The tilting theory of contraction algebras

August, Jenny

doi:10.1016/j.aim.2020.107372

Cited by 15 publications

(25 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…3 The results in Theorem 4.1 (1) and (2) serve as an indication of why τ -tilting finite algebras of infinite and of even wild representation type exist. Examples of τ -tilting finite wild algebras are preprojective algebras of Dynkin type with at least six vertices [19] and wild contraction algebras [5].…”

Section: Remark 42mentioning

confidence: 99%

On band modules and $$\tau $$-tilting finiteness

2021

View full text Add to dashboard Cite

In this paper, motivated by a $$\tau $$ τ -tilting version of the Brauer-Thrall Conjectures, we study general properties of band modules and their endomorphisms in the module category of a finite dimensional algebra. As an application we describe properties of torsion classes containing band modules. Furthermore, we show that a special biserial algebra is $$\tau $$ τ -tilting finite if and only if no band module is a brick. We also recover a criterion for the $$\tau $$ τ -tilting finiteness of Brauer graph algebras in terms of the Brauer graph.

show abstract

Section: Remark 42mentioning

confidence: 99%

On band modules and $$\tau $$-tilting finiteness

2021

View full text Add to dashboard Cite

show abstract

“…After flopping a single curve C i to obtain X + i → Spec R, noncommutative deformation theory associates to this another contraction algebra, written ν i A con . The algebra ν i A con can be intrinsically obtained from A con via a certain mutation procedure, and in fact A con and ν i A con are derived equivalent [Aug20a]. Both A con and ν i A con are finite dimensional algebras [DW16,2.13].…”

Section: And Thus ωI(i)mentioning

confidence: 99%

GV and GW invariants via the enhanced movable cone

Nabijou,

Wemyss

2021

Preprint

View full text Add to dashboard Cite

Given any smooth germ of a threefold flopping contraction, we first give a combinatorial characterisation of which Gopakumar-Vafa (GV) invariants are nonzero, by prescribing multiplicities to the walls in the movable cone. This allows us to describe, and even draw, the critical locus of the associated quantum potential. We prove that the critical locus is the infinite hyperplane arrangement of [IW], and moreover that the quantum potential can be reconstructed from a finite fundamental domain. We then iterate, obtaining a combinatorial description of the matrix which controls the transformation of the non-zero GV invariants under a flop. There are three main ingredients and applications: (1) a construction of flops from simultaneous resolution via cosets, which describes how the dual graph changes, (2) a closed formula which describes the change in dimension of the contraction algebra under flop, and (3) a direct and explicit isomorphism between quantum cohomologies of different crepant resolutions, giving a Coxeter-style, visual proof of the Crepant Transformation Conjecture for isolated cDV singularities.

show abstract

“…In [4] August showed that given two objects M, N ∈ CM(R) with Spec R being a complete local isolated cDV singularity, such that M and N are maximal rigid objects connected through a number of mutations, the contraction algebras End(M ) and End(N ) are derived equivalent. Note that in this setting End(M ) and End(N ) are symmetric algebras.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we generalize the result mentioned above to the setting of a more general Frobenius category than CM(R). Our general course of action and a number of the proofs will be based on those in [4]. We shall use a result from [22] that will give a conflation which is able to replace the exchange sequences you would get from mutations.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Derived equivalences of self-injective 2-Calabi--Yau tilted algebras

Kortegaard¹

2022

Preprint

View full text Add to dashboard Cite

Consider a k-linear Frobenius category E with a projective generator such that the corresponding stable category C is 2-Calabi-Yau, Hom-finite with split idempotents. Let l, m ∈ C be maximal rigid objects with self-injective endomorphism algebras. We will show that their endomorphism algebras C (l, l) and C (m, m) are derived equivalent. Furthermore we will give a description of the two-sided tilting complex which induces this derived equivalence.

show abstract

The tilting theory of contraction algebras

Cited by 15 publications

References 29 publications

On band modules and $$\tau $$-tilting finiteness

On band modules and $$\tau $$-tilting finiteness

GV and GW invariants via the enhanced movable cone

Derived equivalences of self-injective 2-Calabi--Yau tilted algebras

Contact Info

Product

Resources

About