It sounds like the algae is a symptom of something else, too much fertility piling into the lake

Perhaps, run off from pasture, fields, some source of detergents?

If you can cut off the excess fertility pounded into the lake, if you keep extracting the algae or aquatic plants that thrive on eutrophic conditions, that would deplete the excess fertility

A well planted pond is capable of dictating water quality, though you would have the task of extracting surplus plants every couple or three years to stop them from eventually choking the pond

I would make a bet that the homes surrounding the lake have lovely green lawns that have regular generous applications of fertilizer put on them and that all the yard drainage is directed to the pond which might even have been built for that purpose. A lot of retention ponds have the problem. less for natural ponds.

The first thing to do is read the two articles by Norm Meck about green water so you have an inkling of the problem and the myths that surround it. Since the articles are primarily about Koi ponds, not all info is relevant, but it is very good information. The second article discusses a theory about an antibiotic process that may control algae. This is a theory, not proven scientific fact but it is worth investigating with such a large body of water.

Check your local DNR and Cooperative Extension Service to see what local controls exist . The DNR can get very testy.

Once you have an understanding of what is known, you can try some possibilities. The first is to reduce the amount of fertilizer reaching the lake. The second is to place bales of Barley straw in the water and let it rot. This may or may not have an effect but it won't cause the problem to get any worse.

This should be done only with the approval of the HOA. They likely have all sorts of rules about it.

Here is a link that might be useful: Norm Meck's second article.

I agree with Sleepless. Our Extension service has classes on how to keep ponds healthy. It is geared mosly toward farm ponds, but your is big enough to fall into that category.

I believe sleepless and pashta are correct. Our regional multi county area has been working for years to decrease fertilizer and phosphate use/release in central North Carolina to prevent deterioration of stream, river and lake waters. If others in your HOA are concerned, you may be able to effect a change in local lawn treatment.