Let G be a reductive algebraic group defined over an algebraically closed field of characteristic zero and let P be a parabolic subgroup of G. We consider the category of homogeneous vector bundles over the flag variety G / P . This category is equivalent to a category of representations of a certain infinite quiver with relations by a generalisation of a result in [BK]. We prove that both categories are Koszul precisely when the unipotent radical Pu of P is abelian.
We are interested in classifying vector bundles on algebraic varieties. In this paper we consider a homogeneous variety X = G/P, where G is a reductive group over an algebraically closed field k of characteristic zero and P is a parabolic subgroup of G.
We study the category of G-bundles of finite rank on X. This category is equivalent to the category p -mod of finite dimensional integral p -modules, where p is the Lie algebra of P . We describe this category via the category of all finite dimensional representations of some infinite quiver Q following BONDAL and KAPRANOV [BK]. The crucial problem is to find relations R in the path algebra of the quiver Q such that the category (Q, R) -mod of finite dimensional representations of Q which satisfy the relations R is equivalent to p-mod. The main result in this article states that these three equivalent categories are Koszul if and only if the unipotent radical Pu of P is abelian. Thus the relations R are quadratic.
Moreover, we will show in an example that the relations in [BK, Prop. 31 are not correct in general (see also [Hill). Using the structure of the simple modules over the Levi subgroup of P there exists an algorithm which allows to compute particular relations. With this technique we may get all relations only for Bore1 subgroups and minimal parabolic subgroups in GL, (see [Hill for an example). However the general problem of determining all relations for all parabolic subgroups P is still open. The quivers which occur in this context admit a level function s : QO + Q such 1991 Mathematics Subject Classification. Primary: 14M 15; Secondary: 20G 15, 16899. Keywords and phmses. Homogeneous vector bundles, Koszul algebras, representations of parabolic subgroups.
Tame Algebras and Integral Quadratic Forms, LNM 1099, Springer-Fakultat fir Mathematik TI/ Chemnitz