Linear Representations of Partially Ordered Sets and Vector Space Categories

Linear Representations of Partially Ordered Sets and Vector Space Categories......Page 1
Contents......Page 5
Preface......Page 9
Introduction......Page 11
1. General remarks on matrix problems......Page 16
1.1 Matrix problems......Page 18
1.2 Examples......Page 19
2.1 Block matrices and admissible transformations......Page 26
2.2 Additive category of matrix representations......Page 29
2.3 Matrix orbit problems......Page 31
2.4 Posets of width two and their indecomposable representations......Page 33
2.5 A Kleisli category and a bocs......Page 39
2.6 Exercises......Page 43
3. Matrix representations and I-spaces......Page 46
3.1 Reduction functor......Page 48
3.2 Section map......Page 52
3.3 Exercises......Page 55
4.1 Differentiation procedure......Page 58
4.2 Main theorem......Page 60
5.1 Modules over incidence algebras......Page 64
5.2 Reflection duality and sp-injective I-spaces......Page 69
5.3 Subposet induced functors......Page 74
5.4 Exercises......Page 79
6.1 Posets of width two......Page 82
6.2 Gabriel's reduction functor......Page 85
6.3 Exercises......Page 94
7. A differentiation with respect to a minimal element......Page 96
7.1 Exercises......Page 100
8.1 Poset splitting decomposition......Page 102
8.2 A factor category interpretation......Page 105
8.3 Crossed decompositions of posets......Page 107
8.4 Exercises......Page 112
9. A differentiation with respect to a suitable pair of elements......Page 114
9.1 Algorithm of Zavadskij......Page 115
9.2 Main theorem......Page 122
9.3 Section map......Page 123
9.4 A retraction......Page 126
9.5 Functorial interpretations of the algorithm......Page 128
9.6 Differentiation of coordinate vectors......Page 137
10.1 Two theorems of Kleiner......Page 140
10.2 Combinatorial lemmas......Page 146
10.3 Complexity of representation-finite posets......Page 150
10.4 How do we determine sincere posets?......Page 154
10.5 Weakly positive forms and positive roots......Page 167
10.6 On the global dimension of KI*......Page 169
10.7 Tables of sincere I-spaces for representation-finite posets......Page 172
10.8 Exercises......Page 185
11. Irreducible maps and Auslander-Reiten sequences......Page 186
11.1 Irreducible maps and Auslander-Reiten quivers......Page 187
11.2 The Auslander ring......Page 190
11.3 Auslander-Reiten sequences......Page 193
11.4 The Auslander transpose......Page 198
11.5 Auslander-Reiten translate......Page 200
11.6 Prinjective modules over KI*......Page 204
11.7 Auslander-Reiten sequences in the category I-sp......Page 208
11.8 A criterion for finite representation type......Page 212
11.9 Preprojective component construction......Page 214
11.10 Computation algorithm......Page 221
11.11 Dimension vectors......Page 223
11.12 Auslander-Reiten sequences in the category prin(KI*)......Page 226
11.13 A bilinear form and a Coxeter transformation......Page 239
11.14 Exercises......Page 244
12.1 Auslander-Reiten quiver of splitting posets......Page 248
12.2 Differentiation and Auslander-Reiten quivers......Page 257
12.3 Exercises......Page 269
13.1 On lattices over orders......Page 272
13.2 Tiled orders and their covering posets......Page 276
13.3 A covering type functor......Page 280
13.4 Completion functor......Page 286
14.1 Quivers and path algebras......Page 292
14.2 Finite type and wild type......Page 297
14.3 Indecomposable modules over k[t]......Page 302
14.4 Tame representation type......Page 304
14.5 Module varieties......Page 314
14.6 Exercises......Page 319
15. Representation-tame posets......Page 322
15.1 Nazarova theorem......Page 323
15.2 Varieties of prinjective modules......Page 328
15.3 When is the quadratic form of a poset weakly non-negative?......Page 336
15.4 Differentiation of N*-free posets......Page 339
15.5 I-spaces over K[t] and parametrising bimodules......Page 347
15.6 Four subspace problem......Page 356
15.7 N*-free posets are representation-tame......Page 364
15.8 N*-free posets containing NZ......Page 372
15.9 Quadratic forms and roots of N*Z-free posets......Page 378
15.10 Posets of polynomial growth......Page 383
15.11 Tables of sincere posets of polynomial growth......Page 387
15.12 On tameness of tiled orders and infinite posets......Page 397
15.13 Exercises......Page 401
16.1 Matrix representations......Page 402
16.2 Prinjective modules......Page 406
16.3 Adjustment functors......Page 410
16.4 Representation-finite pairs of posets......Page 411
16.5 Representation-tame pairs of posets......Page 413
17. Vector space categories and bimodule matrix problems......Page 418
17.1 Subspace category of a vector space category......Page 419
17.2 Poset type vector space categories......Page 422
17.3 One point extensions and one point coextensions......Page 427
17.4 Right peak algebras and socle projective modules......Page 435
17.5 Embeddings of module categories......Page 445
17.6 Splitting theorem......Page 448
17.7 Representation-finite schurian vector space categories......Page 456
17.8 Stratified posets and related vector space categories......Page 469
17.9 Bipartite bimodule matrix problems......Page 476
17.10 Derivation bimodule matrix problems......Page 484
Bibliography......Page 488
Index......Page 504
List of symbols......Page 510