did you leave it plugged in all night?
ultimately, the cig lighter plug is fuse protected at 20-25a in most vehicles, and loads under that range should be fine.
is there some other part of the story we might need to know in order to help us diagnose your issue?
depending on the wattage and duration, an inverter, a fairly power hungry and ineficient device can drain down a vehicle battery before you know it. most have a low voltage cutoff of around 10v.. but by then your vehicle battery is essentially 'discharged'.
i have seen quite low bats that still had enough umph to make a vehicle crank over and start up, only to cook the alt trying to charge that near dead load. alts are great at putting out rated load, but are not good 'dead battery' chargers. i always top low batteries off with a charger before subjecting an alternator to that kind of brute force load.
what else can you tell us?
soundguy