Yes and no.
It's basically very possible, as the lighting on a LCD screen is usually thanks to a few tiny cold-cathode fluorescent tubes. These tubes start out by generating "black light" inside the tube, then since most people want while light, the tube makers coat the tubes on the inside with a phosphor that emits while light when struck by UV light.
You can get uncoated CCFL tubes, and with a little careful work you could replace the tubes in your LCD display with the UV-emitting tubes.
But that would be kinda dumb, as the color selecting pixels will not do anything interesting withthe UV. Well, maybe the blue pixels will pass a little of the UV, but that's about it. Now if you were a bee and could see UV, you'd see an image of sorts on the screen. But I assume you're not and you can't do you wont.
Now thinking a bit more about your question, are you asking whether the LCD color pixels, instead of being little color filters backlit with white light, could they be color phosphors backlit with UV? Now that might actually work. Probably been thought of and patented though.
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