Øyvind Solberg

Idun Reiten

Aslak Bakke Buan

This book features survey and research papers from The Abel Symposium 2011: Algebras, quivers and representations, held in Balestrand, Norway 2011. It examines a very active research area that has had a growing influence and profound impact in many other areas of mathematics like, commutative algebra, algebraic geometry, algebraic groups and combinatorics. This volume illustrates and extends such connections with algebraic geometry, cluster algebra theory, commutative algebra, dynamical systems and triangulated categories. In addition, it includes contributions on further developments in representation theory of quivers and algebras. Algebras, Quivers and Representations is targeted at researchers and graduate students in algebra, representation theory and triangulate categories. ​

The Abel Symposium 2011

Algebras, Quivers and Representations

Sverre O. Smalø

Maurice Auslander

Auslander made contributions to many parts of algebra, and this 2-volume set (the set ISBN is 0-8218-0679-3, already published) contains a selection of his main work.

Selected Works of Maurice Auslander

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.

Øyvind Solberg

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