The sun doesn't put out UV cuz of the heat, it puts out UV from the Nuclear reactions taking place in it(fusion at the core, varoius fissions/fusions as you get out to the outer layers).
Rather, it may emit UV because of heat too(don't think so, but I'm no expert), but the primary source of the suns radiation is byproducts from the actual fusion and fission, not the heat. the heat itself is one of the byproducts, as heat is EM radiation below infrared.
UV is short wavelength, but actually higher energy band of light. UV is what's above violet(go figure

) Infrared is what is below red, energy wise, though it has longer wavelength. Light goes from 700nm-450nm in wavelength(red to violet) with UV having a shorter wavelength than 450nm and Infrared having a longer wavelength than the red.
Least, that's quick quotes from memory.
As with Lizard king, I'm not a physicist(computer network ing is my specialty), but I am a general nerd.
