O Algebras with Three Simple Modules

0. Introduction. In D], Dyer has introduced the concept of O algebra, this is a basic quasi-hereditary Koszul algebra with an anti-involution, such that its quadratic dual is also quasi-hereditary. Typical examples of such algebras are ones associated with blocks of category O of Bernstein-Gelfand-Gelfand BGG], see also Beilinson-Ginsburg-Soergel BGS]. The present paper grew out of a desire to understand O algebras in the framework of the representation theory of nite-dimensional algebras, in particular, inside the one of quasi-hereditary algebras in the sense of Cline-Parshall-Scott CPS]; see also Dlab-Ringel DR] and Ringel R2]. We consider such algebras with three (two) simple modules, this is the simplest and basic case: any O algebra has a subalgebra of this form. We recall some deenitions and facts in section 1. In section 2 we classify Dyer algebras with three simple modules in terms of quivers and relations (denoted by O(l; m; n)), they are all quadratic. Representation types of O(l; m; n) are determined in section 3. It is proved in section 4 that algebras O(l; m; n) are closed under quadratic duals, and that they are all O algebras; as a corollary the global dimensions are given. In section 5 we give a suucient condition for the existence of an exact Borel subalgebra (in the sense of KK onig K]) of a Dyer algebra, and then use this to compute Kazhdan-Lusztig-Vogan polynomials of O(l; m; n). One of the most important things in representations of a quasi-hereditary algebra is the existence of the characteristic module and the Ringel dual, see R2]. In section 6 we discuss Ringel duals of O(l; m; n; M; B v). This work is done in University Bielefeld, I would like to thank Professor C. M. Ringel, who has introduced me into this topic, and other colleagues, both for helpful conversations and hospitality; I am indebted to the Alexander von Humboldt Foundation for the generous supports.