Personality conflicts between co-workers can create difficulties in the dynamic of your workplace. This is especially true when one person feels resentment toward a colleague, which can create a tense, toxic work environment. When a situation such as this arises, whether this behavior is real or perceived, it must be handled with tact and diplomacy.
If you believe a colleague is treating you with resentment, you should first try to determine his motives. For example, did you receive a promotion for which he was also competing, or might you have possibly done something that seemed innocuous to you but could have proven offensive to him? It's important to put yourself in your co-worker's shoes to determine what could have caused him to feel resentment toward you and realistically determine if you might have contributed to that.
Address this behavior in a forthright manner. If the resentment stems from your promotion, be clear on your new role, and outline your expectations and objectives moving toward the future. Workplace relationships can often evolve and can improve just as they can deteriorate. Clear communication might not solve the problem entirely, but by addressing the issue, you can bring it to the surface and prevent the problem from festering further.
Try to avoid behavior that will further fuel resentment. It might not be your promotion, for example, that is causing resentment, but your subsequent bragging about it. Conversely, don't take the opposite tack and apologize for your success or attempt to ingratiate yourself to co-workers with false humility. If you insist you're undeserving and unworthy of being promoted, they might just agree with you, causing their feelings of resentment to grow.
One simple way to defuse the tension caused by a resentful co-worker is to be nice to him. It's difficult to be angry or resentful toward someone who treats you with kindness, so try to go out of your way to befriend those who are resentful toward you. This strategy, though, might be easier said than done. If your colleague thinks your attempts at kindness are insincere, he might feel more resentment. If possible, be honest, be sincere and be open to understanding a perspective that's different from your own.