Daniel Labardini-Fragoso's research while affiliated with Universidad Nacional Autónoma de México and other places
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Publications (29)
To each triangulation of any surface with marked points on the boundary and orbifold points of order three, we associate a quiver (with loops) with potential whose Jacobian algebra is finite dimensional and gentle. We study the stability scattering diagrams of such gentle algebras and use them to prove that the Caldero–Chapoton map defines a biject...
Let $\boldsymbol{\Sigma}:=(\Sigma,\mathbb{M},\mathbb{P})$ be a marked surface with marked points on the boundary $\mathbb{M}\subset\partial\Sigma\neq\varnothing$, and punctures $\mathbb{P}\subset\Sigma\setminus\partial\Sigma$, and let $T$ be signature zero tagged triangulation of $\boldsymbol{\Sigma}$ in the sense of Fomin-Shapiro-Thurston. In this...
Let $(\Sigma,\mathbb{M},\mathbb{P})$ be a surface with marked points $\mathbb{M}\subseteq \partial\Sigma\neq\varnothing$ and punctures $\mathbb{P}\subseteq\Sigma\setminus\partial\Sigma$. In this paper we show that for every curve $\gamma$ on $\Sigma\setminus\mathbb{P}$, the curve obtained by resolving the kinks of $\gamma$ in any order is uniquely...
This is an extended abstract of my talk at the Oberwolfach Workshop "Representation Theory of Quivers and Finite-Dimensional Algebras" (February 12 - February 18, 2023 ). It is based on a joint work with R. Bennett-Tennenhaus (arXiv:2303.05326).
This is an extended abstract of my talk at the Oberwolfach Workshop ''Cluster Algebras and Related Topics'' (December 8 - 14, 2013). It is based on a joint work with A. Zelevinsky (arXiv:1306.3495).
We study semicontinuous maps on varieties of modules over finite-dimensional algebras. We prove that truncated Euler maps are upper or lower semicontinuous. This implies that $g$-vectors and $E$-invariants of modules are upper semicontinuous. We also discuss inequalities of generic values of some upper semicontinuous maps.
We interpret the Landau-Ginzburg potentials associated to Gross-Hacking-Keel-Kontsevich's partial compactifications of cluster varieties as F-polynomials of projective representations of Jacobian algebras. Along the way, we show that both the projective and the injective representations of Jacobi-finite quivers with potential are well-behaved under...
To each triangulation of any surface with marked points on the boundary and orbifold points of order three, we associate a quiver (with loops) with potential whose Jacobian algebra is finite dimensional and gentle. We study the stability scattering diagrams of such gentle algebras and use them to prove that the Caldero--Chapoton map defines a bijec...
![Bangle functions BT\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/356291726/figure/fig1/AS:1091358991360048@1637211668832/Bangle-functions-BTdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
![A curve (a,τj1,…,τj9,b)\documentclass[12pt]{minimal}...](publication/356291726/figure/fig4/AS:1091358995562503@1637211669076/A-curve-a-tj1-tj9-bdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
![A loop (a,τj1,…,τj8,τj1,a)\documentclass[12pt]{minimal}...](publication/356291726/figure/fig5/AS:1091358995554327@1637211669197/A-loop-a-tj1-tj8-tj1-adocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
We study the affine schemes of modules over gentle algebras. We describe the smooth points of these schemes, and we also analyze their irreducible components in detail. Several of our results generalize formerly known results, e.g. by dropping acyclicity, and by incorporating band modules. A special class of gentle algebras are Jacobian algebras ar...
Skew-gentle algebras are a generalisation of the well-known class of gentle algebras with which they share many common properties. In this work, using non-commutative Gröbner basis theory, we show that these algebras are strong Koszul and that the Koszul dual is again skew-gentle. We give a geometric model of their bounded derived categories in ter...
For algebras of global dimension 2 arising from a cut of the quiver with potential associated with a triangulation of an unpunctured surface, Amiot-Grimeland defined integer-valued functions on the first homology groups of the surfaces. Derived equivalences translate to the existence of automorphisms of surfaces preserving these functions. We gener...
We tackle the classification problem of non-degenerate potentials for quivers arising from triangulations of surfaces in the cases left open by Geiss-Labardini-Schr\"oer. Namely, for once-punctured closed surfaces of positive genus, we show that the quiver of any triangulation admits infinitely many non-degenerate potentials that are pairwise not w...
