UV or ultraviolet light is part of the spectrum of electromagnetic radiation, which also includes radio waves, microwaves, infrared radiation, X-rays, gamma rays – and of course, visible light. These different types of radiation are defined by their wavelengths or frequencies. Radio waves have the longest wavelengths, and gamma rays the shortest.
By definition, what we can see is the visible light range. Anthony Doerr writes in his magnificent novel, All The Light We Cannot See: “What do we call visible light? We call it color. But the electromagnetic spectrum runs to zero in one direction and infinity in the other, so really, children, mathematically, all of light is invisible.” The colours of the rainbow are a very, very small proportion of total spectrum.
As you can see from the diagram, UV light has wavelengths shorter than that of visible light, and our eyes are not sensitive to these wavelengths. So no, we can’t see UV light.
The question then arises: some UV curing lamps seem to generate light that I can see! That’s because broad spectrum UV curing lamps, as the name implies, emit some light which creeps into the visible range – which is why the light that we can see in these instances has a purple-ish hue. But it’s not UV.