9.43 LiePolynomial

Declaration of domains

RN := Fraction Integer

\[\]

FractionInteger

_{Type: Domain}

Lpoly := LiePolynomial(Symbol,RN)

\[\]

LiePolynomial(Symbol,FractionInteger)

_{Type: Domain}

Dpoly := XDPOLY(Symbol,RN)

\[\]

XDistributedPolynomial(Symbol,FractionInteger)

_{Type: Domain}

Lword := LyndonWord Symbol

\[\]

LyndonWordSymbol

_{Type: Domain}

Initialisation

a:Symbol := 'a

\[\]

a

_{Type: Symbol}

b:Symbol := 'b

\[\]

b

_{Type: Symbol}

c:Symbol := 'c

\[\]

c

_{Type: Symbol}

aa: Lpoly := a

\[\]

[a]

_{Type: LiePolynomial(Symbol,Fraction Integer)}

bb: Lpoly := b

\[\]

[b]

_{Type: LiePolynomial(Symbol,Fraction Integer)}

cc: Lpoly := c

\[\]

[c]

_{Type: LiePolynomial(Symbol,Fraction Integer)}

p : Lpoly := [aa,bb]

\[\]

[ab]

_{Type: LiePolynomial(Symbol,Fraction Integer)}

q : Lpoly := [p,bb]

\[\]

[ab2]

_{Type: LiePolynomial(Symbol,Fraction Integer)}

All the Lyndon words of order 4

liste : List Lword := LyndonWordsList([a,b], 4)

\[\]

[[a],[b],[ab],[a2b],[ab2],[a3b],[a2b2],[ab3]]

_{Type: List LyndonWord Symbol}

r: Lpoly := p + q + 3*LiePoly(liste.4)$Lpoly

\[\]

[ab]+3[a2b]+[ab2]

_{Type: LiePolynomial(Symbol,Fraction Integer)}

s:Lpoly := [p,r]

\[\]

-3[a2bab]+[abab2]

_{Type: LiePolynomial(Symbol,Fraction Integer)}

t:Lpoly := s + 2*LiePoly(liste.3) - 5*LiePoly(liste.5)

\[\]

2[ab]-5[ab2]-3[a2bab]+[abab2]

_{Type: LiePolynomial(Symbol,Fraction Integer)}

degree t

\[\]

5

_{Type: PositiveInteger}

mirror t

\[\]

-2[ab]-5[ab2]-3[a2bab]+[abab2]

_{Type: LiePolynomial(Symbol,Fraction Integer)}

Jacobi Relation

Jacobi(p: Lpoly, q: Lpoly, r: Lpoly): Lpoly == [ [p,q]$Lpoly, r] + [

[q,r]$Lpoly, p] + [ [r,p]$Lpoly, q]

Function declaration Jacobi : (
  LiePolynomial(Symbol,  Fraction Integer),
  LiePolynomial(Symbol,Fraction Integer),
  LiePolynomial(Symbol,Fraction Integer)) ->
    LiePolynomial(Symbol,Fraction Integer)
  has been added to workspace.

Void

Tests

test: Lpoly := Jacobi(a,b,b)

\[\]

0

_{Type: LiePolynomial(Symbol,Fraction Integer)}

test: Lpoly := Jacobi(p,q,r)

\[\]

0

_{Type: LiePolynomial(Symbol,Fraction Integer)}

test: Lpoly := Jacobi(r,s,t)

\[\]

0

_{Type: LiePolynomial(Symbol,Fraction Integer)}

Evaluation

eval(p, a, p)$Lpoly

\[\]

[ab2]

_{Type: LiePolynomial(Symbol,Fraction Integer)}

eval(p, [a,b], [2*bb, 3*aa])$Lpoly

\[\]

-6[ab]

_{Type: LiePolynomial(Symbol,Fraction Integer)}

r: Lpoly := [p,c]

\[\]

[abc]+[acb]

_{Type: LiePolynomial(Symbol,Fraction Integer)}

r1: Lpoly := eval(r, [a,b,c], [bb, cc, aa])$Lpoly

\[\]

-[abc]

_{Type: LiePolynomial(Symbol,Fraction Integer)}

r2: Lpoly := eval(r, [a,b,c], [cc, aa, bb])$Lpoly

\[\]

-[acb]

_{Type: LiePolynomial(Symbol,Fraction Integer)}

r + r1 + r2

\[\]

0

_{Type: LiePolynomial(Symbol,Fraction Integer)}