While you cannot change the standard sending address of firstname.lastname@example.org. you are able to change the display name shown to email recipients as well as the email address that any replies are sent to.
To change this, go to Administration > Email Settings. On this form you can set the name shown on alert emails, as well as change the reply address and reply name. This is useful if you want employees to be able to respond via email to alerts they have received.
Once you have made the changes, click on the "Save Settings" button to store your changes.