Blocks of category \(\mathcal O\) double centralizer properties, and Enright's completions (Q2759662)

This is an interesting survey article concerning recent results on the category \(\mathcal O\), obtained by the author jointly with \textit{V. Mazorchuk} [Manuscr. Math. 102, 487-503 (2000; Zbl 1018.17005)], and with \textit{I. H. Slungard} and the reviewer in [J. Algebra 240, 393-412 (2001; Zbl 0980.17003)]. The main results in the paper exhibit the strucure of the blocks of category \(\mathcal O\) or certain related subcategories. Especially, the results of Soergel's double centralizer theorem and Enright's conjecture are obtained in a more general framework. Here the main idea is the use of the dominant dimension, or Auslander's homological context for representation theory of Artin algebras. The surprising thing is that the old notion of dominant dimension (known long time ago) provides a computation-free proof and simplifies the Lie theoretic consideration. NEWLINENEWLINENEWLINEThe modules with a filtration by Verma modules are fundamental objects in the category \(\mathcal O\). In this paper a classification list of representation types of the subcategory of modules having a Verma flag is given, thereby reporting a result of the author jointly with \textit{Th. Brüstle} and \textit{V. Mazorchuk} [Bull. Lond. Math. Soc. 33, 669-681 (2001)].NEWLINENEWLINENEWLINEThe paper starts with recalling basic properties of category \(\mathcal O\) and ends with many interesting references, and it is well-written.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00049].