Cotorsion pairs in categories of quiver representations

Let [Formula: see text] be an abelian model category (in the sense of Hovey). For a large class of quivers, we describe associated abelian model structures on categories of quiver representations with values in [Formula: see text]. This is based on recent work of Holm and Jørgensen on cotorsion pairs in categories of quiver representations. An application on Ding projective and Ding injective representations of quivers over Ding–Chen rings is given.

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MODEL STRUCTURES ON TRIANGULATED CATEGORIES

Glasgow Mathematical Journal ◽

10.1017/s0017089514000299 ◽

2014 ◽

Vol 57 (2) ◽

pp. 263-284 ◽

Cited By ~ 5

Author(s):

XIAOYAN YANG

Keyword(s):

Proper Class ◽

Triangulated Category ◽

Triangulated Categories ◽

General Study ◽

Proper Class Of Triangles ◽

Cotorsion Pairs ◽

One To One ◽

The Relationship ◽

Model Structures

AbstractWe define model structures on a triangulated category with respect to some proper classes of triangles and give a general study of triangulated model structures. We look at the relationship between these model structures and cotorsion pairs with respect to a proper class of triangles on the triangulated category. In particular, we get Hovey's one-to-one correspondence between triangulated model structures and complete cotorsion pairs with respect to a proper class of triangles. Some applications are given.

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On the Null Cone for Quiver Representations Under Reflection Functors

Communications in Algebra ◽

10.1080/00927872.2012.691587 ◽

2013 ◽

Vol 41 (11) ◽

pp. 4104-4115

Author(s):

Claudio Somaini

Keyword(s):

Null Cone ◽

Quiver Representations ◽

Reflection Functors

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New model structures and projective (injective) cotorsion pairs

Journal of Algebra and Its Applications ◽

10.1142/s0219498823500366 ◽

2021 ◽

Author(s):

Aimin Xu

Keyword(s):

Positive Integer ◽

Triangulated Categories ◽

Model Structure ◽

Cotorsion Pair ◽

Flat Dimension ◽

Gorenstein Projective Module ◽

Chain Complexes ◽

Cotorsion Pairs ◽

Gorenstein Injective ◽

Model Structures

Let [Formula: see text] be either the category of [Formula: see text]-modules or the category of chain complexes of [Formula: see text]-modules and [Formula: see text] a cofibrantly generated hereditary abelian model structure on [Formula: see text]. First, we get a new cofibrantly generated model structure on [Formula: see text] related to [Formula: see text] for any positive integer [Formula: see text], and hence, one can get new algebraic triangulated categories. Second, it is shown that any [Formula: see text]-strongly Gorenstein projective module gives rise to a projective cotorsion pair cogenerated by a set. Finally, let [Formula: see text] be an [Formula: see text]-module with finite flat dimension and [Formula: see text] a positive integer, if [Formula: see text] is an exact sequence of [Formula: see text]-modules with every [Formula: see text] Gorenstein injective, then [Formula: see text] is injective.

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On Generalized Minors and Quiver Representations

International Mathematics Research Notices ◽

10.1093/imrn/rny053 ◽

2018 ◽

Vol 2020 (3) ◽

pp. 914-956 ◽

Cited By ~ 1

Author(s):

Dylan Rupel ◽

Salvatore Stella ◽

Harold Williams

Keyword(s):

Finite Type ◽

Dynkin Diagram ◽

Cluster Algebra ◽

Coordinate Ring ◽

Coxeter Element ◽

Quiver Representations ◽

Group Representations ◽

Weight Group ◽

Highest Weight ◽

Bruhat Cell

Abstract The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a subvariety of a Kac–Moody group—the quiver is an orientation of its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat cell. We use this realization to connect representations of the quiver with those of the group. We show that cluster variables of preprojective (resp. postinjective) quiver representations are realized by generalized minors of highest-weight (resp. lowest-weight) group representations, generalizing results of Yang–Zelevinsky in finite type. In type $A_{n}^{\!(1)}$ and finitely many other affine types, we show that cluster variables of regular quiver representations are realized by generalized minors of group representations that are neither highest- nor lowest-weight; we conjecture this holds more generally.

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Donaldson–Thomas invariants versus intersection cohomology of quiver moduli

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2017-0010 ◽

2019 ◽

Vol 2019 (754) ◽

pp. 143-178 ◽

Cited By ~ 5

Author(s):

Sven Meinhardt ◽

Markus Reineke

Keyword(s):

Moduli Space ◽

Stability Condition ◽

Intersection Cohomology ◽

Quiver Representations ◽

Compactly Supported ◽

Generic Stability ◽

Relative Version ◽

Zero Potential ◽

Stable Locus

Abstract The main result of this paper is the statement that the Hodge theoretic Donaldson–Thomas invariant for a quiver with zero potential and a generic stability condition agrees with the compactly supported intersection cohomology of the closure of the stable locus inside the associated coarse moduli space of semistable quiver representations. In fact, we prove an even stronger result relating the Donaldson–Thomas “function” to the intersection complex. The proof of our main result relies on a relative version of the integrality conjecture in Donaldson–Thomas theory. This will be the topic of the second part of the paper, where the relative integrality conjecture will be proven in the motivic context.

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