Functoriality of the BGG Category O

TL;DR

This paper explores the functorial properties of the BGG Category O across various algebraic setups involving Hopf algebras, skew group rings, and tensor products, establishing conditions for highest weight structures and block decompositions.

Contribution

It introduces a unified framework for studying the functoriality of Category O in different algebraic contexts, including Clifford theory and tensor product relations.

Findings

Category O is a highest weight category under certain conditions.

Block decomposition and complete reducibility are established for skew group rings.

Relations between different types of Category O are characterized through equivalent conditions.

Abstract

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.