scholarly journals The Critical CoHA of a Quiver With Potential

2017 ◽  
Vol 68 (2) ◽  
pp. 635-703 ◽  
Author(s):  
Ben Davison
Keyword(s):  
Quiver With Potential

scholarly journals Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials

2012 ◽  
Vol 148 (6) ◽  
pp. 1833-1866 ◽  
Author(s):  
Giovanni Cerulli Irelli ◽  
Daniel Labardini-Fragoso
Keyword(s):  
Linear Independence ◽  
Cluster Algebra ◽  
Arbitrary System ◽  
Triangulated Surfaces ◽  
Jacobian Algebra ◽  
Quiver With Potential ◽  
Ideal Triangulations ◽  
The Given ◽  
Marked Points ◽  
The Ideal

AbstractTo each tagged triangulation of a surface with marked points and non-empty boundary we associate a quiver with potential in such a way that whenever we apply a flip to a tagged triangulation the Jacobian algebra of the quiver with potential (QP) associated to the resulting tagged triangulation is isomorphic to the Jacobian algebra of the QP obtained by mutating the QP of the original one. Furthermore, we show that any two tagged triangulations are related by a sequence of flips compatible with QP-mutation. We also prove that, for each of the QPs constructed, the ideal of the non-completed path algebra generated by the cyclic derivatives is admissible and the corresponding quotient is isomorphic to the Jacobian algebra. These results, which generalize some of the second author’s previous work for ideal triangulations, are then applied to prove properties of cluster monomials, like linear independence, in the cluster algebra associated to the given surface by Fomin, Shapiro and Thurston (with an arbitrary system of coefficients).

scholarly journals Combinatorial Cluster Expansion Formulas from Triangulated Surfaces

10.37236/8351 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Toshiya Yurikusa
Keyword(s):  
Bipartite Graph ◽  
Cluster Expansion ◽  
Independent Sets ◽  
Perfect Matchings ◽  
Cluster Algebras ◽  
Expansion Formula ◽  
Triangulated Surfaces ◽  
Quiver With Potential ◽  
Maximal Independent Sets

We give a cluster expansion formula for cluster algebras with principal coefficients defined from triangulated surfaces in terms of maximal independent sets of angles. Our formula simplifies the cluster expansion formula given by Musiker, Schiffler and Williams in terms of perfect matchings of snake graphs. A key point of our proof is to give a bijection between maximal independent sets of angles in some triangulated polygon and perfect matchings of the corresponding snake graph. Moreover, they also correspond bijectively with perfect matchings of the corresponding bipartite graph and minimal cuts of the corresponding quiver with potential.

scholarly journals Motivic Donaldson–Thomas invariants for the one-loop quiver with potential

2015 ◽  
Vol 19 (5) ◽  
pp. 2535-2555 ◽  
Author(s):  
Ben Davison ◽  
Sven Meinhardt
Keyword(s):  
Quiver With Potential ◽  
The One

scholarly journals On two cohomological Hall algebras

2019 ◽  
Vol 150 (3) ◽  
pp. 1581-1607
Author(s):  
Yaping Yang ◽  
Gufang Zhao
Keyword(s):  
Equivariant Cohomology ◽  
The Other ◽  
Algebra Homomorphism ◽  
Hall Algebra ◽  
Preprojective Algebra ◽  
Positive Part ◽  
Hall Algebras ◽  
Oriented Cohomology ◽  
Quiver With Potential ◽  
Vanishing Cycles

AbstractWe compare two cohomological Hall algebras (CoHA). The first one is the preprojective CoHA introduced in [19] associated with each quiver Q, and each algebraic oriented cohomology theory A. It is defined as the A-homology of the moduli of representations of the preprojective algebra of Q, generalizing the K-theoretic Hall algebra of commuting varieties of Schiffmann-Vasserot [15]. The other one is the critical CoHA defined by Kontsevich-Soibelman associated with each quiver with potentials. It is defined using the equivariant cohomology with compact support with coefficients in the sheaf of vanishing cycles. In the present paper, we show that the critical CoHA, for the quiver with potential of Ginzburg, is isomorphic to the preprojective CoHA as algebras. As applications, we obtain an algebra homomorphism from the positive part of the Yangian to the critical CoHA.

scholarly journals Decorated Marked Surfaces III: The Derived Category of a Decorated Marked Surface

2019 ◽  
Author(s):  
Aslak Bakke Buan ◽  
Yu Qiu ◽  
Yu Zhou
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Stability Conditions ◽  
Finite Dimensional ◽  
Quiver With Potential ◽  
Dg Algebra ◽  
Marked Surface

Abstract We study the Ginzburg dg algebra $\Gamma _{\mathbf {T}}$ associated with the quiver with potential arising from a triangulation $\mathbf {T}$ of a decorated marked surface ${\mathbf {S}}_\bigtriangleup$, in the sense of [22]. We show that there is a canonical way to identify all finite-dimensional derived categories $\operatorname {\mathcal {D}}_{fd}(\Gamma _{\mathbf {T}})$, denoted by $\operatorname {\mathcal {D}}_{fd}({\mathbf {S}}_\bigtriangleup )$. As an application, we show that the spherical twist group $\operatorname {ST}({\mathbf {S}}_\bigtriangleup )$ associated with $\operatorname {\mathcal {D}}_{fd}({\mathbf {S}}_\bigtriangleup )$ acts faithfully on its space of stability conditions.

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