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last edited 11 years ago by test1 |
Edit detail for SandBoxTensorProduct revision 6 of 9
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Editor: Bill Page
Time: 2009/05/13 08:06:58 GMT-7 |
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| Note: first example of a bialgebra | ||
added:
From BillPage Wed May 13 08:06:57 -0700 2009
From: Bill Page
Date: Wed, 13 May 2009 08:06:57 -0700
Subject: first example of a bialgebra
Message-ID: <20090513080657-0700@axiom-wiki.newsynthesis.org>
SandBoxHopfAlgebra
Date: Fri, 15 May 2009 21:03:00 +0200 Franz Lehner wrote:
Attached is a prototype for tensor products. It is free modules over commutative rings.
FreeModule? is defined in poly.spad with a local category. FreeModuleCat? is defined in xpoly.spad. FreeModuleCat? does not require OrderedSet?, but FreeModule? does. FreeModule1? is of category FreeModuleCat?, its Rep however is FreeModule?.
spad
)abbrev category TENSORC TensorProductCategory
TensorProductCategory(R:CommutativeRing, M : Module(R), N : Module(R)):Category == Module(R) with
product: (M, N) -> %
)abbrev category TENSORP TensorProductProperty
TensorProductProperty(R:CommutativeRing, M : Module(R), N : Module(R), _
MxN : TensorProductCategory(R, M, N), S : Module(R)): Category == with
eval: (MxN, (M, N) -> S) -> S
)abbrev package TENSORD TensorProduct
TensorProduct(R : CommutativeRing, B1 : OrderedSet, B2 : OrderedSet, _
M1 : FreeModuleCat(R, B1), M2 : FreeModuleCat(R, B2)): TPcat == TPimp where
TPcat == Join(TensorProductCategory(R,M1,M2),FreeModuleCat(R,Product(B1,B2))) with
if M1 has Algebra(R) and M2 has Algebra(R) then Algebra(R)
TERM1 == Record(k: B1, c: R)
TERM2 == Record(k: B2, c: R)
B1xB2 == Product(B1,B2)
TERM == Record(k: B1xB2, c: R)
TPimp == FreeModule1(R, Product(B1, B2)) add
import Rep, TERM1, TERM2, TERM, B1xB2
product(x1:M1,x2:M2):% ==
zero? x1 or zero? x2 => return 0
ltx1:List TERM1 := ListOfTerms x1
ltx2:List TERM2 := ListOfTerms x2
res : List TERM := []
for s1 in ltx1 repeat
for s2 in ltx2 repeat
res := concat!(res,[makeprod(s1.k, s2.k), s1.c*s2.c]$TERM)
res pretend %
if M1 has Algebra(R) and M2 has Algebra(R) then
(x1:% * x2:%):% ==
res : % := 0
for t1 in ListOfTerms x1 repeat
for t2 in ListOfTerms x2 repeat
-- the coefficients
t1c:R := t1.c
t2c:R := t2.c
-- the basis elements
t1k:B1xB2 := t1.k
t2k:B1xB2 := t2.k
t1a: M1 := monom(selectfirst t1k,1)
t1b: M2 := monom(selectsecond t1k,1)
t2a: M1 := monom(selectfirst t2k,1)
t2b: M2 := monom(selectsecond t2k,1)
res:= res + t1.c*t2.c *product(t1a*t2a,t1b*t2b)
res
spad
Compiling FriCAS source code from file
/var/zope2/var/LatexWiki/7772204117579672776-25px001.spad using
old system compiler.
TENSORC abbreviates category TensorProductCategory
------------------------------------------------------------------------
initializing NRLIB TENSORC for TensorProductCategory
compiling into NRLIB TENSORC
;;; *** |TensorProductCategory| REDEFINED
Time: 0 SEC.
finalizing NRLIB TENSORC
Processing TensorProductCategory for Browser database:
--->-->TensorProductCategory((product (% M N))): Not documented!!!!
--->-->TensorProductCategory(constructor): Not documented!!!!
--->-->TensorProductCategory(): Missing Description
------------------------------------------------------------------------
TensorProductCategory is now explicitly exposed in frame initial
TensorProductCategory will be automatically loaded when needed from
/var/zope2/var/LatexWiki/TENSORC.NRLIB/code
TENSORP abbreviates category TensorProductProperty
------------------------------------------------------------------------
initializing NRLIB TENSORP for TensorProductProperty
compiling into NRLIB TENSORP
;;; *** |TensorProductProperty| REDEFINED
Time: 0 SEC.
finalizing NRLIB TENSORP
Processing TensorProductProperty for Browser database:
--->-->TensorProductProperty((eval (S MxN (Mapping S M N)))): Not documented!!!!
--->-->TensorProductProperty(constructor): Not documented!!!!
--->-->TensorProductProperty(): Missing Description
------------------------------------------------------------------------
TensorProductProperty is now explicitly exposed in frame initial
TensorProductProperty will be automatically loaded when needed from
/var/zope2/var/LatexWiki/TENSORP.NRLIB/code
TENSORD abbreviates package TensorProduct
------------------------------------------------------------------------
initializing NRLIB TENSORD for TensorProduct
compiling into NRLIB TENSORD
importing Rep
importing Record(k: B1,c: R)
importing Record(k: B2,c: R)
importing Record(k: Product(B1,B2),c: R)
importing Product(B1,B2)
compiling exported product : (M1,M2) -> $
Time: 0.09 SEC.
****** Domain: M1 already in scope
augmenting M1: (Algebra R)
****** Domain: M2 already in scope
augmenting M2: (Algebra R)
compiling exported * : ($,$) -> $
Time: 0.08 SEC.
