What is the authority of the Air Boss at an airshow?
Air bosses who orchestrate air shows do not receive special training or certification . Infact FAA also lacks the resources to inspect all aircraft that participate in air shows 

FAA doesn't have certification standards for air bosses . Waivers for customary flight standards are provided to air shows, which then hire people to direct planes in flight.

There is no training in place right now for an air boss

http://www.rgj.com/story/news/2014/03/23/ntsb-hearing-addresses-distance-between-race-course-and-spectators-other-issues/6587627/

However,  typically former air-traffic controllers with a great deal of aviation experience are hired as Air Bosses at Air Shows .

The air boss implements the standards on the FAA waiver

Firms like ICAS provide Air Boss training