We don’t really use manipulators dedicated to rotating, scaling, grabbing etc. in Blender3D (I guess that’s why they don’t have hotkeys), though hotkeys S (scale), R (rotate) and G (translate) is the way to go when using Blender.
Apart from the dedicated manipulator modes that can be clicked in the bottom bar, you could make a hotkey script quite easily in python to let you access these modes using hotkeys.
Also double-clicking the R and G hotkeys brings you into different rotate modes and translate (edge slide) modes.