Section9.43

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## 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]

[a b]

Type: LiePolynomial(Symbol,Fraction Integer)
q : Lpoly := [p,bb]

[a b 2]

Type: LiePolynomial(Symbol,Fraction Integer)

All the Lyndon words of order 4

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

[[a],[b],[a b],[a 2 b],[a b 2],[a 3 b],[a 2 b 2],[a b 3]]

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

[a b]+3[a 2 b]+[a b 2]

Type: LiePolynomial(Symbol,Fraction Integer)
s:Lpoly := [p,r]

-3[a 2 b a b]+[a b a b 2]

Type: LiePolynomial(Symbol,Fraction Integer)
t:Lpoly := s + 2*LiePoly(liste.3) - 5*LiePoly(liste.5)

2[a b]-5[a b 2]-3[a 2 b a b]+[a b a b 2]

Type: LiePolynomial(Symbol,Fraction Integer)
degree t

5

Type: PositiveInteger
mirror t

-2[a b]-5[a b 2]-3[a 2 b a b]+[a b a b 2]

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

[a b 2]

Type: LiePolynomial(Symbol,Fraction Integer)
eval(p, [a,b], [2*bb, 3*aa])$Lpoly

-6[a b]

Type: LiePolynomial(Symbol,Fraction Integer)
r: Lpoly := [p,c]

[a b c]+[a c b]

Type: LiePolynomial(Symbol,Fraction Integer)
r1: Lpoly := eval(r, [a,b,c], [bb, cc, aa])$Lpoly

-[a b c]

Type: LiePolynomial(Symbol,Fraction Integer)
r2: Lpoly := eval(r, [a,b,c], [cc, aa, bb])$Lpoly

-[a c b]

Type: LiePolynomial(Symbol,Fraction Integer)
r + r1 + r2

0

Type: LiePolynomial(Symbol,Fraction Integer)
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