120v variable resistor controlled by 5v?
It sounds like you probably are thinking of controlling the brightness of the lights with your arduino. If that's the case, you want to investigate pulse width modulation (PWM), which as luck would have it, is also something the arduino can do. The reason being that using resistive control, you will waste a lot of power, and generate a lot of troublesome, unwanted heat, whereas PWM is much more efficient.

PWM control over lights of that sort would entail using a switching element, and could be accomplished a couple of different ways. However, with the popularity of the arduino these days, I'd be surprised if there isn't a 'shield' for that.