Google Wave in it's current release version appears to support two types of extensions:

• Robots: Apps that run outside the wave, and are contacted by the wave (mono-directional).
• Gadgets: Apps that live inside the wave and have shared state between wave participants.

For robots the extension libraries are predictably in Python and Java and currently only use the Google App Engine as their application server.

The Gadgets on the other hand are written in HTML, CSS and Javascript, are subject to a public display condition and work primarily on the "wave" object (shared state).

[personal note]
This offers an interesting set of tools for a developer to create 
extensions, which are claimed to be first class citizens within the
wave.  It's reassuring that there is mention of how development has 
not kept to the standard API's but is mostly compatible with them.
This may either mean a forking away from standards in the future, 
or maybe an attempt to get the standards to follow "Wave's" lead?
[/personal note]

A Google Wave robot implementation exists as a Drupal module, containing the 'drubot', and an as yet incomplete sub-module to embed a wave in a sight as a node(?).

There is a Wave plugin for Joomla, but it appears to be a barely functioning stub with no documentation as of yet.

The is a working wave plugin for Wordpress called "Wavr", that easily embeds waves with an entry using the wave-id #.

[personal note]
Notice the difference in plugin development between Wordpress and 
Drupal?  In Drupal the robot (server) was finished first, and in 
Wordpress the embedding(client) functionality was finished first.  I'm 
not trying to say one is better than the other, just pointing out the 
explicit example of how they are both CMS's, but different tools for 
different jobs.  Which is an interesting expression of the developers in 
both communities and how they are focused as much as it is on the tools 
they create.
[/personal note]

References:
http://code.google.com/apis/wave/extensions/gadgets/guide.html
http://code.google.com/apis/wave/extensions/robots/
http://mashable.com/2009/09/08/google-wave-wordpress-plugin/
http://drupal.org/project/wave
http://extensions.joomla.org/extensions/external-contents/social-channel...