Axiomatic framework for the BGG Category O

TL;DR

This paper introduces an axiomatic framework that unifies the structure of the BGG Category O across various infinite-dimensional algebras, including quantum groups and infinitesimal Hecke algebras.

Contribution

It generalizes the definition of Category O with an axiomatic approach, providing conditions for finite length, projectives, and block decomposition applicable to diverse algebraic structures.

Findings

Framework applies to quantum groups and infinitesimal Hecke algebras.

Provides criteria for finite length and block decomposition.

Includes detailed examples and analysis of centers and central characters.

Abstract

The main goal of this paper is to show that a wide variety of infinite-dimensional algebras all share a common structure, including a triangular decomposition and a theory of weights. This structure allows us to define and study the BGG Category O, generalizing previous definitions of it. Having presented our axiomatic framework, we present sufficient conditions that guarantee finite length, enough projectives, and a block decomposition into highest weight categories. The framework is strictly more general than the usual theory of O; this is needed to accommodate (quantized or higher rank) infinitesimal Hecke algebras, in addition to semisimple Lie algebras and their quantum groups. We then present numerous examples, two families of which are studied in detail. These are quantum groups defined using not necessarily the root or weight lattices (for these, we study the center and central characters), and infinitesimal Hecke algebras.