Ringel-Hall algebras and Lusztig's symmetries
The authors of the paper under review have shown [in J. Algebra 251, No. 1, 110-149 (2002; Zbl 1010.16011)] that the Drinfeld double \({\mathcal D}(\Lambda)\) of the Ringel-Hall algebra of a finite-dimensional hereditary algebra \(\Lambda\) augmented by a torus algebra is isomorphic to the quantized enveloping algebra of a generalized Kac-Moody algebra in the sense of \textit{R. Borcherds} [J. Algebra 115, No. 2, 501-512 (1988; Zbl 0644.17010)]. Moreover, the action of Lusztig's symmetries is extended to \({\mathcal D}(\Lambda)\). \textit{B. Sevenhant} and \textit{M. Van den Bergh} [J. Algebra 221, No. 1, 135-160 (1999; Zbl 0955.16016)] as well as \textit{J. Xiao} and \textit{S. Yang} [J. Algebra 241, No. 1, 204-246 (2001; Zbl 0999.17026)] have used Bernstein-Gelfand-Ponomarev reflection functors in order to derive the fundamental properties of Lusztig's symmetries on the Drinfeld double of the composition subalgebra of a Ringel-Hall algebra of \(\Lambda\) augmented by a torus algebra. In the paper under review the authors prove that the Bernstein-Gelfand-Ponomarev reflection functors considered as operators on \({\mathcal D}(\Lambda)\) as well as operators on integrable highest weight \({\mathcal D}(\Lambda)\)-modules coincide with Lusztig's symmetries up to a canonical isomorphism and satisfy the braid group relations. This answers a question of \textit{B. Sevenhant} and \textit{M. Van den Bergh} [op. cit.].
Bernstein-Gelfand-Ponomarev reflection functors
scientific article; zbMATH DE number 1868010