1/ Go and get the PROPER ancillary belt configuration for a 4a-gze... what's been done there is a complete FUCKUP. The alternator is being held on by the bottom swivel bolt and the belt only!!!! You should have the air con compressor there, and the belt going from crank to aircon to alt to idler/tensioner pulley to crank.... lucky for you i have this equipment, at a price.
2/ Judging by the installation of that alternator - it's wired like arse. The alternator won't charge if the charge light isnt wired up... nor if the fuse/fusible link is blown.
3/ Dont even THINK about asking for a toyota genuine part, unless you like having a bleeding arse.
Here is how the alt SHOULD be mounted.... ignoring the MSpaint style drawing on the pic, you can see how the belt on the exhaust side goes arount the air con compressor AND the alternator, very much unlike your setup!!
AND HOLY FUCK I JUST SAW THAT THE IDLER PULLEY IS BEING USED LIKE A FIXED RIBBED PULLEY!!!!!!!!!!!!!!!!!!!!