Functoriality of the BGG Category O

Details Functoriality of the BGG Category O Functoriality of the BGG Category O

2008-11-13

arxiv.org/abs/0811.2073v2

Communications in Algebra 37 (2009), no. 12, 4431-4475

Mathematics

Representation Theory

Final form of a much expanded, improved, and generalized version of a previous preprint - arXiv:math/0504371. Accepted for pub

Scientific paper

This article aims to contribute to the study of algebras with triangular decomposition over a Hopf algebra, as well as the BGG Category O. We study functorial properties of O across various setups. The first setup is over a skew group ring, involving a finite group $\Gamma$ acting on a regular triangular algebra $A$. We develop Clifford theory for $A \rtimes \Gamma$, and obtain results on block decomposition, complete reducibility, and enough projectives. O is shown to be a highest weight category when $A$ satisfies one of the "Conditions (S)"; the BGG Reciprocity formula is slightly different because the duality functor need not preserve each simple module. Next, we turn to tensor products of such skew group rings; such a product is also a skew group ring. We are thus able to relate four different types of Categories O; more precisely, we list several conditions, each of which is equivalent in any one setup, to any other setup - and which yield information about O.

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