Just got an ultraviolet LED in a 10 mm epoxy package that’s water-clear in visible light and slightly fluorescent in its own UV:

The epoxy usually has some fluorescence, but this seems more dramatic than usual. In any event, the die’s wide beam angle shows clearly; the beam along the axis out in front is actually pretty tight.
It’s sitting on the back of a white ceramic tile and the colors came out surprisingly close to real life.
Adding this to an Arduino would follow the same logic as, say, the pager motor: power the LED + resistor + MOSFET from a +5 V external regulator that won’t heat the Arduino board, then define an unused bit in the shift register as, say UV_LED.
It runs at 20 mA and drops around 3.3 V.