Sergey Fomin, Andrei Zelevinsky
In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.
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iQuantum groups and iHopf algebras I: foundation
本文引入i霍普夫代数概念，将Borel量子群上的i霍普夫代数实现为拟分裂通用i量子群，建立了Lusztig辫群作用与i辫群作用的联系，为后续构造对偶典范基奠定基础。
General Presentations of Algebras and Foundations of $τ$-tilting Theory
解释了阿达奇-井山-赖滕“τ-倾斜理论”主要结果与德克森-费“代数的一般表示”第5节结果的关联。
We introduce and start investigating the properties of countably infinite, periodic chains of finite dimensional generalizations of the exceptional Lie algebras: each exceptional Lie algebra (but $\mathbf{g}_{2}$) is part of an infinite family of finite dimensional algebras, which we name "Magic Star" algebras. These algebras have remarkable similarities with many characterizing features of the exceptional Lie algebras.
We provide a graded and quantum version of the category of rooted cluster algebras introduced by Assem, Dupont and Schiffler and show that every graded quantum cluster algebra of infinite rank can be written as a colimit of graded quantum cluster algebras of finite rank. As an application, for each k we construct a graded quantum infinite Grassmannian admitting a cluster algebra structure, extending an earlier construction of the authors for k=2.
We construct quantized versions of generic bases in quantum cluster algebras of finite and affine types. Under the specialization of $q$ and coefficients to 1, these bases are generic bases of finite and affine cluster algebras.
We show that upper cluster algebras need not be finitely generated, answering a question of Berenstein, Fomin and Zelevinsky. Our counter-example is a cluster algebra with B-matrix $\begin{pmatrix} 0 & 3 & -3 \\ -3 & 0 & 3 \\ 3 & -3 & 0 \end{pmatrix}$ and coefficients obeying a genericity condition.
We study monoidal categorifications of certain monoidal subcategories $\mathcal{C}_J$ of finite-dimensional modules over quantum affine algebras, whose cluster algebra structures coincide and arise from the category of finite-dimensional modules over quiver Hecke algebra of type A${}_\infty$. In particular, when the quantum affine algebra is of type A or B, the subcategory coincides with the monoidal category $\mathcal{C}_{\mathfrak{g}}^0$ introduced by Hernandez-Leclerc. As a consequence, the modules corresponding to cluster monomials are real simple modules over quantum affine algebras.
We give a construction of generalized cluster varieties and generalized cluster scattering diagrams for reciprocal generalized cluster algebras, the latter of which were defined by Chekhov and Shapiro. These constructions are analogous to the structures given for ordinary cluster algebras in the work of Gross, Hacking, Keel, and Kontsevich. As a consequence of these constructions, we are also able to construct theta functions for generalized cluster algebras, again in the reciprocal case, and demonstrate a number of their structural properties.
Generalized quantum cluster algebras introduced in [1] are quantum deformation of generalized cluster algebras of geometric types. In this paper, we prove that the Laurent phenomenon holds in these generalized quantum cluster algebras. We also show that upper bounds coincide with the corresponding generalized quantum upper cluster algebras under the "coprimality" condition.
Motivated by the phenomenon that compatible Poisson structures on a cluster algebra play a key role on its quantization (that is, quantum cluster algebra), we introduce the second quantization of a quantum cluster algebra, which means the correspondence between compatible Poisson structures of the quantum cluster algebra and its secondly quantized cluster algebras. Based on this observation, we find that a quantum cluster algebra possesses dual quantum cluster algebras such that their second quantization is essentially the same. As an example, we give the secondly quantized cluster algebra $A_…
We study the cluster automorphism group of a skew-symmetric cluster algebra with geometric coefficients. For this, we introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automorphism group of a gluing free cluster algebra is a subgroup of the cluster automorphism group of its principal part cluster algebra (i.e. the corresponding cluster algebra without coefficients). We show that several classes of cluster algebras with coefficients are gluing free, for example, cluster algebras with principal coefficients, cluster algebras with universal geom…
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.
A quantum symmetric pair consists of a quantum group $\mathbf U$ and its coideal subalgebra ${\mathbf U}^{\imath}_{\boldsymbolς}$ with parameters $\boldsymbolς$ (called an $\imath$quantum group). We initiate a Hall algebra approach for the categorification of $\imath$quantum groups. A universal $\imath$quantum group $\widetilde{\mathbf U}^{\imath}$ is introduced and ${\mathbf U}^{\imath}_{\boldsymbolς}$ is recovered by a central reduction of $\widetilde{\mathbf U}^{\imath}$. The semi-derived Ringel-Hall algebras of the first author and Peng, which are closely related to semi-derived Hall algeb…
Let $A$ be the path algebra of a finite acyclic quiver $Q$ over a finite field. We realize the quantum cluster algebra with principal coefficients associated to $Q$ as a sub-quotient of a certain Hall algebra involving the category of morphisms between projective $A$-modules.
