TweetDeck runs on top of Adobe AIR, but unfortunately, Adobe AIR is currently only released for 32-bit Fedora platforms. Fortunately, it is possible to install the 32-bit version of AIR if the required 32-bit libraries and packages are installed.
I have gathered information from an Adobe KnowledgeBase article and updated it to work for Fedora 13.
Install the following 32-bit libraries using yum. Note this will most likely install a number of dependencies.
$ sudo yum install ld-linux.so.2 gtk2-devel.i686 libdbus-glib-1.so.2 libdbus-glib-1.so.2 libhal.so.1
$ sudo yum install rpm-devel.i686 libXt.so.6 gnome-keyring-devel.i686 libDCOP.so.4
According to Adobe, a conflict occurs with libnss and libxml2 as the 64-bit versions are installed by default; to resolve, install the 32-bit developer libraries.
$ sudo yum install libxml2-devel.i586 nss-devel.i586
Additionally, install the 32-bit libxslt. In their instructions, it was requested to install version 1.1.26, which is the current version as of this article.
$ sudo yum install libxslt.i686
Install Adobe AIR 2 using yum.
$ sudo yum install adobeair
Restart your browser, if it is running. Install TweetDeck from the desktop installation page. I had to press Shift and hit Refresh to get it to work. Press the Download now button. If it works, an Application Install dialog should open with the option to Open. Click the Open button and let it install. It automatically starts TweetDeck but TweetDeck does not seem to work. I had success by closing the window and starting TweetDeck again.