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## 9.42 LieExponentials

a: Symbol := 'a

a

Type: Symbol
b: Symbol := 'b

b

Type: Symbol

Declarations of domains

coef := Fraction(Integer)

FractionInteger

Type: Domain
group := LieExponentials(Symbol, coef, 3)

LieExponentials(Symbol,FractionInteger,3)

Type: Domain
lpoly := LiePolynomial(Symbol, coef)

LiePolynomial(Symbol,FractionInteger)

Type: Domain
poly := XPBWPolynomial(Symbol, coef)

XPBWPolynomial(Symbol,FractionInteger)

Type: Domain

Calculations

ea := exp(a::lpoly)$group

e[a]

Type: LieExponentials(Symbol,Fraction Integer,3)
eb := exp(b::lpoly)$group

e[b]

Type: LieExponentials(Symbol,Fraction Integer,3)
g: group := ea*eb

e[b]e(1 2[a b 2])e[a b]e(1 2[a 2 b])e[a]

Type: LieExponentials(Symbol,Fraction Integer,3)
g :: poly

1+[a]+[b]+1 2[a][a]+[a b]+[b][a]+1 2[b][b]+1 6[a][a][a]+1 2[a 2 b]+[a b][a]+1 2[a b 2]+1 2[b][a][a]+[b][a b]+1 2[b][b][a]+1 6[b][b][b]

Type: XPBWPolynomial(Symbol,Fraction Integer)
log(g)$group

[a]+[b]+1 2[a b]+1 12[a 2 b]+1 12[a b 2]

Type: LiePolynomial(Symbol,Fraction Integer)
g1: group := inv(g)

e(-[b])e(-[a])

Type: LieExponentials(Symbol,Fraction Integer,3)
g*g1

1

Type: LieExponentials(Symbol,Fraction Integer,3)
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