On Arkhipov?s and Enright?s functors

Abstract

We give a description of Arkhipov’s and (Joseph’s and Deodhar-Mathieu’s versions of) Enright’s endofunctors on the category Open image in new window associated with a fixed triangular decomposition of a complex finite-dimensional semi-simple Lie algebra, in terms of (co)approximation functors with respect to suitably chosen injective (resp. projective) modules. We establish some new connections between these functors, for example we show that Arkhipov’s and Joseph’s functors are adjoint to each other. We also give several proofs of braid relations for Arkhipov’s and Enright’s functors.