Yes, very likely! Also go take a look under the hood and see if there is any sign of damage to the belt. It is natural for a new belt to stretch and require retensioning after a short run in period. Factory belts in my opinion are better, so whilst retensioning the belt may solve the issue, if you are using a parts store non OEM belt, consider changing it.
Two other things to note. I mentioned checking for damage to the belt. Whether there is a subtle difference in alternators or if the bearings/shafts can move over time, I am not sure, but I recently had an alternator fitted that had the pulley wheel very slightly offset from the drive pulley on the block. It was barely noticeable but it was sufficient to make the belt jump/move and because it then wasn't straight it would chew itself to bits rubbing against the raised retaining edges of the pulley wheel.
Finally, after sorting out your issue, if the battery has drained that far that it caused the warning light to come on, consider removing it and putting it on a battery tender or taking it to a shop that has one so that you can charge it properly. Charging a dead or very low battery on the car doesn't always work well and if you have no other option but to do it this way you really should take the car out for a really good run (which you might not want to do if your charging system is still suspect). You should also start off with a known healthy charged battery if you want to do any diagnostics on your charging system after retensioning or changing your belt etc.
I have been having some idling issues and I have a thread running in the tech section about rpms dropping when I put my headlights on. There are a several good posts in there from Andre (grocery_getter) and one specifically mentions all the voltages you should be seeing if the system is healthy.
Hope ths helps.
Paul
