Hyundai i10 Alternator Problems

Brand new battery installed 6 weeks later totally dead left it running to see if it would charge it up but switched of engine after 25 mins stil dead on dash board the a warning red light for battery

Problem recently appeared as per title ie when lights are switched on the front fog light tell tale light illuminates - car does not have front fog lights! Hyundai can offer Zeno explanation or estimate to fix they would just strip the car ...

Sounds like fan belt slipping when engaging gears

eps light comes on also while it is on the steering gets very heavy .been to mechanic several times they dont know what the cause is .if i switch engine off it seems to be ok again goes a few days and happens again !