Representations of Algebras and Related Topics

Preface......Page 5
Contents......Page 11
Introduction......Page 13
Motivation......Page 14
2-Calabi–Yau categories and cluster-tilting theory......Page 19
From 3-Calabi–Yau DG-algebras to 2-Calabi–Yau categories......Page 29
Stable categories as generalized cluster categories......Page 44
On the Z-grading on the 3-preprojective algebra......Page 50
References......Page 60
Introduction......Page 67
The stable module category......Page 68
Thick subcategories of stmod(kG)......Page 70
The derived category......Page 73
Rickard idempotent modules and functors......Page 74
Classification of tensor ideal thick subcategories......Page 76
Localising subcategories of StMod(kG)......Page 77
The category KInj(kG)......Page 78
Support for triangulated categories......Page 80
Tensor triangulated categories......Page 82
The local-global principle......Page 83
Stratifying triangulated categories......Page 84
Graded polynomial algebras......Page 85
A BGG correspondence......Page 86
The Koszul construction......Page 87
Quillen stratification......Page 89
Applications......Page 90
Costratification......Page 92
References......Page 93
On cluster theory and quantum dilogarithm identities by Bernhard Keller......Page 97
Introduction......Page 0
Quantum dilogarithm identities from Dynkin quivers, after Reineke......Page 99
Sketch of proof of Reineke's theorem......Page 104
Quantum dilogarithm identities from quivers with potential, after Kontsevich–Soibelman......Page 106
DT-invariants and mutations......Page 108
Compositions of mutations......Page 116
References......Page 125
Introduction......Page 129
Representations of quantum loop algebras......Page 130
Nakajima quiver varieties and irreducible q-characters......Page 137
Tensor structure......Page 151
References......Page 162
Generalities......Page 165
Euler characteristic and stability......Page 172
Triangle singularities......Page 177
Kleinian and Fuchsian singularities......Page 189
Flags of invariant subspaces for nilpotent operators......Page 193
Comments......Page 195
References......Page 196
Introduction......Page 201
Some special cases and examples......Page 203
Defect groups......Page 207
Relative projectivity......Page 212
Brauer's First Main Theorem......Page 220
Source algebras of blocks......Page 223
Fusion systems of blocks......Page 227
Conjectures......Page 231
Symmetric algebras and transfer......Page 239
Separably equivalent algebras......Page 243
Block cohomology......Page 246
Block cohomology and Hochschild cohomology......Page 249
Further remarks and questions......Page 253
References......Page 255
Introduction......Page 263
Module categories......Page 265
Auslander–Reiten components......Page 271
Almost acyclic Auslander–Reiten components......Page 276
Almost cyclic Auslander–Reiten components......Page 281
Tilted algebras......Page 301
Quasitilted algebras......Page 316
Double tilted algebras......Page 333
Generalized double tilted algebras......Page 338
Generalized multicoil algebras......Page 343
Module categories with heart......Page 356
References......Page 360
Classification problems in noncommutative algebraic geometry and representation theory by Izuru Mori......Page 367
Overview......Page 368
Classification of quantum projective planes and noncommutative projective curves......Page 376
Fano algebras......Page 386
Generalizations of AS-Gorenstein algebras......Page 391
The structure of AS-Gorenstein algebras......Page 396
Koszul duality......Page 403
Geometric classifications revisited......Page 409
References......Page 415
Introduction......Page 419
Periodicities of exchange matrices and seeds......Page 421
Examples......Page 425
Restriction/Extension Theorem......Page 430
T-systems and Y-systems......Page 433
Dilogarithm identities......Page 444
References......Page 453
Introduction......Page 457
Tame and wild algebras......Page 459
Integral quadratic forms......Page 462
Tits form of an algebra......Page 464
Simply connected algebras......Page 469
Critical, pg-critical and hypercritical algebras......Page 472
Coil algebras......Page 477
Strongly simply connected algebras of polynomial growth......Page 480
Tame strongly simply connected algebras......Page 488
Tame algebras and partially ordered sets......Page 497
Tits forms with maximal roots......Page 501
References......Page 506
Introduction......Page 513
Preliminaries: Words......Page 516
Barification......Page 518
The barbell algebras......Page 520
The wind wheel algebras......Page 521
Proof of Theorem 1.1......Page 522
Proof of Theorem 1.2......Page 524
Proof of Theorem 1.3......Page 527
Further examples......Page 529
The cycle algebras......Page 532
The wind wheel algebras......Page 534
The Auslander–Reiten quilt of a wind wheel......Page 548
The Auslander–Reiten quiver of a barbell......Page 562
Sectional paths......Page 565
Further minimal representation-infinite algebras......Page 567
Barification may change the representation type......Page 568
Accessible representations......Page 569
Semigroup algebras......Page 570
References......Page 571
Introduction......Page 573
Preliminaries, basic facts and notation......Page 577
The Grothendieck group and the composition length vector of a comodule......Page 588
The left-valued Gabriel quiver of a basic coalgebra......Page 590
Irreducible morphisms and the Auslander–Reiten quiver of a coalgebra......Page 594
Two concepts of tameness and wildness for a K-coalgebra......Page 598
Quivers, profinite bound quivers, path coalgebras and locally nilpotent representations......Page 626
Euler coalgebras, Cartan matrices and Euler characteristic......Page 646
Almost split sequences in comodule categories......Page 658
References......Page 666
Introduction......Page 673
Varieties of representations and modules......Page 676
Types of singularities......Page 679
Orbit closures of small dimensions......Page 683
Orbit closures which are hypersurfaces......Page 686
Characterizations of degenerations......Page 688
Hom-order......Page 694
Transversal slices......Page 703
Desingularizations and unibranch varieties......Page 706
Hom-controlled exact functors......Page 713
Schubert varieties......Page 715
Equations of orbit closures......Page 718
Tangent spaces......Page 721
Singularities in codimension one......Page 725
Singularities for degenerations of codimension two......Page 728
Generic singularities......Page 732
References......Page 734
List of Contributors......Page 739