Maths - Group Programs - Martin Baker

Maths - Program

There are a number of open source programs that can work with groups. I have used Axiom, how to install Axiom here.

I have put user input in red:

(1) -> a: Symbol := 'a
             (1)  a
             Type: Symbol

(2) -> b: Symbol := 'b

 (2)  b
             Type: Symbol

(3) -> coef := Fraction(Integer)

 (3)  Fraction Integer
             Type: Domain

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

(5) -> lpoly := LiePolynomial(Symbol, coef)

 (5)  LiePolynomial(Symbol,Fraction Integer)
             Type: Domain

(6) -> poly := XPBWPolynomial(Symbol, coef)

 (6)  XPBWPolynomial(Symbol,Fraction Integer)
             Type: Domain

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

(8) -> eb := exp(b::lpoly)$group

         [b]
   (8)  e
             Type: LieExponentials(Symbol,Fraction Integer,3)

(9) -> g: group := ea*eb

             1     2        1   2
             - [a b ]       - [a b]
         [b] 2        [a b] 2       [a]
   (9)  e   e        e     e       e
             Type: LieExponentials(Symbol,Fraction Integer,3)

(10) -> g :: poly

 (10)
                    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)

(11) -> log(g)$group

                   1          1   2      1     2
 (11)  [a] + [b] + - [a b] + -- [a b] + -- [a b ]
                   2         12         12
             Type: LiePolynomial(Symbol,Fraction Integer)

(12) -> g1: group := inv(g)

          - [b] - [a]
     (12)  e     e
             Type: LieExponentials(Symbol,Fraction Integer,3)

(13) -> g*g1

 (13)  1
             Type: LieExponentials(Symbol,Fraction Integer,3)

EuclideanSpace

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