BGP-Reflection Functors and Lusztigʹs Symmetries: A Ringel–Hall Algebra Approach to Quantum Groups,

According to the canonical isomorphisms between the Ringel–Hall algebras (composition algebras) and the quantum groups, we deduce Lusztigʹs symmetries T″i, 1, i I, by applying the Bernstein–Gelfand–Ponomarev reflection functors to the Drinfeld doubles of Ringel–Hall algebras. The fundamental properties of T″i, 1 including the following can be obtained conceptually. (1) T″i, 1, i I induce automorphisms of the quantum groups Uq( ) and on the integrable modules. (2) T″i, 1, i I satisfy the braid group relations. This extends and completes the results of B. Sevenhant and M. Van den Bergh (1999, J. Algebra221, 135–160).