BGP-reflection functors and Lusztig's symmetries: A Ringel-Hall algebra approach to quantum groups

Abstract

According to the canonical isomorphisms between the Ringel-Hall algebras (composition algebras) and the quantum groups, we deduce Lusztig's 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(g) 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). © 2001 Academic Press.