Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors

Abstract

I nt r oduc t i onReflection functors were introduced into the representation theory of quivers by Bernstein, Gelfand and Ponomarev in their work on the 4-subspace problem and on

Fourier-Mukai transforms

“…Most algebraists probably became aware of the existence non-trivial derived equivalences when Happel showed that "tilting" (as introduced by Brenner and Butler [15]) leads to a derived equivalence between finite dimensional algebras [32]. This was generalized by Rickard who worked out the Morita theory for derived categories of rings [61,62].…”

Derived categories and tilting

“…. The now classical theory (based on homological algebra but avoiding derived categories) is due to: Brenner-Butler [14], who first proved the 'tilting theorem', Happel-Ringel [35], who improved the theorem and defined tilted algebras, Bongartz [13], who streamlined the theory, and Miyashita [55], who generalized it to tilting modules of projective dimension > 1. The use of derived categories goes back to D. Happel [31].…”

Non-commutative Crepant Resolutions: Scenes From Categorical Geometry

“…A complex T as in condition (iii) is called a tilting complex for Λ, and Γ is tilted from Λ. Tilting complexes appeared first in the form of tilting modules, as part of Brenner and Butler's [BB80] study of the reflection functors of Bernšteȋn, Gel ′ fand, and Ponomarev [BGP73]. (The word was chosen to illustrate their effect on the vectors in a root system, namely a change of basis that tilts the axes relative to the positive roots.)…”