Your DLP type SLA printer will need to manage 2 basic tasks in order to print. The first is to adjust the Z-axis for each layer and the second is to send the appropriate slice image to the projector. This is pretty simple stuff, but unfortunately it is beyond the ability of a simple Arduino. It is within the realm of possibility for a RaspberryPi (they come with HDMI ports), but no one has written the software yet. So at this point the best solution is a Windows machine running Creation Workshop.
The Windows machine USB’s to an arduino based board (I use a ramps 1.4, but any will do) to interpret the gcode and control the z-axis stepper. If you have an old Windows box, or an old laptop lying around you may already have what you need, but if you don’t there is a perfect solution. Finally a good use for a Windows tablet! Perhaps because no one has found another good use they are practically giving Windows Tablets away. You need one which has an HDMI output and will drive two displays. You also need a usb port to connect to the ramps board. I also recommend you get one with 2gb of ram. I started with a Winbook TW700 which worked well except that if you had a large or complex model it tended to crash while printing. This wouldn’t be such a big deal except that it failed with the projector blasting pure white. If you weren’t right there when it happened this tended to cook off all the resin in the build vat.
It made an interesting brick like print, not to mention an awful smell. I only did this a couple of times before I upgraded to the Winbook TW802 which has double the memory and hasn’t crashed yet. This one cost $159, the TW700 was only $59, less than a retail license of Win8.