We study deformations of cluster algebras with several quantum parameters, called toroidal cluster algebras, which naturally appear in the study of Grothendieck rings of representations of quantum affine algebras. In this context, we construct toroidal Grothendieck rings and we establish these are flat deformations of Grothendieck rings. We prove that for a family of monoidal categories $\mathscr{C}_1$ of simply-laced quantum affine algebras categorifying finite-type cluster algebras, the toroidal Grothendieck ring has a natural structure of a toroidal cluster algebra.
We survey some recent constructions of cluster algebra structures on coordinate rings of unipotent subgroups and unipotent cells of Kac-Moody groups. We also review a quantized version of these results.
Let C be the category of finite-dimensional representations of a quantum affine algebra of simply-laced type. We introduce certain monoidal subcategories C_l (l integer) of C and we study their Grothendieck rings using cluster algebras.
Let $Q$ be a finite acyclic valued quiver. We give the cluster multiplication formulas in the quantum cluster algebra of $Q$ with arbitrary coefficients, by applying certain quotients of derived Hall subalgebras of $Q$. These formulas can be viewed as the quantum version of the cluster multiplication theorem in the classical cluster algebra proved by Caldero-Keller for finite type, Hubery for affine type and Xiao-Xu for acyclic quivers.
The objective of the present paper is to prove cluster multiplication theorem in the quantum cluster algebra of type $A_{2}^{(2)}$. As corollaries, we obtain bar-invariant $\mathbb{Z}[q^{\pm\frac{1}{2}}]$-bases established in [6], and naturally deduce the positivity of the elements in these bases. One bar-invariant basis as the triangular basis of this quantum cluster algebra is also explicitly described.
A cluster tilted algebra is known to be gentle if and only if it is cluster tilted of Dynkin type $\bbA$ or Euclidean type $\tilde{\bbA}$. We classify all finite dimensional algebras which are derived equivalent to gentle cluster tilted algebras.
It is well known that the relation-extensions of tilted algebras are cluster-tilted algebras. In this paper, we extend the result to silted algebras and prove some extension of silted algebras are cluster-tilted algebras.
This is an introductory survey on cluster algebras and their (additive) categorification using derived categories of Ginzburg algebras. After a gentle introduction to cluster combinatorics, we review important examples of coordinate rings admitting a cluster algebra structure. We then present the general definition of a cluster algebra and describe the interplay between cluster variables, coefficients, c-vectors and g-vectors. We show how c-vectors appear in the study of quantum cluster algebras and their links to the quantum dilogarithm. We then present the framework of additive categorificat…
In \cite{rupel3},the authors defined algebra homomorphisms from the dual Ringel-Hall algebra of certain hereditary abelian category $\mathcal{A}$ to an appropriate $q$-polynomial algebra. In the case that $\mathcal{A}$ is the representation category of an acyclic quiver, we give an alternative proof by using the cluster multiplication formulas in \cite{DX}. Moreover, if the underlying graph of $Q$ is bipartite and the matrix $B$ associated to the quiver $Q$ is of full rank, we show that the image of the algebra homomorphisms is in the corresponding quantum cluster algebra.
We give an introduction to the theory of cluster categories and cluster tilted algebras. We include some background on the theory of cluster algebras, and discuss the interplay with cluster categories and cluster tilted algebras.
Cluster algebras generated by projective cluster variables
该研究提出由投射丛变量生成的下界丛代数概念，证明无环条件下其与丛代数一致并可构造丛代数的基，且$A_n$、$\tilde{A}_n$型无系数丛代数均等于对应下界丛代数。
We show that in case a cluster algebra coincides with its upper cluster algebra and the cluster algebra admits a grading with finite dimensional homogeneous components, the corresponding Berenstein-Zelevinsky quantum cluster algebra can be viewed as a flat deformation of the classical cluster algebra.
Cluster-tilted algebras are trivial extensions of tilted algebras. This correspondence induces a surjective map from tilted algebras to cluster-tilted algebras. If B is a cluster-tilted algebra, we use the fibre of B under this map to study the module category of B. In particular, we introduce the notion of reflections of tilted algebras and define an algorithm that constructs the transjective component of the Auslander-Reiten quiver of cluster-tilted algebras of tree type.
We provide a technique to find a cluster-tilting object having a given cluster-tilted algebra as endomorphism ring in the finite type case.
We give a complete description of all special biserial cluster-tilted algebras over a finite dimensional hereditary algebra H over an algebraically closed field K.
Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. To a cluster algebra of simply laced Dynkin type one can associate the cluster category. Any cluster of the cluster algebra corresponds to a tilting object in the cluster category. The cluster tilted algebra is the algebra of endomorphisms of that tilting object. Viewing the cluster tilted algebra as a path algebra of a quiver with relations, we prove in this paper that the quiver of the cluster tilted algebra is equal to the cluster diagram. We study also the relations. As an application of…
We introduce admissible group actions on cluster algebras, cluster categories and quivers with potential and study the resulting orbit spaces. The orbit space of the cluster algebra has the structure of a generalized cluster algebra. This generalized cluster structure is different from those introduced by Chekhov-Shapiro and Lam-Pylyavskyy. For group actions on cluster algebras from surfaces, we describe the generalized cluster structure of the orbit space in terms of a triangulated orbifold. In this case, we give a complete list of exchange polynomials, and we classify the algebras of rank 1 …
A cluster algebra is unistructural if the set of its cluster variables determines its clusters and seeds. It is conjectured that all cluster algebras are unistructural. In this paper, we show that any cluster algebra arising from a triangulation of a marked surface without punctures is unistructural. Our proof relies on the existence of a positive basis known as the bracelet basis, and on the skein relations. We also prove that a cluster algebra defined from a disjoint union of quivers is unistructural if and only if the cluster algebras defined from the connected components of the quiver are …
In this paper the relationship between iterated tilted algebras and cluster-tilted algebras and relation-extensions is studied. In the Dynkin case, it is shown that the relationship is very strong and combinatorial.
The cluster-tilted algebras have been introduced by Buan, Marsh and Reiten, they are the endomorphism rings of cluster-tilting objects $T$ in cluster categories; we call such an algebra cluster-concealed in case $T$ is obtained from a preprojective tilting module. For example, all representation-finite cluster-tilted algebras are cluster-concealed. If $C$ is a representation-finite cluster-tilted algebra, then the indecomposable $C$-modules are shown to be determined by their dimension vectors. For a general cluster-tilted algebra $C$, we are going to describe the dimension vectors of the inde…
We are going to show that the representation dimension of a cluster-concealed algebra B is 3. We compute its representation dimension by showing an explicit Auslander generator for the cluster-tilted algebra.
We introduce and study a class of Iwanaga-Gorenstein algebras defined via quivers with relations associated with symmetrizable Cartan matrices. These algebras generalize the path algebras of quivers associated with symmetric Cartan matrices. We also define a corresponding class of generalized preprojective algebras. Without any assumption on the ground field, we obtain new representation-theoretic realizations of all finite root systems.
We compute the Hochschild cohomology groups of the cluster-tilted algebras of finite representation type.
We introduce a class of non-commutative algebras that carry a non-commutative (geometric) cluster structure which are generated by identical copies of generalized Weyl algebras. Equivalent conditions for the finiteness of the set of the cluster variables of these cluster structures are provided. Some combinatorial data, called \textit{cluster strands,} arising from the cluster structure are used to construct irreducible representations of generalized Weyl algebras.
We give a characterization of $n$-cluster tilting subcategories of representation-directed algebras based on the $n$-Auslander-Reiten translations. As an application we classify acyclic Nakayama algebras with homogeneous relations which admit an $n$-cluster tilting subcategory. Finally, we classify Nakayama algebras of global dimension $d<\infty$ which admit a $d$-cluster tilting subcategory.
It is conjectured by Ibrahim Assem, Ralf Schiffler and Vasilisa Shramchenko in "Cluster Automorphisms and Compatibility of Cluster Variables" that every cluster algebra is unistructural, that is to say, that the set of cluster variables determines uniquely the cluster algebra structure. In other words, there exists a unique decomposition of the set of cluster variables into clusters. This conjecture has been proven to hold true for algebras of type Dynkin or rank 2 by Assem, Schiffler and Shramchenko. The aim of this paper is to prove it for algebras of type ${\widetilde{\mathbb{A}}}$. We use …
A cluster automorphism is a $\mathbb{Z}$-algebra automorphism of a cluster algebra $\mathcal A$ satisfying that it sends a cluster to another and commutes with mutations. Chang and Schiffler conjectured that a cluster automorphism of $\mathcal A$ is just a $\mathbb{Z}$-algebra homomorphism of a cluster algebra sending a cluster to another. The aim of this article is to prove this conjecture.
In this work we study the connection between iterated tilted algebras and m-cluster tilted algebras. We show that an iterated tilted algebra induces an m-cluster tilted algebra. This m-cluster tilted algebra can be seen as a trivial extension of another iterated tilted algebra which is derived equivalent to the original one. We give a procedure to find this new iterated tilted algebra. These m-cluster tilted algebras are quotients of higher relation extensions.
