Would it be possible to use a solid state relay as a dimmer switch?
You need a random, not zero-crossing, switching SSR and you will need to get the zero crossings into your Arduino through something like an optoisolator, so you can time the delays to triggering on each half-cycle. 

The latter would look something like this: 







simulate this circuit &ndash; Schematic created using CircuitLab

Everything to the left of the optoisolator on the above schematic and the output terminals of the SSR are at mains potential, and proper precautions must be taken for safety. Best to have someone local have a look at it to make sure it's safe before applying mains power- you could hurt yourself or permanently damage your computer or other property.