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topological transformation group (Definition)

Let $ G$ be a topological group and $ X$ any topological space. We say that $ G$ is a topological transformation group of $ X$ if $ G$ acts on $ X$ continuously, in the following sense:

  1. there is a continuous function $ \alpha:G\times X\to X$, where $ G\times X$ is given the product topology
  2. $ \alpha(1,x)=x$, and
  3. $ \alpha(g_1g_2,x)=\alpha(g_1,\alpha(g_2,x))$.

The function $ \alpha$ is called the (left) action of $ G$ on $ X$. When there is no confusion, $ \alpha(g,x)$ is simply written $ gx$, so that the two conditions above read $ 1x=x$ and $ (g_1g_2)x=g_1(g_2x)$.

If a topological transformation group $ G$ on $ X$ is effective, then $ G$ can be viewed as a group of homeomorphisms on $ X$: simply define $ h_g:X\to X$ by $ h_g(x)=gx$ for each $ g\in G$ so that $ h_g$ is the identity function precisely when $ g=1$.

Some Examples.

  1. Let $ X=\mathbb{R}^n$, and $ G$ be the group of $ n\times n$ matrices over $ \mathbb{R}$. Clearly $ X$ and $ G$ are both topological spaces with the usual topology. Furthermore, $ G$ is a topological group. $ G$ acts on $ X$ continuous if we view elements of $ X$ as column vectors and take the action to be the matrix multiplication on the left.
  2. If $ G$ is a topological group, $ G$ can be considered a topological transformation group on itself. There are many continuous actions that can be defined on $ G$. For example, $ \alpha:G\times G\to G$ given by $ \alpha(g,x)=gx$ is one such action. It is continuous, and satisfies the two action axioms. $ G$ is also effective with respect to $ \alpha$.



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Also defines:  effective topological transformation group
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Cross-references: axioms, matrix multiplication, column vectors, acts on, usual topology, matrices, identity function, homeomorphisms, group, effective, action, function, product topology, continuous function, topological space, topological group

This is version 2 of topological transformation group, born on 2007-02-23, modified 2007-02-23.
Object id is 8955, canonical name is TopologicalTransformationGroup.
Accessed 737 times total.

Classification:
AMS MSC: 22F05 (Topological groups, Lie groups :: Noncompact transformation groups :: General theory of group and pseudogroup actions)
 54H15 (General topology :: Connections with other structures, applications :: Transformation groups and semigroups)
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