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SandBoxHopfAlgebra
  
last edited 11 years ago by test1

Edit detail for SandBoxHopfAlgebra revision 3 of 6
1 2 3 4 5 6
Editor: Bill Page
Time: 2009/05/13 11:13:48 GMT-7
Note: first example 
added:
q:=p+3
Delta(q^2)

On Fri, May 15, 2009 at 03:23:33AM +0200, Franz Lehner wrote:

Now it more or less works, included is a first example of a bialgebra (just too lazy to write up the antipode):

axiom
)library TENSORC TENSORP TENSORD

   TensorProductCategory is now explicitly exposed in frame initial
   TensorProductCategory will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/TENSORC.NRLIB/code
   TensorProductProperty is now explicitly exposed in frame initial
   TensorProductProperty will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/TENSORP.NRLIB/code
   TensorProduct is now explicitly exposed in frame initial
   TensorProduct will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/TENSORD.NRLIB/code

spad
)abbrev category COALG CoAlgebra
CoAlgebra(R : CommutativeRing, RR : TensorProductCategory(R, %, %)) : Category == _
        Module(R) with
    Delta : % -> RR
       ++ delta is comultiplication
    counit: % -> R
       ++ counit
)abbrev category BIALG BiAlgebra
BiAlgebra(R : CommutativeRing, RR : TensorProductCategory(R, %, %)) : Category == _
     Join(Algebra(R), CoAlgebra(R, RR))
)abbrev category HOPFALG HopfAlgebra
HopfAlgebra(R : CommutativeRing, RR : TensorProductCategory(R, %, %)) : Category _
     == BiAlgebra(R, RR) with
    S : % -> %
       ++ the antipode
)abbrev domain PHALG PolyHopfAlgebra
PolyHopfAlgebra(R: CommutativeRing,x: Symbol): C == T where
    S == Variable x
    FM == FreeMonoid Symbol
    PxP == TensorProduct(R, FM, FM, %, %) 
--    C == Join(FreeModule(R, FM),HopfAlgebra(R, PxP))
    C == Join(UnivariatePolynomialCategory(R), _
         FreeModuleCat(R, FM), BiAlgebra(R, PxP))

    TERM == Record(k: FM,c: R)

    T == SparseUnivariatePolynomial(R) add
           Rep:= SparseUnivariatePolynomial(R) 

           monom(a:FM,r:R):% == 
                one? a => monomial(r, 0@NonNegativeInteger)
                monomial(r, nthExpon(a, 1@Integer))

--           monomial(r:R, n:NonNegativeInteger) == monomial(r,n)@Rep

           coerce(p:%):OutputForm == outputForm(p,outputForm x)$Rep

           Delta1(n:NonNegativeInteger):PxP == 
                res: PxP := 0
                nn:List NonNegativeInteger := [k for k in 0..n]
                for k1 in nn for k2 in reverse nn repeat
                     res:= res + binomial(n,k1)*product(monomial(1@R,k1),monomial(1@R,k2))$PxP
                res

           ListOfTerms(p:%):List TERM ==
               res:List TERM := []
               while not zero? p repeat
                  m:FM := (variable()$S)**degree p
                  m1:TERM := [m,leadingCoefficient p]
                  res:=concat!(res, m1)
                  p:=reductum p
               res

           Delta(p:%):PxP == 
                zero? p => 0
                lt:R := leadingCoefficient p
                lt * Delta1(degree p) + Delta reductum p

           counit(p:%):R == coefficient(p,0)
spad
   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/8300278871242681918-25px002.spad using 
      old system compiler.
   COALG abbreviates category CoAlgebra 
------------------------------------------------------------------------
   initializing NRLIB COALG for CoAlgebra 
   compiling into NRLIB COALG 

;;;     ***       |CoAlgebra| REDEFINED
Time: 0 SEC.

   finalizing NRLIB COALG 
   Processing CoAlgebra for Browser database:
--------(Delta (RR %))---------
--------(counit (R %))---------
--->-->CoAlgebra(constructor): Not documented!!!!
--->-->CoAlgebra(): Missing Description
------------------------------------------------------------------------
   CoAlgebra is now explicitly exposed in frame initial 
   CoAlgebra will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/COALG.NRLIB/code

