Yes you can. Tweetdeck will leave an icon in the system tray (Windows) or the dock (mac) at all times.

You can control what happens when you minimise or close the TweetDeck window by using the options under Settings > General

• Minimising the widow can either "Hide to tray" or 'Minimise to taskbar"
• Closing the window can either "Hide to tray" or "Close the application"

If you choose to hide TweetDeck to the tray, or minimise to the taskbar, it will continue to run and will keep receiving updates and notifying you.

If you have opted for closing the window to hide TweetDeck to the tray, in order to close down TweetDeck completely, you must right-click on the icon in the tray/dock and select Exit.