****** Domain: M1 already in scope
augmenting M1: (Algebra R)
****** Domain: M2 already in scope
augmenting M2: (Algebra R)
(time taken in buildFunctor: 1)
;;; *** |TensorProduct| REDEFINED
;;; *** |TensorProduct| REDEFINED
Time: 0.02 SEC.
Warnings:
[1] not known that (OrderedSet) is of mode (CATEGORY domain (IF (has B1 (Finite)) (IF (has B2 (Finite)) (ATTRIBUTE (Finite)) noBranch) noBranch) (IF (has B1 (Monoid)) (IF (has B2 (Monoid)) (ATTRIBUTE (Monoid)) noBranch) noBranch) (IF (has B1 (AbelianMonoid)) (IF (has B2 (AbelianMonoid)) (ATTRIBUTE (AbelianMonoid)) noBranch) noBranch) (IF (has B1 (CancellationAbelianMonoid)) (IF (has B2 (CancellationAbelianMonoid)) (ATTRIBUTE (CancellationAbelianMonoid)) noBranch) noBranch) (IF (has B1 (Group)) (IF (has B2 (Group)) (ATTRIBUTE (Group)) noBranch) noBranch) (IF (has B1 (AbelianGroup)) (IF (has B2 (AbelianGroup)) (ATTRIBUTE (AbelianGroup)) noBranch) noBranch) (IF (has B1 (OrderedAbelianMonoidSup)) (IF (has B2 (OrderedAbelianMonoidSup)) (ATTRIBUTE (OrderedAbelianMonoidSup)) noBranch) noBranch) (IF (has B1 (OrderedSet)) (IF (has B2 (OrderedSet)) (ATTRIBUTE (OrderedSet)) noBranch) noBranch) (SIGNATURE makeprod ($ B1 B2)) (SIGNATURE selectfirst (B1 $)) (SIGNATURE selectsecond (B2 $)))
Cumulative Statistics for Constructor TensorProduct
Time: 0.19 seconds
--------------non extending category----------------------
.. TensorProduct(#1,#2,#3,#4,#5) of cat
(|Join| (|TensorProductCategory| |#1| |#4| |#5|)
(|FreeModuleCat| |#1| (|Product| |#2| |#3|))
(CATEGORY |package|
(IF (|has| |#4| (|Algebra| |#1|))
(IF (|has| |#5| (|Algebra| |#1|))
(ATTRIBUTE (|Algebra| |#1|)) |noBranch|)
|noBranch|))) has no ?*? : (Product(#2,#3),#1) -> %
finalizing NRLIB TENSORD
Processing TensorProduct for Browser database:
--->-->TensorProduct(): Missing Description
------------------------------------------------------------------------
TensorProduct is now explicitly exposed in frame initial
TensorProduct will be automatically loaded when needed from
/var/zope2/var/LatexWiki/TENSORD.NRLIB/codeaxiom
M:=FreeModule1(Integer,Symbol)
 | (1) |
Type: Domain
axiom
N:=FreeModule1(Integer,Symbol)
 | (2) |
Type: Domain
axiom
a1:='a1::M
 | (3) |
Type: FreeModule1?(Integer,Symbol)
axiom
a2:='a2::M
 | (4) |
Type: FreeModule1?(Integer,Symbol)
axiom
b1:='b1::N
 | (5) |
Type: FreeModule1?(Integer,Symbol)
axiom
b2:='b2::N
 | (6) |
Type: FreeModule1?(Integer,Symbol)
axiom
MxN:=TensorProduct(Integer,Symbol,Symbol,M,N);
Type: Domain
axiom
t:=product(a1+a2,b1+b2)$MxN;
Type: TensorProduct?(Integer,Symbol,Symbol,FreeModule1?(Integer,Symbol),FreeModule1?(Integer,Symbol))
axiom
t
 | (7) |
Type: TensorProduct?(Integer,Symbol,Symbol,FreeModule1?(Integer,Symbol),FreeModule1?(Integer,Symbol))
axiom
leadingMonomial t
 | (8) |
Type: Product(Symbol,Symbol)
axiom
numberOfMonomials t
 | (9) |
Type: PositiveInteger?
Demonstrating the axioms of the tensor product:
axiom
x:M
Type: Void
axiom
y:M
Type: Void
axiom
u:M
Type: Void
axiom
p:=2*x+3*u
 | (10) |
Type: FreeModule1?(Integer,Symbol)
axiom
q:=5*x+7*y+11*u
 | (11) |
Type: FreeModule1?(Integer,Symbol)
axiom
MxM:=TensorProduct(Integer,Symbol,Symbol,M,M);
Type: Domain
axiom
r:=product(p,q)$MxM
 | (12) |
Type: TensorProduct?(Integer,Symbol,Symbol,FreeModule1?(Integer,Symbol),FreeModule1?(Integer,Symbol))
axiom
w:= 13*y+17*y+19*u
 | (13) |
Type: FreeModule1?(Integer,Symbol)
axiom
test( product(p+q,w)$MxM = product(p,w)$MxM + product(q,w)$MxM )
 | (14) |
Type: Boolean
axiom
test( product(p,q+w)$MxM = product(p,q)$MxM + product(p,w)$MxM )
 | (15) |
Type: Boolean
axiom
test( product(p,23*w)$MxM = 23*product(p,w)$MxM )
 | (16) |
Type: Boolean
axiom
test( product(23*p,w)$MxM = 23*product(p,w)$MxM )
 | (17) |
Type: Boolean
first example of a bialgebra --Bill Page, Wed, 13 May 2009 08:06:57 -0700 reply
SandBoxHopfAlgebra?