  The goal of this paper is to construct quantum analogues of Chevalley groups
inside completions of quantum groups or, more precisely, inside completions of
Hall algebras of finitary categories. In particular, we obtain pentagonal and
other identities in the quantum Chevalley groups which generalize their
classical counterparts and explain Faddeev-Volkov quantum dilogarithmic
identities and their recent generalizations due to Keller
