Lie Exponentials

a: Symbol := 'a
  a
                       Type: Symbol

b: Symbol := 'b
  b
                       Type: Symbol

Declarations of domains

coef := Fraction(Integer)
  Fraction Integer
                       Type: Domain

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

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

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

Calculations

ea := exp(a::lpoly)$group
   [a]
  e
                       Type: LieExponentials(Symbol,Fraction Integer,3)

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

g: group := ea*eb
       1     2        1   2
       - [a b ]       - [a b]
   [b] 2        [a b] 2       [a]
  e   e        e     e       e
                       Type: LieExponentials(Symbol,Fraction Integer,3)

g :: poly
                   1                           1          1
   1 + [a] + [b] + - [a][a] + [a b] + [b][a] + - [b][b] + - [a][a][a]
                   2                           2          6
 +
   1   2                1     2    1                        1
   - [a b] + [a b][a] + - [a b ] + - [b][a][a] + [b][a b] + - [b][b][a]
   2                    2          2                        2
 +
   1
   - [b][b][b]
   6
                      Type: XPBWPolynomial(Symbol,Fraction Integer)

log(g)$group
              1          1   2      1     2
  [a] + [b] + - [a b] + -- [a b] + -- [a b ]
              2         12         12
                      Type: LiePolynomial(Symbol,Fraction Integer)

g1: group := inv(g)
   - [b] - [a]
  e     e
                      Type: LieExponentials(Symbol,Fraction Integer,3)

g*g1
  1
                      Type: LieExponentials(Symbol,Fraction Integer,3)

See Also:

• )show LieExponentials