We classify all finite dimensional algebras which are derived equivalent to m-cluster tilted algebras of type A.
In this paper, we characterize all the finite dimensional algebras that are m-cluster tilted algebras of type A tilde. We show that these algebras are gentle and we give an explicit description of their quivers with relations.
A cluster is a finite set of generators of a cluster algebra. The Laurent Phenomenon of Fomin and Zelevinsky says that any element of a cluster algebra can be written as a Laurent polynomial in terms of any cluster. The upper cluster algebra of a cluster algebra is the ring of rational functions that can be written as a Laurent polynomial in every cluster of the cluster algebra. By the Laurent phenomenon a cluster algebra is always contained in its upper cluster algebra, but they are not always equal. In 2014 it was conjectured that the equality of the cluster algebra and upper cluster algebra…
We show that the cluster complex of an arbitrary hereditary artin algebra has the structure of an abstract simplicial polytope. In particular, the cluster-tilting objects form one equivalence class under mutation.
Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be recovered from the data of the corresponding quiver representation category. This also provides some explicit formulas for cluster variables.
We show that for cluster algebras associated with finite quivers without oriented cycles (with no coefficients), a seed is determined by its cluster, as conjectured by Fomin and Zelevinsky.We also obtain an interpretation of the monomial in the denominator of a non-polynomial cluster variable in terms of the composition factors of an indecomposable exceptional module over an associated hereditary algebra.
We formalize the way in which one can think about cluster algebras of infinite rank by showing that every rooted cluster algebra of infinite rank can be written as a colimit of rooted cluster algebras of finite rank. Relying on the proof of the posivity conjecture for skew-symmetric cluster algebras (of finite rank) by Lee and Schiffler, it follows as a direct consequence that the positivity conjecture holds for cluster algebras of infinite rank. Furthermore, we give a sufficient and necessary condition for a ring homomorphism between cluster algebras to give rise to a rooted cluster morphism …
We construct the intermediate coverings of cluster-tilted algebras by defining the generalized cluster categories. These generalized cluster categories are Calabi-Yau triangulated categories with fraction CY-dimension and have also cluster tilting objects (subcategories). Furthermore we study the representations of these intermediate coverings of cluster-tilted algebras.
In the theory of generalized cluster algebras, we build the so-called cluster formula and $D$-matrix pattern. Then as applications, some fundamental conjectures of generalized cluster algebras are solved affirmatively.
We study the module category of a certain Galois covering of a cluster-tilted algebra which we call the cluster repetitive algebra. Our main result compares the module categories of the cluster repetitive algebra of a tilted algebra C and the repetitive algebra of C, in the sense of Hughes and Waschbuesch.
In this paper, we characterize all the finite dimensional algebras that are derived equivalent to an m-cluster tilted algebra of type A tilde. This generalizes a result of Bobonski and Buan [9].
Non-singular weighted surface algebras satisfy the necessary condition found in [6] for existence of cluster tilting modules. We show that any such algebra whose Gabriel quiver is bipartite, has a module satisfying the necessary ext vanishing condition. We show that it is 3-cluster tilting precisely for the non-singular triangular or spherical algebras but not for any other weighted surface algebra with bipartite Gabriel quiver.
The article concerns the subalgebra U_v^+(w) of the quantized universal enveloping algebra of the complex Lie algebra sl_{n+1} associated with a particular Weyl group element of length 2n. We verify that U_v^+(w) can be endowed with the structure of a quantum cluster algebra of type A_n. The quantum cluster algebra is a deformation of the ordinary cluster algebra Geiss-Leclerc-Schroeer attached to w using the representation theory of the preprojective algebra. Furthermore, we prove that the quantum cluster variables are, up to a power of v, elements in the dual of Lusztig's canonical basis und…
We show how a cluster-tilted algebra of finite representation type is related to the corresponding tilted algebra, in the case of algebras defined over an algebraically closed field.
This is a concise introduction to Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers in the acyclic case. We review the definition of cluster algebras (geometric, without coefficients), construct the cluster category and present the bijection between cluster variables and rigid indecomposable objects of the cluster category.
The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra A of finite type can be realized as a Hall algebra, called the exceptional Hall algebra, of the cluster category. This realization provides a natural basis for A. We prove new results and formulate conjectures on `good basis' properties, positivity, denominator theorems and toric degenerations.
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 use the maximal faces of the $m$-cluster complex of type A to describe the m-cluster tilted algebras of type A as quivers with relations. We then classify connected components of m-cluster tilted algebras of type A up to derived equivalence using tilting complexes directly related to the combinatorics of the m-cluster complex of type A. This generalizes a result of Buan and Vatne.
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