MathAction SandBoxHopfAlgebra

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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):

The line below is no longer needed, since the constructors are included in FriCAS

)library TENSORC TENSORP TENSORD

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(1) -> <spad>

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)abbrev category COALG CoAlgebra
CoAlgebra(R : CommutativeRing, RR : TensorProductCategory(R, %, %)) : Category == _
        Module(R) with
    delta : % -> RR
       ++ delta is comultiplication
    counit: % -> R
       ++ counit

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)abbrev category BIALG BiAlgebra
BiAlgebra(R : CommutativeRing, RR : TensorProductCategory(R, %, %)) : Category == _
     Join(Algebra(R), CoAlgebra(R, RR))

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)abbrev category HOPFALG HopfAlgebra
HopfAlgebra(R : CommutativeRing, RR : TensorProductCategory(R, %, %)) : Category _
     == BiAlgebra(R, RR) with
    S : % -> %
       ++ the antipode

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)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), _
         FreeModuleCategory(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)*tensor(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>

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Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/1189790658020391427-25px001.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:
--->-->CoAlgebra(constructor): Not documented!!!!
--------(delta (RR %))---------
--------(counit (R %))---------
--->-->CoAlgebra(): Missing Description
; compiling file "/var/aw/var/LatexWiki/COALG.NRLIB/COALG.lsp" (written 16 DEC 2024 10:18:41 PM):

; wrote /var/aw/var/LatexWiki/COALG.NRLIB/COALG.fasl
; compilation finished in 0:00:00.004
------------------------------------------------------------------------
   CoAlgebra is now explicitly exposed in frame initial 
   CoAlgebra will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/COALG.NRLIB/COALG

   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
; compiling file "/var/aw/var/LatexWiki/BIALG.NRLIB/BIALG.lsp" (written 16 DEC 2024 10:18:41 PM):

; wrote /var/aw/var/LatexWiki/BIALG.NRLIB/BIALG.fasl
; compilation finished in 0:00:00.004
------------------------------------------------------------------------
   BiAlgebra is now explicitly exposed in frame initial 
   BiAlgebra will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/BIALG.NRLIB/BIALG

   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:
--->/usr/local/lib/fricas/target/x86_64-linux-gnu/../../src/algebra/HOPFALG.spad-->HopfAlgebra(constructor): Not documented!!!!
--------(S (% %))---------
--->/usr/local/lib/fricas/target/x86_64-linux-gnu/../../src/algebra/HOPFALG.spad-->HopfAlgebra(): Missing Description
; compiling file "/var/aw/var/LatexWiki/HOPFALG.NRLIB/HOPFALG.lsp" (written 16 DEC 2024 10:18:41 PM):

; wrote /var/aw/var/LatexWiki/HOPFALG.NRLIB/HOPFALG.fasl
; compilation finished in 0:00:00.000
------------------------------------------------------------------------
   HopfAlgebra is now explicitly exposed in frame initial 
   HopfAlgebra will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/HOPFALG.NRLIB/HOPFALG

   PHALG abbreviates domain PolyHopfAlgebra 
------------------------------------------------------------------------
   initializing NRLIB PHALG for PolyHopfAlgebra 
   compiling into NRLIB PHALG 
   compiling local monom : (FreeMonoid Symbol,R) -> %
Time: 0.03 SEC.

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

   compiling local delta1 : NonNegativeInteger -> TensorProduct(R,FreeMonoid Symbol,FreeMonoid Symbol,%,%)
Time: 0.01 SEC.

   compiling exported listOfTerms : % -> List Record(k: FreeMonoid Symbol,c: R)
Time: 0 SEC.

   compiling exported delta : % -> TensorProduct(R,FreeMonoid Symbol,FreeMonoid Symbol,%,%)
Time: 0 SEC.

   compiling exported counit : % -> R
Time: 0 SEC.

****** Domain: % already in scope
augmenting %: (CharacteristicNonZero)
****** Domain: R already in scope
augmenting R: (PolynomialFactorizationExplicit)
****** Domain: R already in scope
augmenting R: (ConvertibleTo (InputForm))
****** Domain: (SingletonAsOrderedSet) already in scope
augmenting (SingletonAsOrderedSet): (ConvertibleTo (InputForm))
****** Domain: R already in scope
augmenting R: (ConvertibleTo (Pattern (Float)))
****** Domain: (SingletonAsOrderedSet) already in scope
augmenting (SingletonAsOrderedSet): (ConvertibleTo (Pattern (Float)))
****** Domain: R already in scope
augmenting R: (ConvertibleTo (Pattern (Integer)))
****** Domain: (SingletonAsOrderedSet) already in scope
augmenting (SingletonAsOrderedSet): (ConvertibleTo (Pattern (Integer)))
****** Domain: R already in scope
augmenting R: (Hashable)
****** Domain: R already in scope
augmenting R: (LinearlyExplicitOver (Integer))
****** Domain: R already in scope
augmenting R: (PartialDifferentialRing (Symbol))
****** Domain: R already in scope
augmenting R: (PatternMatchable (Float))
****** Domain: (SingletonAsOrderedSet) already in scope
augmenting (SingletonAsOrderedSet): (PatternMatchable (Float))
****** Domain: R already in scope
augmenting R: (PatternMatchable (Integer))
****** Domain: (SingletonAsOrderedSet) already in scope
augmenting (SingletonAsOrderedSet): (PatternMatchable (Integer))
****** Domain: R already in scope
augmenting R: (Algebra (Fraction (Integer)))
****** Domain: R already in scope
augmenting R: (Comparable)
****** Domain: R already in scope
augmenting R: (EntireRing)
****** 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: (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:  77240)

;;;     ***       |PolyHopfAlgebra| REDEFINED

;;;     ***       |PolyHopfAlgebra| REDEFINED
Time: 0.10 SEC.

