A decorated surface S is an oriented surface, with or without boundary, and a finite set {s 1,..., s n} of special points on the boundary, considered modulo isotopy. Let G be a split reductive group over [special characters omitted].

No abstract uploaded!

Let A be a finite dimensional algebra and D^b(A) be the bounded derived category of finitely generated left A-modules. In this paper we consider lengths of compact exceptional objects in D^b(A), proving a sufficient condition such that these lengths are bounded by the number of isomorphism classes of simple A-modules. Moreover, we show that algebras satisfying this condition is bounded derived simple.

In this article we study a conjecture of Geiss-Leclerc-Schröer, which is an analogue of a classical conjecture of Lusztig in the Weyl group case. It concerns the relation between canonical basis and semi-canonical basis through the characteristic cycles. We formulate an approach to this conjecture and prove it for type quiver. In the general type A case, we reduce the conjecture to show that certain nearby cycles have vanishing Euler characteristic.

Let $A = \bigoplus _{i \geqslant 0} A_i$ be a graded locally finite $k$-algebra where $A_0$ is a finite-dimensional algebra whose finitistic dimension is 0. In this paper we develop a generalized Koszul theory preserving many classical results, and show an explicit correspondence between this generalized theory and the classical theory. Applications in representations of certain categories and extension algebras of standard modules of standardly stratified algebras are described.