We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras.
Furthermore, we obtain direct formulas for F-polynomials and g-vectors and show that F-polynomials have constant term equal to 1. As an application, we compute the Euler-Poincaré characteristic of quiver Grassmannians in Dynkin type $A$ and affine Dynkin type $\tilde A$.
There are several examples in which algebraic properties of Jacobian algebras from (unpunctured) Riemann surfaces can be computed from the geometry of the Riemann surface. In this work, we compute the dimension of the Hochschild cohomology groups of any Jacobian algebra from unpunctured Riemann surfaces. In those expressions appear geometric objects of the triangulated surface, namely: the number of internal triangles and certain types of boundaries. Moreover, we give geometric conditions on the triangulated surface $(S,M,\mathbb T)$ such that the Gerstenhaber algebra $\operatorname{HH}^*(A_{\…
In a previous paper, the author compute the dimension of Hochschild cohomology groups of Jacobian algebras from (unpunctured) triangulated surfaces, and gave a geometric interpretation of those numbers in terms of the number of internal triangles, the number of vertices and the existence of certain kind of boundaries. The aim of this note is computing the cyclic (co)homology and the Hochschild homology of the same family of algebras and giving an interpretation of those dimensions through elements of the triangulated surface.
We give the quantum Laurent expansion formula for the quantum cluster algebras from unpunctured orbifolds with arbitrary coefficients and quantization. As an application, positivity for such class of quantum cluster algebras is given. For technical reasons, it will always be assumed that the weights of the orbifold points are 2.
We study quantum cluster algebras from unpunctured surfaces with arbitrary coefficients and quantization. We first give a new proof of the Laurent expansion formulas for commutative cluster algebras from unpunctured surfaces, we then give the quantum Laurent expansion formulas for the quantum cluster algebras. Particularly, this gives a combinatorial proof of the positivity for such class of quantum cluster algebras.
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 exists an automorphism of the surface that makes their associated functions coincide. In the present paper we generalize the constructions and results of Amiot-Grimeland to the setting of arbitrarily punctured surfaces. As an application, we show that…
We extend a T-path expansion formula for arcs on an unpunctured surface to the case of arcs on a once-punctured polygon and use this formula to give a combinatorial proof that cluster monomials form the atomic basis of a cluster algebra of type D.
In this article, gentle algebras are realised as tiling algebras, which are associated to partial triangulations of unpunctured surfaces with marked points on the boundary. This notion of tiling algebras generalise the notion of Jacobian algebras of triangulations of surfaces and the notion of surface algebras. We use this description to give a geometric model of the module category of any gentle algebra.
We introduce several commutative rings, the snake rings, that have strong connections to cluster algebras. The elements of these rings are residue classes of unions of certain labeled graphs that were used to construct canonical bases in the theory of cluster algebras. We obtain several rings by varying the conditions on the structure as well as the labelling of the graphs. The most restrictive form of this ring is isomorphic to the ring $\mathbb{Z}[x,y]$ of polynomials in two variables over the integers. A more general form contains all cluster algebras of unpunctured surface type. The defini…
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 arising from triangulations of unpunctured marked surfaces. For these we obtain a bijection between the set of generically tau-reduced decorated irreducible components and the set of laminations of the surface. As an application, we get that the set of…
In the context of representation theory of finite dimensional algebras, string algebras have been extensively studied and most aspects of their representation theory are well-understood. One exception to this is the classification of extensions between indecomposable modules. In this paper we explicitly describe such extensions for a class of string algebras, namely gentle algebras associated to surface triangulations. These algebras arise as Jacobian algebras of unpunctured surfaces. We relate the extension spaces of indecomposable modules to crossings of arcs in the surface and give explicit…
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 3, and is realized as a Caldero-Chapoton algebra of a quiver with relations naturally associated to any triangulation of the alluded polygon. The realization is done by defining for every arc $j$ on the polygon with orbifold point a representation $M(j)$ of the referred quiver with relations, and by proving that for every triangulation $τ$ and every arc $j\inτ$, the product of the Caldero-Chapo…
In this paper we study the derived equivalences between surface algebras, introduced by David-Roesler and Schiffler. Each surface algebra arises from a cut of an ideal triangulation of an unpunctured marked Riemann surface with boundary. A cut can be regarded as a grading on the Jacobian algebra of the quiver with potential (Q,W) associated with the triangulation. Fixing a set $ε$ of generators of the fundamental group of the surface, we associate to any cut $d$ a weight $w^ε(d)\in\mathbb Z^{2g+b}$, where $g$ is the genus of $S$ and $b$ the number of boundary components. The main result of the…
We study cluster algebras arising from cluster tubes. We obtain categorical interpretations for $g$-vectors, $c$-vectors and denominator vectors for cluster algebras of type $\mathrm{C}$ with respect to arbitrary initial seeds. In particular, a denominator theorem has been proved, which enables us to establish the linearly independence of denominator vectors of cluster variables from the same cluster for cluster algebras of type $\mathrm{A}\mathrm{B}\mathrm{C}$. This strengthens the link between cluster tubes and cluster algebras of type $\mathrm{C}$ initiated by Buan, Marsh and Vatne.