We realize Derksen-Weyman-Zelevinsky's mutations of representations as densely-defined regular maps on representation spaces, and study the generic values of Caldero-Chapoton functions with coefficients, giving, for instance, a sufficient combinatorial condition for their linear independence. For a quiver with potential $(Q,S)$, we show that if $k$...
Skew-gentle algebras are a generalisation of the well-known class of gentle algebras with which they share many common properties. In this work, using non-commutative Gr\"obner basis theory, we show that these algebras are Koszul and that the Koszul dual is again skew-gentle. We give a geometric model of their bounded derived categories in terms of...
We study the affine schemes of modules over gentle algebras. We describe the smooth points of these schemes, and we also analyze their irreducible components in detail. Several of our results generalize formerly known results, e.g. by dropping acyclicity, and by incorporating band modules. A special class of gentle algebras are Jacobian algebras ar...
![Leftσ\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/280104574/figure/fig5/AS:960034980761611@1605901584486/Leftsdocumentclass12ptminimal-usepackageamsmath-usepackagewasysym_Q320.jpg)
We present a definition of mutations of species with potential that can be
applied to the species realizations of any skew-symmetrizable matrix B over
cyclic Galois extensions E/F whose base field F has a primitive [E:F]-th root
of unity. After providing an example of a globally unfoldable
skew-symmetrizable matrix whose species realizations do not...
We realize a family of generalized cluster algebras as Caldero-Chapoton algebras of quivers with relations. Each member of this family arises from an unpunctured polygon with one orbifold point of order 2, and is realized as a Caldero-Chapoton algebra of a quiver with relations naturally associated to a special triangulation of the alluded polygon....
Let (\Sigma,M,O) be a surface with marked points and order-2 orbifold points which is either unpunctured or once-punctured closed, and let \omega be a function from O to {1,4}. For each triangulation \tau of (\Sigma,M,O) we construct a cochain complex C^\bullet(\tau,\omega) with coefficients in the field with two elements. A colored triangulation o...
For each algebra of global dimension 2 arising from the quiver with potential associated to a triangulation of an unpunctured surface, Amiot-Grimeland have defined an integer-valued function on the first singular homology group of the surface, and have proved that two such algebras of global dimension 2 are derived equivalent precisely when there e...
We show that the representation type of the Jacobian algebra P(Q,S)
associated to a 2-acyclic quiver Q with non-degenerate potential S is invariant
under QP-mutations. We prove that, apart from very few exceptions, P(Q,S) is of
tame representation type if and only if Q is of finite mutation type. We also
show that most quivers Q of finite mutation...
Motivated by the mutation theory of quivers with potentials developed by
Derksen-Weyman-Zelevinsky, and the representation-theoretic approach to cluster
algebras it provides, we propose a mutation theory of species with potentials
for species that arise from skew-symmetrizable matrices that admit a
skew-symmetrizer with pairwise coprime diagonal en...
In this survey article we give a brief account of constructions and results
concerning the quivers with potentials associated to triangulations of surfaces
with marked points. Besides the fact that the mutations of these quivers with
potentials are compatible with the flips of triangulations, we mention some
recent results on the representation typ...
Motivated by the representation theory of quivers with potentials introduced
by Derksen, Weyman and Zelevinsky and by work of Caldero and Chapoton, who gave
explicit formulae for the cluster variables of Dynkin quivers, we associate a
Caldero-Chapoton algebra to any (possibly infinite dimensional) basic algebra.
By definition, the Caldero-Chapoton...
We prove that the quivers with potentials associated to triangulations of
surfaces with marked points, and possibly empty boundary, are non-degenerate,
provided the underlying surface with marked points is not a closed sphere with
exactly 5 punctures. This is done by explicitly defining the QPs that
correspond to tagged triangulations and proving t...
Fomin-Zelevinsky conjectured that in any cluster algebra, the cluster
monomials are linearly independent and that the exchange graph and cluster
complex are independent of the choice of coefficients. We confirm these
conjectures for all skew-symmetric cluster algebras.
To 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 QP
associated to the resulting tagged triangulation is isomorphic to the Jacobian
algebra of the QP obtained by mutating the QP o...
