Journal Fur Die Reine Und Angewandte Mathematik Finite Dimensional Algebras and Highest Weight Categories ')

This paper continues the program begun by us in [8j2), [9] (see also [15], [18]) in which the authors have begun to exploit in the modular representation theory of semisimple algebraic groups some of the powerful techniques of the theory of derived categories. As noted in the above references, the inspiration for this work comes both from geometry, in the form of the classic algebraic work of Bernstein-Beilinson-Deligne [l] on singular spaces and perverse sheaves, and from the tilting theory of finite dimensional algebras [2], [3], [13], [14]. The present paper broadens and extends this connection with finite dimensional algebra representation theory into a central theme. We begin in Section 1 by completing the results of [9], $ 1, which dealt with " recollement " of triangulated categories in the sense of [l]. We apply this work in Section 2 to the situation of module categories. While [9], 9 3, treats the case of the natural exact functor Db(mod-B)-+ Db(mod-A) of derived categories arising when B is a quotient ring of A, we consider in this paper the " dual " situation in which A is a centralizer ring A = End(eB) g eBe, e E B an idempotent. We remark that our interest in this situation was first kindled by Green's treatment of the Schur algebra in [12], 9 6. Also, it turns out to lit very well with the stratification theory begun in [9]. In Section 3, we define the unifying concept of a highest weight category. Although we obtain this notion by abstracting from the classical representation theory of semisimple groups (or Lie algebras), other examples given in [16], ?$j 5, 6, indicate that such categories arise in many (perhaps surprising) situations, including quiver algebras and constructible and perverse sheaves. Theorems 3. 4 and 3. 6 relate the theory of highest weight categories to the theory of quasi-hereditary algebras (introduced in [18]). Th ese results especially appear to provide a strong link between the representation theory of finite dimensional algebras and that of semisimple groups and Lie algebras. Theorems 3. 5 and 3. 9 indicate how the recollement setup works for the derived categories associated to highest weight categories. In particular, Theorem 3. 9 ') Research supported in part by N.S.F. ') This paper extended some results of Happel [14] and broadened their scope into the beginnings of a Morita theory for derived categories. We record here …