On Arkhipov's and Enright's functors.
This article studies various endofunctors of the Bernstein-Gelfand-Gelfand category \(\mathcal O\) of a semisimple complex Lie algebra. There are three types of results: Connections between different functors are established, for instance by proving adjointness. It is shown that some functors satisfy braid relations. Certain Lie theoretically functors are identified with functors defined abstractly in terms of Auslander-Reiten approximation theory. The main functors studied are Arkhipov's functors (which compose to a functor used by \textit{W. Soergel} [Represent. Theory 2, No. 13, 432--448 (1998; Zbl 0964.17018)] to prove Ringel self-duality of category \(\mathcal O\)), Enright's functors (in Joseph's version, thus extending results of [\textit{S. König, V. Mazorchuk}, Trans. Am. Math. Soc. 354, No. 7, 2725--2743 (2002; Zbl 1007.17004)]) and a new class of functors called partial (co-)approximations. Arkhipov's and Enright's functors (in Joseph's version) both are identified with certain partial (co-)approximations and then related to each other by adjointness. Arkhipov's functors then are shown to satisfy braid relations. As shown elsewhere (see for instance [\textit{V. Mazorchuk, C. Stroppel}, Trans. Am. Math. Soc. 357, No. 7, 2939--2973 (2005; Zbl 1095.17001)] and [\textit{C. Stroppel}, Duke Math. J. 126, No. 3, 547--596 (2005; Zbl 1112.17010)]), these results are also of interest in the context of categorification of knot invariants.
scientific article; zbMATH DE number 2156049
scientific article