Graded and Non-graded Kazhdan-lusztig Theories

Let %‘A be the category of finite dimensional right modules for a quasi-hereditary algebra A. In the context of various types of Kazhdan-Lusztig theories, we study both the homological dual A! = Extt, (A/ rad(A), A/ rad(A)) and the graded algebra grA = @rad(A)j/rad(A)j+‘. For example, we investigate a condition introduced in [6], and here called (SKL’), which guarantees that A!’ E gr A. A strengthening of (SKL’) leads to the notion of a strong Kazbdan-Lusttig theory (SKL) for ‘%‘A. We show that (SKL) behaves well with respect to recollement and we relate this notion to that of a graded Kazhdan-Lusztig theory. In particular, we prove that the category Utriv associated to a complex semisimple Lie algebra satisfies the (SKL) condition. Finally, we present an example showing that, even in very favorable circumstances, strong properties of the algebras A! and gr A may not imply similar properties for A. Let A be a quasi-hereditary algebra over an algebraically closed field k. Let ‘ZA be the category of finite dimensional right A-modules. In general, the homological dual A! = ExtkA(A/ rad(A), A/ rad(A)) is not a quasi-hereditary algebra. However, when %‘A has a Kazhdan-Lusztig theory in the sense of [5], a main result of [6] established that A! is quasi-hereditary. Another main result in [6] presented a strengthening of the condition for a Kazhdan-Lusztig theory. This new condition, here called (SKL’), is a parity requirement, involving, for example, Ext%A (-, L), for L an irreducible A-module, as applied to the radical series of standard modules. The (SKL’) condition suffices to guarantee that the category %‘if of finite dimensional graded right A!-modules has a graded Kazhdan-Lusztig theory (and, consequently, A! is Koszul). As remarked in [6, (3.10)] (see also [14]), the stronger (SKL’) condition does hold in at least the classical case when VA s @h-iv, the principal block for the Bernstein-Gelfand-Gelfand category B associated with a complex semisimple Lie algebra. Thus, (SKL’) illustrates a 1991 Mathematics Subject Classificatzon. 20G05. Research supported in part by the National Science Foundation