Axiomatic Framework for the Bgg

We present a general setup in which one can define an algebra with a regular triangular decomposition. This setup incorporates several important examples in representation theory, including semisimple, Kac-Moody, contragredient, and Borcherds Lie algebras, the Virasoro algebra, and quantum groups. In all these cases, the “Cartan” subalgebra is a commutative cocommutative Hopf algebra; we show that this forces some of the assumptions to hold. Under one additional hypothesis, the BGG category O splits into blocks, each of which is a highest weight category. 1. Setup and results Given a (split) semisimple Lie algebra g, the BGG category O is a widely studied object in representation theory. We present an algebraic framework in which one can study the category O. This incorporates several well-known examples in representation theory, as mentioned above. We work throughout over a ground field k of characteristic zero. We define N0 = N ∪ {0}. We work over an associative k-algebra A, having the following properties. (1) The multiplication map : B− ⊗k H ⊗k B+ → A is an isomorphism (the surjectivity is the triangular decomposition; the injectivity is the PBW property), where B± and H are associative k-subalgebras of A, all with unit. (2) There is a linear map ad : H → Endk(A), so that adh preserves each of H,B±, for all h ∈ H (identifying them with their respective images in A). Moreover, H ⊗B± are k-subalgebras of A. (3) The set of weights G := Homk−alg(H, k) contains an abelian group P = ⊕ α∈∆ Zα, freely generated by a finite set ∆ of simple roots; we call it the root lattice, and denote the identity by e = 0. Date: June 23, 2009. 2000 Mathematics Subject Classification. Primary: 16D90; Secondary: 16W30, 17B10. 1