BGP-reflection functors and Lusztig's symmetries: a Ringel-Hall algebra approach to quantum groups
This paper contributes a very interesting Ringel-Hall algebra approach to quantum groups through BGP-reflection functors and Lusztig's symmetries. In this paper, the authors deduce Lusztig's symmetries \(T_{i,1}''\), \(i\in I\), by applying the Bernstein-Gelfand-Ponomarev reflection functors to the Drinfeld doubles of Ringel-Hall algebras, according to the canonical isomorphism between the Ringel-Hall algebras (composition algebras) and the quantum groups. The fundamental properties of \(T_{i,1}''\) including the following are obtained conceptually, that is, (1) \(T_{i,1}''\), \(i\in I\), induce automorphisms of the quantum groups \(U_q(g)\) on the integrable modules; (2) \(T_{i,1}''\), \(i\in I\), satisfy the braid group relations. This extends and completes the results of \textit{B. Sevenhant} and \textit{M. Van den Bergh} [J. Algebra 221, 135--160 (1999; Zbl 0955.16016)].
scientific article; zbMATH DE number 1658814