Øyvind Solberg

Idun Reiten

Edward L. Green

Oyvind Solberg

Koszul rings are graded rings which have played an important role in algebraic topology, algebraic geometry, noncommutative algebraic geometry, and in the theory of quantum groups. One aspect of the theory is to compare the module theory for a Koszul ring and its Koszul dual. There are dualities between subcategories of graded modules; the Koszul modules. When $\Lambda$ is an artin algebra and $T$ is a cotilting $\Lambda$-module, the functor $\mathrm{Hom}_\Lambda(\,T)$ induces a duality between certain subcategories of the finitely generated modules over $\Lambda$ and $\mathrm{End}_\Lambda(T)$. The purpose of this paper is to develop a unified approach to both the Koszul duality and the duality for cotilting modules. This theory specializes to these two cases and also contains interesting new examples. The starting point for the theory is a positively $\mathbb{Z}$-graded ring $\Lambda=\Lambda_0+\Lambda_1+\Lambda_2+\cdots$ and a (Wakamatsu) cotilting $\Lambda_0$-module $T$, satisfying additional assumptions. The theory gives a duality between certain subcategories of the finitely generated graded modules generated in degree zero over $\Lambda$ on one hand and over the Yoneda algebra $\oplus_{i\geq 0} \mathrm{Ext}^i_\Lambda(T,T)$ on the other hand.

Dualities on Generalized Koszul Algebras

Birge Huisgen-Zimmermann

This refereed collection of research papers and survey articles reflects the interplay of finite-dimensional algebras with other areas (algebraic geometry, homological algebra, and the theory of quantum groups). Current trends are presented from the discussions at the AMS-IMS-SIAM Joint Summer Research Conference at the University of Washington (Seattle). The volume features several excellent expository articles which will introduce the beginning researcher to cutting-edge topics in representation theory. The book will also provide inspiration to researchers in related areas, as it includes original papers spanning a broad spectrum of representation theory. It's features include: work outlining significant progress on long-standing open problems; survey articles offering both overviews and introductions to various subfields of the topic; and, expositions reflecting the interplay between the representation theory of algebras and other fields.

Edward L. Green

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