   BIALG abbreviates category BiAlgebra 
------------------------------------------------------------------------
   initializing NRLIB BIALG for BiAlgebra 
   compiling into NRLIB BIALG 

;;;     ***       |BiAlgebra| REDEFINED
Time: 0 SEC.

   finalizing NRLIB BIALG 
   Processing BiAlgebra for Browser database:
--->-->BiAlgebra(): Missing Description
------------------------------------------------------------------------
   BiAlgebra is now explicitly exposed in frame initial 
   BiAlgebra will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/BIALG.NRLIB/code

   HOPFALG abbreviates category HopfAlgebra 
------------------------------------------------------------------------
   initializing NRLIB HOPFALG for HopfAlgebra 
   compiling into NRLIB HOPFALG 

;;;     ***       |HopfAlgebra| REDEFINED
Time: 0 SEC.

   finalizing NRLIB HOPFALG 
   Processing HopfAlgebra for Browser database:
--------(S (% %))---------
--->-->HopfAlgebra(constructor): Not documented!!!!
--->-->HopfAlgebra(): Missing Description
------------------------------------------------------------------------
   HopfAlgebra is now explicitly exposed in frame initial 
   HopfAlgebra will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/HOPFALG.NRLIB/code

   PHALG abbreviates domain PolyHopfAlgebra 
------------------------------------------------------------------------
   initializing NRLIB PHALG for PolyHopfAlgebra 
   compiling into NRLIB PHALG 
   compiling exported monom : (FreeMonoid Symbol,R) -> $
Time: 0.12 SEC.

   compiling exported coerce : $ -> OutputForm
Time: 0 SEC.

   compiling local Delta1 : NonNegativeInteger -> TensorProduct(R,FreeMonoid Symbol,FreeMonoid Symbol,$,$)
Time: 0.03 SEC.

   compiling exported ListOfTerms : $ -> List Record(k: FreeMonoid Symbol,c: R)
Time: 0.04 SEC.

   compiling exported Delta : $ -> TensorProduct(R,FreeMonoid Symbol,FreeMonoid Symbol,$,$)
Time: 0.06 SEC.

   compiling exported counit : $ -> R
Time: 0.01 SEC.

augmenting (SingletonAsOrderedSet): (ConvertibleTo (InputForm))
****** Domain: R already in scope
augmenting R: (ConvertibleTo (InputForm))
augmenting (SingletonAsOrderedSet): (ConvertibleTo (Pattern (Float)))
****** Domain: R already in scope
augmenting R: (ConvertibleTo (Pattern (Float)))
augmenting (SingletonAsOrderedSet): (ConvertibleTo (Pattern (Integer)))
****** Domain: R already in scope
augmenting R: (ConvertibleTo (Pattern (Integer)))
augmenting (SingletonAsOrderedSet): (PatternMatchable (Float))
****** Domain: R already in scope
augmenting R: (PatternMatchable (Float))
augmenting (SingletonAsOrderedSet): (PatternMatchable (Integer))
****** Domain: R already in scope
augmenting R: (PatternMatchable (Integer))
****** Domain: $ already in scope
augmenting $: (CharacteristicNonZero)
****** Domain: R already in scope
augmenting R: (PolynomialFactorizationExplicit)
****** Domain: $ already in scope
augmenting $: (CharacteristicNonZero)
****** Domain: R already in scope
augmenting R: (PolynomialFactorizationExplicit)
****** Domain: R already in scope
augmenting R: (Algebra (Fraction (Integer)))
****** Domain: R already in scope
augmenting R: (Algebra (Fraction (Integer)))
****** Domain: R already in scope
augmenting R: (Field)
****** Domain: R already in scope
augmenting R: (GcdDomain)
****** Domain: R already in scope
augmenting R: (IntegralDomain)
****** Domain: R already in scope
augmenting R: (LinearlyExplicitRingOver (Integer))
****** Domain: R already in scope
augmenting R: (OrderedSet)
****** Domain: R already in scope
augmenting R: (PartialDifferentialRing (Symbol))
****** Domain: R already in scope
augmenting R: (PolynomialFactorizationExplicit)
****** Domain: R already in scope
augmenting R: (RetractableTo (Fraction (Integer)))
****** Domain: R already in scope
augmenting R: (RetractableTo (Integer))
****** Domain: R already in scope
augmenting R: (StepThrough)
(time taken in buildFunctor:  49)

;;;     ***       |PolyHopfAlgebra| REDEFINED

;;;     ***       |PolyHopfAlgebra| REDEFINED
Time: 0.60 SEC.

