MacBook Air mid 2011 external Display resoltion > 1080p which adapter?
Mac OS X should detect the supported resolutions when you connect the display. Your problem might indeed be related to the adapter and cable you are using. According to the HDMI 1.4 spec, standard HDMI cables only support resolutions up to 1080i.

I suggest you use the DisplayPort or DVI port on your monitor instead, and buy the appropriate adapter for one of those.