Ringel–Hall algebras and Lusztigʹs symmetries

Let be the double Ringel–Hall algebra of a finite dimensional hereditary algebra Λ. The present paper shows that the BGP-reflection operators of coincide with Lusztigʹs symmetries (up to canonical isomorphisms) and satisfy the braid group relations on the whole double Ringel–Hall algebra . This answers a question of Sevenhant and Van den Bergh [J. Algebra 221 (1999) 135–160].