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)abbrev domain LIE AssociatedLieAlgebra
++ Author: J. Grabmeier
++ Date Created: 07 March 1991
++ Date Last Updated: 14 June 1991
++ Basic Operations: *,**,+,-
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: associated Liealgebra
++ References:
++ Description:
++ AssociatedLieAlgebra takes an algebra \spad{A}
++ and uses \spadfun{*$A} to define the
++ Lie bracket \spad{a*b := (a *$A b - b *$A a)} (commutator). Note that
++ the notation \spad{[a,b]} cannot be used due to
++ restrictions of the current compiler.
++ This domain only gives a Lie algebra if the
++ Jacobi-identity \spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} holds
++ for all \spad{a},\spad{b},\spad{c} in \spad{A}.
++ This relation can be checked by
++ \spad{lieAdmissible?()$A}.
++
++ If the underlying algebra is of type
++ \spadtype{FramedNonAssociativeAlgebra(R)} (i.e. a non
++ associative algebra over R which is a free \spad{R}-module of finite
++ rank, together with a fixed \spad{R}-module basis), then the same
++ is true for the associated Lie algebra.
++ Also, if the underlying algebra is of type
++ \spadtype{FiniteRankNonAssociativeAlgebra(R)} (i.e. a non
++ associative algebra over R which is a free R-module of finite
++ rank), then the same is true for the associated Lie algebra.
AssociatedLieAlgebra(R:CommutativeRing,A:NonAssociativeAlgebra R):
public == private where
public ==> Join (NonAssociativeAlgebra R, CoercibleTo A) with
coerce : A -> %
++ coerce(a) coerces the element \spad{a} of the algebra \spad{A}
++ to an element of the Lie
++ algebra \spadtype{AssociatedLieAlgebra}(R,A).
if A has FramedNonAssociativeAlgebra(R) then
FramedNonAssociativeAlgebra(R)
if A has FiniteRankNonAssociativeAlgebra(R) then
FiniteRankNonAssociativeAlgebra(R)
private ==> A add
(a:%) * (b:%) == per(rep a * rep b - rep b * rep a)
coerce(a:%):A == rep a
coerce(a:A):% == per a
(a:%) ** (n:PositiveInteger) ==
n = 1 => a
0
)abbrev domain JORDAN AssociatedJordanAlgebra
++ Author: J. Grabmeier
++ Date Created: 14 June 1991
++ Date Last Updated: 14 June 1991
++ Basic Operations: *,**,+,-
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: associated Jordan algebra
++ References:
++ Description:
++ AssociatedJordanAlgebra takes an algebra \spad{A} and uses \spadfun{*$A}
++ to define the new multiplications \spad{a*b := (a *$A b + b *$A a)/2}
++ (anticommutator).
++ The usual notation \spad{{a,b}_+} cannot be used due to
++ restrictions in the current language.
++ This domain only gives a Jordan algebra if the
++ Jordan-identity \spad{(a*b)*c + (b*c)*a + (c*a)*b = 0} holds
++ for all \spad{a},\spad{b},\spad{c} in \spad{A}.
++ This relation can be checked by
++ \spadfun{jordanAdmissible?()$A}.
++
++ If the underlying algebra is of type
++ \spadtype{FramedNonAssociativeAlgebra(R)} (i.e. a non
++ associative algebra over R which is a free R-module of finite
++ rank, together with a fixed R-module basis), then the same
++ is true for the associated Jordan algebra.
++ Moreover, if the underlying algebra is of type
++ \spadtype{FiniteRankNonAssociativeAlgebra(R)} (i.e. a non
++ associative algebra over R which is a free R-module of finite
++ rank), then the same true for the associated Jordan algebra.
AssociatedJordanAlgebra(R:CommutativeRing,A:NonAssociativeAlgebra R):
public == private where
public ==> Join (NonAssociativeAlgebra R, CoercibleTo A) with
coerce : A -> %
++ coerce(a) coerces the element \spad{a} of the algebra \spad{A}
++ to an element of the Jordan algebra
++ \spadtype{AssociatedJordanAlgebra}(R,A).
if A has FramedNonAssociativeAlgebra(R) then _
FramedNonAssociativeAlgebra(R)
if A has FiniteRankNonAssociativeAlgebra(R) then _
FiniteRankNonAssociativeAlgebra(R)
private ==> A add
oneHalf : R := recip(1$R + 1$R) :: R
(a:%) * (b:%) ==
characteristic$R = 2 => error
"constructor must not be called with Ring of characteristic 2"
per((rep a * rep b + rep b * rep a) * oneHalf)
coerce(a:%):A == rep a
coerce(a:A):% == per a
(a:%) ** (n:PositiveInteger) == a
)abbrev domain LSQM LieSquareMatrix
++ Author: J. Grabmeier
++ Date Created: 07 March 1991
++ Date Last Updated: 08 March 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ LieSquareMatrix(n,R) implements the Lie algebra of the n by n
++ matrices over the commutative ring R.
++ The Lie bracket (commutator) of the algebra is given by
++ \spad{a*b := (a *$SQMATRIX(n,R) b - b *$SQMATRIX(n,R) a)},
++ where \spadfun{*$SQMATRIX(n,R)} is the usual matrix multiplication.
LieSquareMatrix(n,R): Exports == Implementation where
n : PositiveInteger
R : CommutativeRing
Row ==> DirectProduct(n,R)
Col ==> DirectProduct(n,R)
Exports ==> Join(SquareMatrixCategory(n,R,Row,Col), CoercibleTo Matrix R,_
FramedNonAssociativeAlgebra R) --with
Implementation ==> AssociatedLieAlgebra (R,SquareMatrix(n, R)) add
-- local functions
n2 : PositiveInteger := n*n
convDM : DirectProduct(n2,R) -> %
conv : DirectProduct(n2,R) -> SquareMatrix(n,R)
--++ converts n2-vector to (n,n)-matrix row by row
conv v ==
cond : Matrix(R) := new(n,n,0$R)$Matrix(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)
squareMatrix(cond)$SquareMatrix(n, R)
coordinates(a:%,b:Vector(%)):Vector(R) ==
-- only valid for b canonicalBasis
res : Vector R := new(n2,0$R)
z : Integer := 0
for i in 1..n repeat
for j in 1..n repeat
z := z+1
res.z := elt(a,i,j)$%
res
convDM v ==
per(conv(v)::Rep)
basis() ==
n2 : PositiveInteger := n*n
ldp : List DirectProduct(n2,R) :=
[unitVector(i::PositiveInteger)$DirectProduct(n2,R) for i in 1..n2]
res:Vector % := vector map(convDM,_
ldp)$ListFunctions2(DirectProduct(n2,R), %)
someBasis() == basis()
rank() == n*n
-- transpose: % -> %
-- ++ computes the transpose of a matrix
-- squareMatrix: Matrix R -> %
-- ++ converts a Matrix to a LieSquareMatrix
-- coerce: % -> Matrix R
-- ++ converts a LieSquareMatrix to a Matrix
-- symdecomp : % -> Record(sym:%,antisym:%)
-- if R has commutative("*") then
-- minorsVect: -> Vector(Union(R,"uncomputed")) --range: 1..2**n-1
-- if R has commutative("*") then central
-- if R has commutative("*") and R has unitsKnown then unitsKnown
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