This paper is a representation-theoretic extension of Part I. It has been inspired by three recent developments: surface cluster algebras studied by Fomin-Shapiro-Thurston, the mutation theory of quivers with potentials initiated by Derksen-Weyman-Zelevinsky, and string modules associated to arcs on unpunctured surfaces by Assem-Brustle-Charbonneau...
If a convex body C has modular and irreducible face lattice (and is not strictly convex), there is a face-preserving homeomorphism from C to a section of a cone of hermitian matrices over R, C, or H, or C has dimension 8, 14 or 26. 1. Introduction. Let C be a convex body in R n. A subset F of C is a face of C if every open interval in C that contai...
We attempt to relate two recent developments: cluster algebras associated to triangulations of surfaces by Fomin–Shapiro–Thurston,
and quivers with potentials (QPs) and their mutations introduced by Derksen–Weyman–Zelevinsky. To each ideal triangulation
of a bordered surface with marked points, we associate a QP, in such a way that whenever two ide...
Citations
... We note that Theorem 1.1 can be easily adapted to using the algebras H (b, c). In the case b = c = 2, the algebra H (b, c) coincides with a construction in [25] where the ordinary Caldero-Chapoton functions are shown to give cluster variables of a generalized cluster algebra. ...
... This point of view has been fruitful in providing new results on derived equivalences of gentle algebras [LP20,APS23,Opp,CJS,JSW]. Related geometric models have also been used to study gentle algebras [DRS12, CPS19, LZ21, BS21, BL22, CS23], Brauer graph algebras [OZ22] and skew-gentle algebras [AB22,LFSV22,HZZ23,Ami23]. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/272563066634304-1441995507957_Q64/Sibylle-Schroll.jpg)
![[object Object]](https://i1.rgstatic.net/ii/profile.image/282300752384002-1444317152383_Q64/Yadira-Valdivieso-Diaz.jpg)
... On the one hand, Wald and Waschbüsch [69] used the classification of modules over string algebras to obtain a similar classification 1 for the special-biserial algebras introduced by Skowroński and Waschbüsch [66]. Representations for other interesting classes of algebras have also been studied in terms of string algebras; see for example [14,17,22,26,46]. On the other hand, the gentle algebras introduced by Assem and Skowroński [2] form a particular class of better-behaved string algebras. ...
... For generalized cluster algebras, however, so far there have been only a couple of works providing such representation-theoretic expressions for cluster variables, cf. [37,47]. ...
... We remark that the recursion in Proposition 1.5 has already been achieved in the skewsymmetric case for any reflection, not necessarily at sink or source, of any quiver by Derksen-Weyman-Zelevinsky [9,10]. Extending their theory, especially obtaining Caldero-Chapoton type formulas, to the skew-symmetrizable case in full generality remains an open problem; see for example [1,7,22,23,27,28]. ...
... On the other hand, we also give a geometric description of the notion of admissible cuts of [12], which complete the previous description of [22,1]. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/273872620617728-1442307729057_Q64/Claire-Amiot.jpg)
... We remark that the recursion in Proposition 1.5 has already been achieved in the skewsymmetric case for any reflection, not necessarily at sink or source, of any quiver by Derksen-Weyman-Zelevinsky [9,10]. Extending their theory, especially obtaining Caldero-Chapoton type formulas, to the skew-symmetrizable case in full generality remains an open problem; see for example [1,7,22,23,27,28]. ...
... We may assume that m i (P 0 ) = m i (Q 0 ) ≥ 0. Then π ′ π(P 0 ) = {P 0 } and π ′ π(Q 0 ) = {Q 0 } by Theorem 9.22. By Lemma 11.4, we have (30) R∈π ′ π(P 0 ) y T (R) ...
... Note that 2-Calabi-Yau tilted algebra arising from marked surfaces are tame algebras (cf. [GLaS16]) and it is clear that skew-gentle algebras are tame algebras. According to [PYK23], tame algebras are g-tame. ...
Reference: τ -TILTING GRAPHS AND QUOTIENTS
... We remark that the recursion in Proposition 1.5 has already been achieved in the skewsymmetric case for any reflection, not necessarily at sink or source, of any quiver by Derksen-Weyman-Zelevinsky [9,10]. Extending their theory, especially obtaining Caldero-Chapoton type formulas, to the skew-symmetrizable case in full generality remains an open problem; see for example [1,7,22,23,27,28]. ...
