"I am not a man."
Translation:Ben erkek değilim.
It's a great question. In this sentence, erkek refers merely to a quality, a category, not some definite or indefinite man. It simply expresses the gender of the speaker. In a way, in this role, it even resembles an adjective. Cf: Ben zengin değilim. As number and definiteness aren't important in such sentences, 'bir', being a marker of number (among its other functions), isn't used in them.
As for definiteness, it doesn't show itself in this kind of sentence if the definite noun is singular. The hearer has to infer it from the context.
I'm a policeman. → Ben polisim.
I'm the policeman. → Ben polisim.
We're policemen. → Biz polisiz.
We're the policemen. → Biz polisleriz.
Thank you very much indeed. By the way - these kinds of sentences or this kind of sentence. I hope you don't mind me saying. English must be a nightmare. Maybe it's a typo!