   Warnings: 
      [1] not known that (OrderedSet) is of mode (CATEGORY domain (SIGNATURE * ($ (Symbol) $)) (SIGNATURE * ($ $ (Symbol))) (SIGNATURE ** ($ (Symbol) (NonNegativeInteger))) (SIGNATURE hclf ($ $ $)) (SIGNATURE hcrf ($ $ $)) (SIGNATURE lquo ((Union $ failed) $ $)) (SIGNATURE rquo ((Union $ failed) $ $)) (SIGNATURE divide ((Union (Record (: lm $) (: rm $)) failed) $ $)) (SIGNATURE overlap ((Record (: lm $) (: mm $) (: rm $)) $ $)) (SIGNATURE size ((NonNegativeInteger) $)) (SIGNATURE factors ((List (Record (: gen (Symbol)) (: exp (NonNegativeInteger)))) $)) (SIGNATURE nthExpon ((NonNegativeInteger) $ (Integer))) (SIGNATURE nthFactor ((Symbol) $ (Integer))) (SIGNATURE mapExpon ($ (Mapping (NonNegativeInteger) (NonNegativeInteger)) $)) (SIGNATURE mapGen ($ (Mapping (Symbol) (Symbol)) $)) (IF (has (Symbol) (OrderedSet)) (ATTRIBUTE (OrderedSet)) noBranch))
      [2] Delta1: not known that (OrderedSet) is of mode (CATEGORY domain (SIGNATURE * ($ (Symbol) $)) (SIGNATURE * ($ $ (Symbol))) (SIGNATURE ** ($ (Symbol) (NonNegativeInteger))) (SIGNATURE hclf ($ $ $)) (SIGNATURE hcrf ($ $ $)) (SIGNATURE lquo ((Union $ failed) $ $)) (SIGNATURE rquo ((Union $ failed) $ $)) (SIGNATURE divide ((Union (Record (: lm $) (: rm $)) failed) $ $)) (SIGNATURE overlap ((Record (: lm $) (: mm $) (: rm $)) $ $)) (SIGNATURE size ((NonNegativeInteger) $)) (SIGNATURE factors ((List (Record (: gen (Symbol)) (: exp (NonNegativeInteger)))) $)) (SIGNATURE nthExpon ((NonNegativeInteger) $ (Integer))) (SIGNATURE nthFactor ((Symbol) $ (Integer))) (SIGNATURE mapExpon ($ (Mapping (NonNegativeInteger) (NonNegativeInteger)) $)) (SIGNATURE mapGen ($ (Mapping (Symbol) (Symbol)) $)) (IF (has (Symbol) (OrderedSet)) (ATTRIBUTE (OrderedSet)) noBranch))
      [3] Delta: not known that (OrderedSet) is of mode (CATEGORY domain (SIGNATURE * ($ (Symbol) $)) (SIGNATURE * ($ $ (Symbol))) (SIGNATURE ** ($ (Symbol) (NonNegativeInteger))) (SIGNATURE hclf ($ $ $)) (SIGNATURE hcrf ($ $ $)) (SIGNATURE lquo ((Union $ failed) $ $)) (SIGNATURE rquo ((Union $ failed) $ $)) (SIGNATURE divide ((Union (Record (: lm $) (: rm $)) failed) $ $)) (SIGNATURE overlap ((Record (: lm $) (: mm $) (: rm $)) $ $)) (SIGNATURE size ((NonNegativeInteger) $)) (SIGNATURE factors ((List (Record (: gen (Symbol)) (: exp (NonNegativeInteger)))) $)) (SIGNATURE nthExpon ((NonNegativeInteger) $ (Integer))) (SIGNATURE nthFactor ((Symbol) $ (Integer))) (SIGNATURE mapExpon ($ (Mapping (NonNegativeInteger) (NonNegativeInteger)) $)) (SIGNATURE mapGen ($ (Mapping (Symbol) (Symbol)) $)) (IF (has (Symbol) (OrderedSet)) (ATTRIBUTE (OrderedSet)) noBranch))

