**Contemporary Mathematics**

Volume: 758;
2020;
298 pp;
Softcover

MSC: Primary 16; 18;

**Print ISBN: 978-1-4704-5131-8
Product Code: CONM/758**

List Price: $120.00

AMS Member Price: $96.00

MAA Member Price: $108.00

**Electronic ISBN: 978-1-4704-5699-3
Product Code: CONM/758.E**

List Price: $120.00

AMS Member Price: $96.00

MAA Member Price: $108.00

# Representation Theory and Beyond

Share this page *Edited by *
*Jan Šťovíček; Jan Trlifaj*

This volume contains the proceedings of the Workshop and 18th
International Conference on Representations of Algebras (ICRA 2018)
held from August 8–17, 2018, in Prague, Czech Republic.

It presents several themes of contemporary representation theory
together with some new tools, such as stable \(\infty\)-categories,
stable derivators, and contramodules.

In the first part, expanded lecture notes of four courses delivered
at the workshop are presented, covering the representation theory of
finite sets with correspondences, geometric theory of quiver
Grassmannians, recent applications of contramodules to tilting theory,
as well as symmetries in the representation theory over an abstract
stable homotopy theory.

The second part consists of six more-advanced papers based on
plenary talks of the conference, presenting selected topics from
contemporary representation theory: recollements and purity, maximal
green sequences, cohomological Hall algebras, Hochschild cohomology of
associative algebras, cohomology of local selfinjective algebras, and
the higher Auslander–Reiten theory studied via homotopy theory.

#### Readership

Graduate students and research mathematicians interested in representation theory of algebras.

# Table of Contents

## Representation Theory and Beyond

- Cover Cover11
- Title page iii4
- Contents vii8
- Preface ix10
- Workshop Lecture Notes 112
- Contramodules and their applications to tilting theory 314
- Representations of finite sets and correspondences 3142
- 1. Introduction 3142
- 2. The representation theory of categories 3243
- 3. Correspondences and relations 3546
- 4. The algebra of permuted orders 3748
- 5. The simple modules for the essential algebra 3950
- 6. Correspondence functors 4051
- 7. Finiteness properties 4253
- 8. Stability 4657
- 9. Correspondence functors and lattices 4758
- 10. Fundamental functors 4960
- 11. The simple correspondence functors 5162
- References 5465

- Three lectures on quiver Grassmannians 5768
- Introduction 5768
- 1. Notations 5970
- 2. Quiver Grassmannians 6374
- 3. Degeneration of 𝑄–representations: Bongartz’s theorem and applications to quiver Grassmannians 6677
- 4. Examples of quiver Grassmannians 6677
- 5. Quiver Grassmannians of type 𝐴 7384
- 6. Quiver Grassmannians of Dynkin type 7889
- 7. Cell decomposition and property (S) 7990
- 8. The cluster multiplication formula 8394
- 9. Exercises 8495
- Acknowledgments 8798
- References 8798

- Higher symmetries in abstract stable homotopy theories 91102
- Conference Proceedings Papers 195206
- Cohomology of some local selfinjective algebras 197208
- On the cohomological Hall algebra of the Kronecker quiver 213224
- The recollements of purity 227238
- Higher Auslander algebras of type 𝔸 and the higher Waldhausen 𝖲-constructions 249260
- A survey on maximal green sequences 267278
- 1. Introduction 267278
- 2. Mutation and green mutation 268279
- 3. Applications 270281
- 4. Existence and properties 273284
- 5. Muller’s theorem via representation theory 275286
- 6. Comparing the proofs 278289
- Appendix A. Maximal chains of torsion classes, by Laurent Demonet 278289
- Acknowledgments 280291
- References 281292

- The Gerstenhaber bracket in Hochschild cohomology: Methods and examples 287298
- Back Cover Back Cover1312