Axiomatic framework for the BGG Category O

We introduce a general axiomatic framework for algebras with triangular decomposition, which allows for a systematic study of the Bernstein-Gelfand-Gelfand Category $\mathcal{O}$. The framework is stated via three relatively simple axioms; algebras satisfying them are termed "regular triangular algebras (RTAs)". These encompass a large class of algebras in the literature, including (a) generalized Weyl algebras, (b) symmetrizable Kac-Moody Lie algebras $\mathfrak{g}$, (c) quantum groups $U_q(\mathfrak{g})$ 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 study the BGG Category $\mathcal{O}$ over an RTA. In order to work with general RTAs - and also bypass the use of central characters - we introduce conditions termed the "Conditions (S)", under which distinguished subcategories of Category $\mathcal{O}$ possess desirable homological properties, including: (a) being a finite length, abelian, self-dual category; (b) having enough projectives/injectives; or (c) being a highest weight category satisfying BGG Reciprocity. We discuss whether the above examples satisfy the various Conditions (S). We also discuss two new examples of RTAs that cannot be studied using previous theories of Category $\mathcal{O}$, but require the full scope of our framework. These include the first construction of algebras for which the "root lattice" is non-abelian.