Axiomatic Framework for the Bgg Category O

In this paper we introduce a general axiomatic framework for algebras with triangular decomposition, which allows for a systematic study of the Bernstein-Gelfand-Gelfand Category O. Our axiomatic framework can be stated via three relatively simple axioms, and it encompasses a very large class of algebras studied in the literature. We term the algebras satisfying our axioms as regular triangular algebras (RTAs); these include (a) generalized Weyl algebras, (b) symmetrizable Kac-Moody Lie algebras g, (c) quantum groups Uq(g) over “lattices with possible torsion”, (d) infinitesimal Hecke algebras, (e) higher rank Virasoro algebras, and others. In order to incorporate these special cases under a common setting, our theory distinguishes between roots and weights, and does not require the Cartan subalgebra to be a Hopf algebra. We also allow RTAs to have roots in arbitrary monoids rather than root lattices, and the roots of the Borel subalgebras to lie in cones with respect to a strict subalgebra of the Cartan subalgebra. These relaxations of the triangular structure have not been explored in the literature. We then define and study the BGG Category O over an arbitrary RTA. In order to work with general RTAs – and also bypass the use of central characters – we introduce certain conditions (termed the Conditions (S)), under which distinguished subcategories of Category O, termed “blocks”, possess desirable homological properties including: (a) being a finite length, abelian, self-dual category; (b) having enough projectives and injectives; or (c) being a highest weight category satisfying BGG Reciprocity. We discuss the above examples and whether they satisfy the various Conditions (S). We also discuss two new examples of RTAs that cannot be studied by using previous theories of Category O, but require the full scope of our framework. These include the first construction of a family of algebras for which the “root lattice” is non-abelian.