The standard technique for this is to drive a pair of insulated gate bipolar junction transistors (IGBTs) with a sine table in the microprocesor. By varying the duty cycle of the drive, a sine output is approximatated. You can eliminate the capcitor in a split phase induction motor by driving one output 90 degrees from the other.

There are lots of application notes on this at motorola and international rectifier's web sites. Also try 'variable frequency drive' in your search engine.