Very little is known about Baumansky Alliance, its origins, or exact relations towards most other factions in the Metro. It is an alliance between three stations on the Arbatsko-Pokrovskaya Line - Baumanskaya (from which the Alliance gets its name), Elektrozavodskaya and Semyonovskaya.
The Alliance's main trade is recreating and refurbishing Pre-War technology from the surface and selling it to the other factions and stations 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 fully dependent on them. It is also confirmed that the station of Elektrozavodskaya, named for the electric light bulb factory on the surface nearby, runs a power plant, generating and supplying electricity to the Metro. Given Hansa's advanced technology and relatively wealthy living conditions, it is likely they buy a considerable amount of the Alliance's products.
This important trade relationship ensures that the Alliance is not fully absorbed by Hansa, unlike other smaller factions, and remains independent. Since the Alliance's goods are arguably in high demand across the whole Metro, it can also be assumed that they have few enemies.
A lot of what is known about the Baumansky Alliance is speculative, however, as neither Artyom nor Hunter visit the Alliance in the novels or the games, though some information on them is in the Metro Universe Book Series.