I was recently promoted to be the head of a division, overseeing some of my former colleagues that I used to socialize with. How can I now establish my authority and signal the changed nature of our relationship?

You need to quickly reset the context of your relationship with your former peers, ideally by meeting with them to discuss the changes and your expectations of them. As part of the carrot approach, you can use your familiarity with them to discuss how you'll support their goals and aspirations, perhaps by using a line like, "Who better than me to be your boss if you perform well?" If appropriate, establish yourself as their champion.