Qunhua Liu

Schur Algebras of Classical Groups II

We study Schur algebras of classical groups over an algebraically closed field of characteristic different from 2. We prove that Schur algebras are generalized Schur algebras (in Donkins sense) in types A, C, and D, while this does not hold in type B. Consequently Schur algebras of types A, C, and D are integral quasi-hereditary by Donkin [7, 9]. By using the coalgebra approach we put Schur algebras of a fixed classical group into a certain inverse system. We find that the corresponding hyperalgebra is contained in the inverse limit as a subalgebra. Moreover in types A, C, and D, the surjections in the inverse systems are compatible with the integral quasi-hereditary structure of Schur algebras.

The double Ringel-Hall algebra on a hereditary abelian finitary length category

In this paper, we study the category ℋ(ρ) of semi-stable coherent sheaves of a fixed slope ρ over a weighted projective curve. This category has nice properties: it is a hereditary abelian finitary length category. We will define the Ringel-Hall algebra of ℋ(ρ) and relate it to generalized Kac-Moody Lie algebras. Finally we obtain the Kac type theorem to describe the indecomposable objects in this category, i.e. the indecomposable semi-stable sheaves.