Dynamical combinatorics and torsion classes |
| |
| Authors: | Emily Barnard Gordana Todorov Shijie Zhu |
| |
| Affiliation: | 1. DePaul University, Department of Mathematical Sciences, 2320 N Kenmore Ave, Chicago, IL 60614, USA;2. Northeastern University, Department of Mathematics, 360 Huntington Avenue, 567 Lake Hall, Boston, MA 02115, USA;3. The University of Iowa, Department of Mathematics, 14 MacLean Hall, Iowa City, IA 52242, USA;1. Shizuoka University, Japan;2. Western Washington University, United States of America;1. Institutt for matematiske fag, NTNU, N-7491 Trondheim, Norway;2. Department of Mathematics, Indiana University, Bloomington, IN 47405, USA;3. Department of Mathematics, Texas A & M University, College Station, TX 77843, USA;1. Department of Mathematics, The University of Uath, Salt Lake City, UT 84112, USA;2. Google LLC, 1600 Amphitheater Parkway, Mountain View, CA, USA |
| |
| Abstract: | For finite semidistributive lattices the map κ gives a bijection between the sets of completely join-irreducible elements and completely meet-irreducible elements.Here we study the κ-map in the context of torsion classes. It is well-known that the lattice of torsion classes for an artin algebra is semidistributive, but in general it is far from finite. We show the κ-map is well-defined on the set of completely join-irreducible elements, even when the lattice of torsion classes is infinite. We then extend κ to a map on torsion classes which have canonical join representations given by the special torsion classes associated to the minimal extending modules introduced by the first and third authors and A. Carroll in 2019.For hereditary algebras, we show that the extended κ-map on torsion classes is essentially the same as Ringel's ?-map on wide subcategories. Also in the hereditary case, we relate the square of κ to the Auslander-Reiten translation. |
| |
| Keywords: | Lattice Torsion class Kappa map Auslander-Reiten translation Minimal extending module Wide subcategory |
| 本文献已被 ScienceDirect 等数据库收录！ |
|