   Cumulative Statistics for Constructor PolyHopfAlgebra
      Time: 0.86 seconds

--------------non extending category----------------------
.. PolyHopfAlgebra(#1,#2) of cat 
(|Join| (|UnivariatePolynomialCategory| |#1|)
        (|FreeModuleCat| |#1| (|FreeMonoid| (|Symbol|)))
        (|BiAlgebra| |#1|
            (|TensorProduct| |#1| (|FreeMonoid| (|Symbol|))
                (|FreeMonoid| (|Symbol|)) $ $)))    has no  outputForm : (%,OutputForm) -> OutputForm 
   finalizing NRLIB PHALG 
   Processing PolyHopfAlgebra for Browser database:
--->-->PolyHopfAlgebra(): Missing Description
------------------------------------------------------------------------
   PolyHopfAlgebra is now explicitly exposed in frame initial 
   PolyHopfAlgebra will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/PHALG.NRLIB/code

axiom
P:=PolyHopfAlgebra(Integer,'u)
LatexWiki Image(1)
Type: Domain
axiom
p:=monomial(1,1)$P
LatexWiki Image(2)
Type: PolyHopfAlgebra?(Integer,u)
axiom
Delta p
LatexWiki Image(3)
Type: TensorProduct?(Integer,FreeMonoid? Symbol,FreeMonoid? Symbol,PolyHopfAlgebra?(Integer,u),PolyHopfAlgebra?(Integer,u))
axiom
Delta(p^2)
LatexWiki Image(4)
Type: TensorProduct?(Integer,FreeMonoid? Symbol,FreeMonoid? Symbol,PolyHopfAlgebra?(Integer,u),PolyHopfAlgebra?(Integer,u))
axiom
Delta(p)^2
LatexWiki Image(5)
Type: TensorProduct?(Integer,FreeMonoid? Symbol,FreeMonoid? Symbol,PolyHopfAlgebra?(Integer,u),PolyHopfAlgebra?(Integer,u))
axiom
q:=p+3
LatexWiki Image(6)
Type: PolyHopfAlgebra?(Integer,u)
axiom
Delta(q^2)
LatexWiki Image(7)
Type: TensorProduct?(Integer,FreeMonoid? Symbol,FreeMonoid? Symbol,PolyHopfAlgebra?(Integer,u),PolyHopfAlgebra?(Integer,u))

Please comment.

Next comes the old problem come with iterated structures. Is it in principle possible to have: 

  TensorProduct(R:CommutativeRing,LB:List OrderedSet, LM:List FreeModuleCat(R))?

Since the coproduct is coassociative, we might want to define "powers", but I realize that tensor powers are actually feasible.

Actually the basis is not redundant; in the attached example the implementation of the Coalgebra does not use the basis which it pretends to the tensor product. Still exporting a basis would be a nice thing.

More questions:

  1. How far are we from having unicode in output?
  2. CoAlgebra? or Coalgebra?
  3. Delta or coproduct? epsilon or counit? S or antipode? or both?
  4. I do not see which domains should use TensorProductProperty?, the target module S is not part of the domain. Anyways morphisms have to be implemented in packages ("TensorProductFunctions2?" or something like that)
  5. For now TensorProduct? relies on FreeModuleCat?. To make use of the existing applicable domains we have to add FreeModuleCat? everywhere. Extend would be good ... First of course comes a cleanup of FreeModule? itself. The example I cooked up in the attachment is rather ugly and fragile.
  6. The compiler does not warn about missing exports. (this is a really nice feature of the aldor compiler) I can add all categories I like, they pass the compiler and )sh domain shows all the operations exported by the categories. The only way to detect missing ones is by runtime errors.
  7. ceterum censeo Aldorem esse liberandam (ok, that suffices for now)