I have four 40W flourescent bulbs on adjustable chains.

What would grow well under them? I live in an apartment with only east facing windows, and no room left on the balcony, so whatever I try has to be capable of growing to maturity with only the artificial light. My first thoughts were lettuces, radishes, and other short cycle veggies, but I'm open to suggestions.

Also, specific variety recommendations would be appreciated, as I'm quite the beginner.

Thanks in advance,
Allan

You could grow a little batch of lettuce with those lights. But lettuce gets leggy and frail unless the light is bright enough.

Outdoor garden space is so much more efficient than lights. Maybe use them to start successive crops of fast-bolting herbs such as basil and cilantro, and come midsummer to begin new batches of cool-weather plants such as lettuce, chard, and kale for the fall. People in other forums such as vegetable gardening and herbs do this sort of thing regularly and can advise on the best times to start different sorts of crops, and you can also work backwards from the information on your seed packet. You can also start perennial herbs (chives, thyme, sage, burnet, French sorrel) or other garden plants for fall planting.

During the fall and winter, when they'd otherwise be idle, you could use them to grow lettuces and some herbs, though a cold frame on the balcony might be a more efficient way to accomplish the same thing.