Incredibly little is known about this faction, not when or who it was formed by, or who they are allied and hostile to. All that is known is that it is an alliance between three stations on the Arbatsko-Pokrovskaya Line - Baumanskaya (from which the Alliance gets its name), Elektrozavodskaya and Semyonovskaya.
What is known is that their main trade its to recreate and refurbish Pre-War technology from the surface, selling them to the other factions of the Metro. It is likely they are under a protection pact with Hansa, in return for technology, but it appears they are powerful enough to not be dependant on them. It is also confirmed that the station of Elektrozavoskaya, named for the electric light bulb factory nearby, on the surface, runs a power plant, generating and supplying electricity to the Metro - again, mainly Hansa. Given Hansa's advanced technology and relatively wealthy lifestyles, it is likely they buy most of their technology and receive electricity from the Baumansky Alliance.
Both of these necessary trades ensure that the Alliance is not fully absorbed by Hansa, like other smaller factions, and remains independant. Since the whole Metro relies on them, it can also be assumed they have few enemies, for if a faction were to attack them, they would find all their electricity suddenly cut off, leading to ruin.
However, a lot of what is known about the Baumansky Alliance is still speculative, as neither Artyom nor Hunter visit the Alliance in the novel or the games, though some information on them is in the Metro Universe Book Series.