Bridgeland's Hall algebras of tilted algebras and Lusztig's symmetries

For each finite dimensional algebra A of finite global dimension over a finite field, Bridgeland considered the Ringel–Hall algebra of 2-cyclic complexes of projective A-modules, and achieved an algebra called Bridgeland's Hall algebra by taking some localization and reduction. In this paper, we show that there exist homomorphisms of algebras from quantum groups to Bridgeland's Hall algebras of tilted algebras. In particular, for APR-tilting, this homomorphism provides a realization of Lusztig's symmetries of quantum groups via Bridgeland's Hall algebras.