,z,k,entry)$Matrix(R)
u : Integer := n*n
w : Integer := n*(n+1)
c := n2 + 2*n + 1
for j in 1..n repeat
u := u+1
w := w+1
for k in 1..n repeat
entry : R := 0
enter : R := 0
for i in 1..n repeat
entry := entry + elt(x,i)*elt(gamma.k,j,i)
enter := enter + elt(x,i)*elt(gamma.k,i,j)
setelt(cond,u,k,entry)$Matrix(R)
setelt(cond,w,k,enter)$Matrix(R)
setelt(cond,c,j, elt(x,j))
rank(cond)$(M R)
leftRank x ==
cond : Matrix(R) := new(n,n,0$R)
for k in 1..n repeat
for j in 1..n repeat
entry : R := 0
for i in 1..n repeat
entry := entry + elt(x,i)*elt(gamma.k,i,j)
setelt(cond,j,k,entry)$Matrix(R)
rank(cond)$(M R)
rightRank x ==
cond : Matrix(R) := new(n,n,0$R)
for k in 1..n repeat
for j in 1..n repeat
entry : R := 0
for i in 1..n repeat
entry := entry + elt(x,i)*elt(gamma.k,j,i)
setelt(cond,j,k,entry)$Matrix(R)
rank(cond)$(M R)
if R has EuclideanDomain then
basis va ==
v : V A := remove(zero?, va)$(V A)
v : V A := removeDuplicates v
empty? v => [0$A]
m : Matrix R := coerce(coordinates(v.1))$(Matrix R)
for i in 2..maxIndex v repeat
m := horizConcat(m,coerce(coordinates(v.i))$(Matrix R) )
m := rowEchelon m
lj : List Integer := []
h : Integer := 1
mRI : Integer := maxRowIndex m
mCI : Integer := maxColIndex m
finished? : Boolean := false
j : Integer := 1
while not finished? repeat
not zero? m(h,j) => -- corner found
lj := cons(j,lj)
h := mRI
while zero? m(h,j) repeat h := h-1
finished? := (h = mRI)
if not finished? then h := h+1
if j < mCI then
j := j + 1
else
finished? := true
[v.j for j in reverse lj]
)abbrev package SCPKG StructuralConstantsPackage
++ Authors: J. Grabmeier
++ Date Created: 02 April 1992
++ Date Last Updated: 14 April 1992
++ Basic Operations:
++ Related Constructors: AlgebraPackage, AlgebraGivenByStructuralConstants
++ Also See:
++ AMS Classifications:
++ Keywords: structural constants
++ Reference:
++ Description:
++ StructuralConstantsPackage provides functions creating
++ structural constants from a multiplication tables or a basis
++ of a matrix algebra and other useful functions in this context.
StructuralConstantsPackage(R:Field): public == private where
L ==> List
S ==> Symbol
FRAC ==> Fraction
POLY ==> Polynomial
V ==> Vector
M ==> Matrix
REC ==> Record(particular: Union(V R,"failed"),basis: List V R)
LSMP ==> LinearSystemMatrixPackage(R,V R,V R, M R)
public ==> with
-- what we really want to have here is a matrix over
-- linear polynomials in the list of symbols, having arbitrary
-- coefficients from a ring extension of R, e.g. FRAC POLY R.
structuralConstants : (L S, M FRAC POLY R) -> V M FRAC POLY R
++ structuralConstants(ls,mt) determines the structural constants
++ of an algebra with generators ls and multiplication table mt, the
++ entries of which must be given as linear polynomials in the
++ indeterminates given by ls. The result is in particular useful
++ as fourth argument for \spadtype{AlgebraGivenByStructuralConstants}
++ and \spadtype{GenericNonAssociativeAlgebra}.
structuralConstants : (L S, M POLY R) -> V M POLY R
++ structuralConstants(ls,mt) determines the structural constants
++ of an algebra with generators ls and multiplication table mt, the
++ entries of which must be given as linear polynomials in the
++ indeterminates given by ls. The result is in particular useful
++ as fourth argument for \spadtype{AlgebraGivenByStructuralConstants}
++ and \spadtype{GenericNonAssociativeAlgebra}.
structuralConstants: L M R -> V M R
++ structuralConstants(basis) takes the basis of a matrix
++ algebra, e.g. the result of \spadfun{basisOfCentroid} and calculates
++ the structural constants.
