Re: alternator thick cable always hot wj overland
So it seems that there is an electrical resistance at the alternator connection. I would remove the wire lug and check for corrosion/dirt at the connection, maybe even the lug nut is loose. The voltage with engine off is ok (eventhough a fully charged battery gives around 12.7V, 30 minutes after engine shutdown), the engine running voltage should be 13 +V, but this can have to do with the alternator wire connection.
But is the wire hotter then the alternator itself, because the alternator is mounted to the engine and heats up because of the engine heat and also due to the electrical generation.