  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.