   Warnings: 
      [1] not known that (OrderedSet) is of mode (CATEGORY domain (SIGNATURE * (% (Symbol) %)) (SIGNATURE * (% % (Symbol))) (SIGNATURE ^ (% (Symbol) (NonNegativeInteger))) (SIGNATURE first ((Symbol) %)) (SIGNATURE rest (% %)) (SIGNATURE mirror (% %)) (SIGNATURE hclf (% % %)) (SIGNATURE hcrf (% % %)) (SIGNATURE lquo ((Union % failed) % %)) (SIGNATURE rquo ((Union % failed) % %)) (SIGNATURE lquo ((Union % failed) % (Symbol))) (SIGNATURE rquo ((Union % failed) % (Symbol))) (SIGNATURE divide ((Union (Record (: lm %) (: rm %)) failed) % %)) (SIGNATURE overlap ((Record (: lm %) (: mm %) (: rm %)) % %)) (SIGNATURE size ((NonNegativeInteger) %)) (SIGNATURE length ((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) (BasicType)) (SIGNATURE varList ((List (Symbol)) %)) noBranch) (IF (has (Symbol) (Comparable)) (ATTRIBUTE (Comparable)) noBranch) (IF (has (Symbol) (OrderedSet)) (PROGN (ATTRIBUTE (OrderedMonoid)) (SIGNATURE lexico ((Boolean) % %))) noBranch))
      [2] delta1: not known that (OrderedSet) is of mode (CATEGORY domain (SIGNATURE * (% (Symbol) %)) (SIGNATURE * (% % (Symbol))) (SIGNATURE ^ (% (Symbol) (NonNegativeInteger))) (SIGNATURE first ((Symbol) %)) (SIGNATURE rest (% %)) (SIGNATURE mirror (% %)) (SIGNATURE hclf (% % %)) (SIGNATURE hcrf (% % %)) (SIGNATURE lquo ((Union % failed) % %)) (SIGNATURE rquo ((Union % failed) % %)) (SIGNATURE lquo ((Union % failed) % (Symbol))) (SIGNATURE rquo ((Union % failed) % (Symbol))) (SIGNATURE divide ((Union (Record (: lm %) (: rm %)) failed) % %)) (SIGNATURE overlap ((Record (: lm %) (: mm %) (: rm %)) % %)) (SIGNATURE size ((NonNegativeInteger) %)) (SIGNATURE length ((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) (BasicType)) (SIGNATURE varList ((List (Symbol)) %)) noBranch) (IF (has (Symbol) (Comparable)) (ATTRIBUTE (Comparable)) noBranch) (IF (has (Symbol) (OrderedSet)) (PROGN (ATTRIBUTE (OrderedMonoid)) (SIGNATURE lexico ((Boolean) % %))) noBranch))

   Cumulative Statistics for Constructor PolyHopfAlgebra
      Time: 0.16 seconds

--------------non extending category----------------------
.. PolyHopfAlgebra(#1,#2) of cat 
(|Join| (|UnivariatePolynomialCategory| |#1|)
        (|FreeModuleCategory| |#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
; compiling file "/var/aw/var/LatexWiki/PHALG.NRLIB/PHALG.lsp" (written 16 DEC 2024 10:18:41 PM):

; wrote /var/aw/var/LatexWiki/PHALG.NRLIB/PHALG.fasl
; compilation finished in 0:00:00.028
------------------------------------------------------------------------
   PolyHopfAlgebra is now explicitly exposed in frame initial 
   PolyHopfAlgebra will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/PHALG.NRLIB/PHALG

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P:=PolyHopfAlgebra(Integer,'u)

$\label{eq1}\hbox{\axiomType{PolyHopfAlgebra}\ } \left({\hbox{\axiomType{Integer}\ } , \: u}\right)$

(1)

Type: Type

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p:=monomial(1,1)$P

$\label{eq2}u$

(2)

Type: PolyHopfAlgebra?(Integer,u)

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delta p

$\label{eq3}{u \otimes 1}+{1 \otimes u}$

(3)

Type: TensorProduct?(Integer,FreeMonoid(Symbol),FreeMonoid(Symbol),PolyHopfAlgebra?(Integer,u),PolyHopfAlgebra?(Integer,u))

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delta(p^2)

$\label{eq4}{{{u}^{2}}\otimes 1}+{2 \ {u \otimes u}}+{1 \otimes{{u}^{2}}}$

(4)

Type: TensorProduct?(Integer,FreeMonoid(Symbol),FreeMonoid(Symbol),PolyHopfAlgebra?(Integer,u),PolyHopfAlgebra?(Integer,u))

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delta(p)^2

Function:  monomial : (Integer, FreeMonoid(Symbol)) -> % is missing from domain: PolyHopfAlgebra(Integer,u)
   Internal Error
   The function monomial with signature (Integer)(FreeMonoid (Symbol)) is 
      missing from domain 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 FreeModuleCategory(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.

Subject: Be Bold !!

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