Disconnect the big wire at the alternator output terminal and see if the light goes out. If it does, you've got a shorted diode bridge. Even a "new" alternator can have bad circuits. A parasitic drain in the regulator (the two small wires) usually takes several days to kill a fully charged battery. One faulty diode will still let the alternator charge, but the max output will be less than 1/3 of its rated capacity.
Jerry

"It is better to be silent and be thought a fool than to speak and eliminate all doubt!" - Abraham Lincoln
Cringe and wail in fear, Eloi- - - - -we Morlocks are on the hunt!
There is nothing noble in being superior to your fellow man; true nobility is being superior to your former self. - Ernest Hemingway
Love your enemies and drive 'em nuts!