Affine highest weight categories and affine quasihereditary algebras

Koenig and Xi introduced {\em affine cellular algebras}. Kleshchev and Loubert showed that an important class of {\em infinite dimensional} algebras, the KLR algebras $R(\Gamma)$ of finite Lie type $\Gamma$, are (graded) affine cellular; in fact, the corresponding affine cell ideals are idempotent. This additional property is reminiscent of the properties of {\em quasihereditary algebras} of Cline-Parshall-Scott in a {\em finite dimensional} situation. A fundamental result of Cline-Parshall-Scott says that a finite dimensional algebra $A$ is quasihereditary if and only if the category of finite dimensional $A$-modules is a {\em highest weight category}. On the other hand, S. Kato and Brundan-Kleshchev-McNamara proved that the category of {\em finitely generated graded} $R(\Gamma)$-modules has many features reminiscent of those of a highest weight category. The goal of this paper is to axiomatize and study the notions of an {\em affine quasihereditary algebra} and an {\em affine highest weight category}. In particular, we prove an affine analogue of the Cline-Parshall-Scott Theorem. We also develop {\em stratified} versions of these notions

Controlled random walk with a target site

We consider a simple random walk W_i in 1 or 2 dimensions, in which the walker may choose to stand still for a limited time. The time horizon is n, the maximum consecutive time steps which can be spent standing still is m_n and the goal is to maximize P(W_n=0). We show that for dimension 1, if m_n grows faster than (\log n)^{2+\gamma} for some \gamma>0, there is a strategy for each n such that P(W_n = 0) approaches 1. For dimension 2, if m_n grows faster than a positive power of n then there are strategies keeping P(W_n=0) bounded away from 0.Comment: 7 page

Representation theory and cohomology of Khovanov-Lauda-Rouquier algebras

This expository paper is based on the lectures given at the program `Modular Representation Theory of Finite and $p$-adic Groups' at the National University of Singapore. We are concerned with recent results on representation theory and cohomology of KLR algebras, with emphasis on standard module theory.Comment: arXiv admin note: text overlap with arXiv:1210.655