Definition
A transformation group is a triple \((X,T, \pi)\) where
- \(X\) is a topological space,
- \(T\) is a topological-group, and
- \(\pi\) is a continuous map of \(X \times T \to X\).
\((X,T, \pi)\) satisfy the following:
- \(\pi(x,e) = x ~ (x \in X, e \text{ the identity of } T)\) and
- \(\pi(\pi(x,s),t) = \pi(x, st) ~ (x \in X, s,t \in T)\).
Synonyms:
- transformation group ↔ flow
- topological space ↔ phase space
- topological group ↔ phase group or acting group
Typical assumptions
- The phase space \(X\) is Hausdorff.
- Often assume the topology on \(T\) is discrete (because the topology is mostly irrelevant).
- Typically assume that \(X\) is compact.
Maps
\(\pi^t\)
Each \(t \in T\) defines a continuous map \(\pi^t\) of \(X \to X\) by \[ \pi^t(x) = \pi(x,t). \] If \(t,s \in T\), it is immediate that \(\pi^s \pi^t = \pi^{st}\); in particular, \(\pi^{t^{-1}}\pi^t = \pi^e\), the identity map of \(X\), so each \(\pi^t\) is a homeomorphism of \(X\) onto itself, with \(\pi^{t^{-1}} = (\pi^t)^{-1}\).
Axioms
- \((x,t) \mapsto xt\) is continuous
- \(xe = x\)
- \((xs)t = x(st)\)
Extensions
Let \((X,T)\) and \((Y,T)\) be flows with the same acting group. A homomorphism from \(X\) to \(Y\) is a continuous map \(\pi:X \to Y\) such that \(\pi(xt)=\pi(x)t\) for all \(x \in X, t \in T\).
If there is a homomorphism \(\pi\) from \(X\) onto \(Y\), we say that \(T\) is a factor of \(X\) and \(X\) is an extension of \(Y\).
Notation
- flows will be used over transformation group
- \(\pi(x,y)\) will be written as \(xt\)
- Regard \(t \in T\) as “acting on \(X\)”.
- Usually write \((X,T)\) or even just \(X\) if the group \(T\) is understood.
- An element \(t \in T\) will be identified with the homeomorphism of \(X\) it defines (\(\pi^t\)).
Notes
The topology of the group is really not that important - it is the action of the group of homeomorphisms that is interesting.