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SandBoxHopfAlgebra

Edit detail for SandBoxHopfAlgebra revision 6 of 6

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Editor: test1 Time: 2013/04/25 20:16:57 GMT+0
Note:

changed:
-\begin{axiom}

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


changed:
-\end{axiom}


changed:
-                     res:= res + binomial(n,k1)*product(monomial(1@R,k1),monomial(1@R,k2))$PxP
                     res:= res + binomial(n,k1)*tensor(monomial(1@R,k1),monomial(1@R,k2))$PxP

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

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), _
         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

   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 04 APR 2022 07:18:32 PM):

; /var/aw/var/LatexWiki/COALG.NRLIB/COALG.fasl written
; compilation finished in 0:00:00.005
------------------------------------------------------------------------
   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 04 APR 2022 07:18:32 PM):

; /var/aw/var/LatexWiki/BIALG.NRLIB/BIALG.fasl written
; compilation finished in 0:00:00.003
------------------------------------------------------------------------
   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 04 APR 2022 07:18:32 PM):

; /var/aw/var/LatexWiki/HOPFALG.NRLIB/HOPFALG.fasl written
; compilation finished in 0:00:00.003
------------------------------------------------------------------------
   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: 0.08 SEC.

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

   compiling local delta1 : NonNegativeInteger -> TensorProduct(R,FreeMonoid Symbol,FreeMonoid Symbol,$,$)
Time: 0.04 SEC.

   compiling exported listOfTerms : $ -> List Record(k: FreeMonoid Symbol,c: R)
Time: 0.01 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: (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:  200)

;;;     ***       |PolyHopfAlgebra| REDEFINED

;;;     ***       |PolyHopfAlgebra| REDEFINED
Time: 0.24 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: 0.37 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 04 APR 2022 07:18:32 PM):

; /var/aw/var/LatexWiki/PHALG.NRLIB/PHALG.fasl written
; compilation finished in 0:00:00.053
------------------------------------------------------------------------
   PolyHopfAlgebra is now explicitly exposed in frame initial 
   PolyHopfAlgebra will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/PHALG.NRLIB/PHALG

fricas

P:=PolyHopfAlgebra(Integer,'u)

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

(1)

Type: Type

fricas

p:=monomial(1,1)$P

$\label{eq2}u$

(2)

Type: PolyHopfAlgebra?(Integer,u)

fricas

delta p

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

(3)

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

fricas

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))

fricas

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.