++ Note, that the it is not checked, whether basis really is a
++ basis of a matrix algebra.
coordinates: (M R, L M R) -> V R
++ coordinates(a,[v1,...,vn]) returns the coordinates of \spad{a}
++ with respect to the \spad{R}-module basis \spad{v1},...,\spad{vn}.
private ==> add
matrix2Vector: M R -> V R
matrix2Vector m ==
lili : L L R := listOfLists m
--li : L R := reduce(concat, listOfLists m)
li : L R := reduce(concat, lili)
construct(li)$(V R)
coordinates(x,b) ==
m : NonNegativeInteger := (maxIndex b) :: NonNegativeInteger
n : NonNegativeInteger := nrows(b.1) * ncols(b.1)
transitionMatrix : Matrix R := new(n,m,0$R)$Matrix(R)
for i in 1..m repeat
setColumn!(transitionMatrix,i,matrix2Vector(b.i))
res : REC := solve(transitionMatrix,matrix2Vector(x))$LSMP
if (not every?(zero?$R,first res.basis)) then
error("coordinates: the second argument is linearly dependent")
(res.particular case "failed") =>
error("coordinates: first argument is not in linear span of _
second argument")
(res.particular) :: (Vector R)
structuralConstants b ==
--n := rank()
-- be careful with the possibility that b is not a basis
m : NonNegativeInteger := (maxIndex b) :: NonNegativeInteger
sC : Vector Matrix R := [new(m,m,0$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
structuralConstants(ls:L S, mt: M POLY R) ==
nn := #(ls)
nrows(mt) ~= nn or ncols(mt) ~= nn =>
error "structuralConstants: size of second argument does not _
agree with number of generators"
gamma : L M POLY R := []
lscopy : L S := copy ls
while not null lscopy repeat
mat : M POLY R := new(nn,nn,0)
s : S := first lscopy
for i in 1..nn repeat
for j in 1..nn repeat
p := qelt(mt,i,j)
totalDegree(p,ls) > 1 =>
error "structuralConstants: entries of second argument _
must be linear polynomials in the generators"
if (c := coefficient(p, s, 1) ) ~= 0 then qsetelt!(mat,i,j,c)
gamma := cons(mat, gamma)
lscopy := rest lscopy
vector reverse gamma
structuralConstants(ls:L S, mt: M FRAC POLY R) ==
nn := #(ls)
nrows(mt) ~= nn or ncols(mt) ~= nn =>
error "structuralConstants: size of second argument does not _
agree with number of generators"
gamma : L M FRAC(POLY R) := []
lscopy : L S := copy ls
while not null lscopy repeat
mat : M FRAC(POLY R) := new(nn,nn,0)
s : S := first lscopy
for i in 1..nn repeat
for j in 1..nn repeat
r := qelt(mt,i,j)
q := denom(r)
totalDegree(q,ls) ~= 0 =>
error "structuralConstants: entries of second argument _
must be (linear) polynomials in the generators"
p := numer(r)
totalDegree(p,ls) > 1 =>
error "structuralConstants: entries of second argument _
must be linear polynomials in the generators"
if (c := coefficient(p, s, 1) ) ~= 0 then qsetelt!(mat,i,j,c/q)
gamma := cons(mat, gamma)
lscopy := rest lscopy
vector reverse gamma
)abbrev package FRNAAF2 FramedNonAssociativeAlgebraFunctions2
++ Author: Johannes Grabmeier
++ Date Created: 28 February 1992
++ Date Last Updated: 28 February 1992
++ Basic Operations: map
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: non-associative algebra
++ References:
++ Description:
++ FramedNonAssociativeAlgebraFunctions2 implements functions between
++ two framed non associative algebra domains defined over different rings.
++ The function map is used to coerce between algebras over different
++ domains having the same structural constants.
FramedNonAssociativeAlgebraFunctions2(AR,R,AS,S) : Exports ==
Implementation where
R : CommutativeRing
S : CommutativeRing
AR : FramedNonAssociativeAlgebra R
AS : FramedNonAssociativeAlgebra S
V ==> Vector
Exports ==> with
map: (R -> S, AR) -> AS
++ map(f,u) maps f onto the coordinates of u to get an element
++ in \spad{AS} via identification of the basis of \spad{AR}
++ as beginning part of the basis of \spad{AS}.
Implementation ==> add
map(fn : R -> S, u : AR): AS ==
rank()$AR > rank()$AS => error("map: ranks of algebras do not fit")
vr : V R := coordinates u
vs : V S := map(fn,vr)$VectorFunctions2(R,S)
rank()$AR = rank()$AS => represents(vs)$AS
ba := basis()$AS
represents(vs,[ba.i for i in 1..rank()$AR])
|