Can I change the transformation manipulators with hotkeys?

By default, Blender does not. However, a hotkey can be created for just about any function in Blender. I can’t give an exact step by step procedure for you because I don’t have access to my computer at the moment. You can check out this video about Creating Custom Hotkeys in Blender and try to follow that though 🙂

If you can’t figure it out from there though, I’d be more than happy to give you a step by step guide in a few days once I’m back at my computer!