We give a construction that takes a simple linear algebraic group $G$ over a field and produces a commutative, unital, and simple non-associative algebra $A$ over that field. Two attractions of this construction are that (1) when $G$ has type $E_8$, the algebra $A$ is obtained by adjoining a unit to the 3875-dimensional representation and (2) it is effective, in that the product operation on $A$ can be implemented on a computer. A description of the algebra in the $E_8$ case has been requested for some time, and interest has been increased by the recent proof that $E_8$ is the full automorphis…
The Grothendieck groups of the categories of finitely generated modules and finitely generated projective modules over a tower of algebras can be endowed with (co)algebra structures that, in many cases of interest, give rise to a dual pair of Hopf algebras. Moreover, given a dual pair of Hopf algebras, one can construct an algebra called the Heisenberg double, which is a generalization of the classical Heisenberg algebra. The aim of this paper is to study Heisenberg doubles arising from towers of algebras in this manner. First, we develop the basic representation theory of such Heisenberg doub…
We give a complete classification of all algebras appearing as endomorphism algebras of maximal rigid objects in standard 2-Calabi-Yau categories of finite type. Such categories are equivalent to certain orbit categories of derived categories of Dynkin algebras. It turns out that with one exception, all the algebras that occur are $2$-Calabi-Yau-tilted, and therefore appear in an earlier classification by Bertani-Økland and Oppermann. We explain this phenomenon by investigating the subcategories generated by rigid objects in standard 2-Calabi-Yau categories of finite type.
We characterize Y/T-system type difference equations arising from cluster algebras by triples of matrices, which we call T-data, that have a certain symplectic property. We show that all mutation loops are essentially obtained from T-data, which generalizes the general solution for period 1 quivers given by Fordy and Marsh. We also show that any T-datum associated with a periodic Y/T-system has the simultaneous positivity. As an application, we propose a version of Nahm's conjecture from a viewpoint of cluster algebras. We conjecture that given a periodic T/Y-system of a certain type, we have …
We describe poles and the corresponding residual automorphic representations of Eisenstein series attached to maximal parabolic subgroups whose unipotent radicals admit Jordan algebra structure.
本文在Artin代数的背景下从不同的角度研究了Morita环$Λ_{(ϕ,ψ)}=\bigl({smallmatrix} A &_AN_B_BM_A & B {smallmatrix}\bigr)$.首先研究了Morita环的双模同态$ϕ$和$ψ$为零时模范畴的协变有限、反变有限和函有限子范畴。进一步，当$ϕ$和$ψ$均为零时，我们利用$A$和$B$的整体维数给出了Morita环$Λ_{(0,0)}$（作为Artin代数）的整体维数的界.我们用一些例子来说明我们的界限。最后我们研究了Morita环是GorensteinArtin代数的条件，并利用$A=N=M=B=Λ$确定了Morita环上的所有Gorenstein-投射模，其中$Λ$是Artin代数.
We develop new techniques to classify basic algebras of blocks of finite groups over algebraically closed fields of prime characteristic. We apply these techniques to simplify and extend previous classifications by Linckelmann, Murphy and Sambale. In particular, we fully classify blocks with 16-dimensional basic algebra.
Poset topology and homological invariants of algebras arising in algebraic combinatorics
我们提出了一个美丽的组合拓扑和同调代数之间的相互作用的一类幺半群，自然出现在代数组合。我们探索了这种相互作用的几个应用。例如，我们用非交换代数给出了团复形的Leray数的一个新的解释。简历我们提出了一个巨大的相互作用，其中包括组合拓扑和一个在组合代数中具有自然特征的单元类的代数同系物。我们探索这种相互作用的多种应用。同样，我们引入了一种新的Leray d'un déctique nombre的解释，它是关于一个非对易代数的全局维数的。
We give a simple description of the closure of the nilpotent orbits appearing as associated varieties of admissible affine vertex algebras in terms of primitive ideals.
Let $G$ be a complex, connected, reductive, algebraic group, and $χ:\mathbb{C}^\times \to G$ be a fixed cocharacter that defines a grading on $\mathfrak{g}$, the Lie algebra of $G$. Let $G_0$ be the centralizer of $χ(\mathbb{C}^\times)$. In this paper, we study $G_0$-equivariant parity sheaves on $\mathfrak{g}_n$, under some assumptions on the field $\Bbbk$ and the group $G$. The assumption on $G$ holds for $GL_n$ and for any $G$, it recovers results of Lusztig in characteristic $0$. The main result is that every parity sheaf occurs as a direct summand of the parabolic induction of some cuspid…
We define two classes of representations of quivers over arbitrary fields, called monomorphic representations and epimorphic representations. We show that every representation has a unique maximal nilpotent subrepresentation and the associated quotient is always monomorphic, and every representation has a unique maximal epimorphic subrepresentation and the associated quotient is always nilpotent. The uniquenesses of such subrepresenations imply two identities in the Ringel-Hall algebra. By applying Reineke's integration map, we obtain two identities in the corresponding quantum torus.
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 bijection between reachable $τ$-rigid pairs and cluster monomials of the generalized cluster algebra associated to the surface by Chekhov and Shapiro.
