05 July 2006

Defining your own Lie Group

Today I was asked how to define your own Lie Group using my functional differential geometry software (see this post and subsequent posts). Basically, you have to provide five things:
  1. Chi : G -> R^n. This is the coordinate function on the group manifold.
  2. Chi^-1 : R^n -> G. The inverse of Chi.
  3. e \in G. The identity element.
  4. inverse: G -> G. The inverse function (on the group manifold: g -> g^-1).
  5. *: GxG -> G. The group multiplication function.
1&2 are provided through a <chart> object. Note carefully the signatures of these functions; in particular, note that only Chi and Chi^-1 deal with R^n while all others deal directly with group elements.

See lie-group-SO3.ss for an example of how this is done. It's a bit tricky, because the <chart> (which contains Chi and Chi^-1) must take and produce objects of the <lie-group-element> class, which require the <lie-group> class for one of the slots; but the <lie-group> class requires the <chart> object, so they must be recursively defined.

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