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In the present paper, we first give a characterization for Bongartz completion in |$\tau $| -tilting theory via |$c$| -vectors. Motivated by this characterization, we give the definition of Bongartz completion in cluster algebras using |$c$| -vectors. Then we prove the existence and uniqueness of Bongartz completion in cluster algebras. We also prove that Bongartz completion admits certain commutativity. We give two applications for Bongartz completion in cluster algebras. As the first application, we prove the full subquiver of the exchange quiver (or known as oriented exchange graph) of a cluster algebra |$\mathcal A$| whose vertices consist of the seeds of |$\mathcal A$| containing particular cluster variables is isomorphic to the exchange quiver of another cluster algebra. As the second application, we prove that in a |$Y$| -pattern over a universal semifield, each |$Y$| -seed (up to a |$Y$| -seed equivalence) is uniquely determined by the negative |$y$| -variables in this |$Y$| -seed. [ABSTRACT FROM AUTHOR]

We show that associated with any |$n$| -dimensional Bott–Samelson variety of a complex semi-simple Lie group |$G$|⁠ , one has |$2^n$| Poisson brackets on the polynomial algebra |$A={\mathbb{C}}[z_1, \ldots , z_n]$|⁠ , each an iterated Poisson Ore extension and one of them a symmetric Poisson Cauchon–Goodearl–Letzter (CGL) extension in the sense of Goodearl–Yakimov. We express the Poisson brackets in terms of root strings and structure constants of the Lie algebra of |$G$|⁠. It follows that the coordinate rings of all generalized Bruhat cells have presentations as symmetric Poisson CGL extensions. The paper establishes the foundation on generalized Bruhat cells and sets the stage for their applications to integrable systems, cluster algebras, total positivity, and toric degenerations of Poisson varieties, some of which are discussed in the Introduction. [ABSTRACT FROM AUTHOR]

Let |$\textrm{coh}\ \mathbb{X}$| be the category of coherent sheaves over a weighted projective line |$\mathbb{X}$| and let |$D^b(\textrm{coh}\ \mathbb{X})$| be its bounded derived category. The present paper focuses on the study of the right and left mutation functors arising in |$D^b(\textrm{coh}\ \mathbb{X})$| attached to certain line bundles. As applications, we first show that these mutation functors give rise to simple reflections for the Weyl group of the star-shaped quiver |${Q}$| associated with |$\mathbb{X}$|⁠. By further dealing with the Ringel–Hall algebra of |$\mathbb{X}$|⁠ , we show that these functors provide a realization for Tits' automorphisms of the Kac–Moody algebra |${\mathfrak g}_{Q}$| associated with |${Q}$|⁠ , as well as for Lusztig's symmetries of the quantum enveloping algebra of |${\mathfrak g}_{Q}$|⁠. [ABSTRACT FROM AUTHOR]

We develop a general approach to finding combinatorial models for cluster algebras. The approach is to construct a labeled graph called a framework. When a framework is constructed with certain properties, the result is a model incorporating information about exchange matrices, principal coefficients, g-vectors, and g-vector fans. The idea behind frameworks arises from Cambrian combinatorics and sortable elements, and in this paper, we use sortable elements to construct a framework for any cluster algebra with an acyclic initial exchange matrix. This Cambrian framework yields a model of the entire exchange graph when the cluster algebra is of finite type. Outside of finite type, the Cambrian framework models only part of the exchange graph. In a forthcoming paper, we extend the Cambrian construction to produce a complete framework for a cluster algebra whose associated Cartan matrix is of affine type. [ABSTRACT FROM AUTHOR]

We first study a new family of graded quiver varieties together with a new t-deformation of the associated Grothendieck rings. This provides the geometric foundations for a joint paper by Yoshiyuki Kimura and the author. We further generalize the result of that paper to any acyclic quantum cluster algebra with arbitrary nondegenerate coefficients. In particular, we obtain the generic basis, the dual Poincaré–Birkhoff–Witt basis, and the dual canonical basis. The method consists in a correction technique, which works for general quantum cluster algebras. [ABSTRACT FROM PUBLISHER]

This paper concerns cluster algebras with principal coefficients associated to bordered surfaces (S,M), and is a companion to a concurrent work of the authors with Schiffler [23]. Given any (generalized) arc or loop in the surface—with or without self-intersections—we associate an element of (the fraction field of) , using products of elements of PSL2. We prove an abstract combinatorial result which gives a formula for the number of matchings of a snake or band graph in terms of an appropriate product of 2×2 matrices. We then use this formula to prove that our matrix formulas for arcs and loops agree with the combinatorial formulas for arcs and loops in terms of matchings, which were given in [22, 23]. Finally, we use our matrix formulas to prove skein relations for the cluster algebra elements associated to arcs and loops. Our matrix formulas and skein relations generalize prior work of Fock and Goncharov [13, 12, 14], who worked in the coefficient-free case. The results of this paper will be used in [23] in order to show that certain collections of arcs and loops comprise a vector-space basis for . [ABSTRACT FROM PUBLISHER]

The aim of this paper is two-fold: (1) to introduce four systems of equations called M-systems and dual M-systems of types An and Bn, respectively; (2) to make a connection between M-systems (dual M-systems) and cluster algebras, and prove that the Hernandez-Leclerc conjecture is true for minimal affinizations of types An and Bn. [ABSTRACT FROM AUTHOR]

This paper develops techniques for producing presentations of upper cluster algebras. These techniques are suited to computer implementation, and will always succeed when the upper cluster algebra is totally coprime and finitely generated. We include several examples of presentations produced by these methods. [ABSTRACT FROM AUTHOR]

It is known that the set of irreducible components of nilpotent varieties provides a geometric realization of the crystal basis for quantum groups. For each reduced expression of a Weyl group element, Geiß, Leclerc, and Schröer have recently given a parametrization of irreducible components of nilpotent varieties in studying cluster algebras. In this paper, we show that their parametrization coincides with Lusztig's parametrization of the canonical basis. [ABSTRACT FROM PUBLISHER]

We define a new generalized class of cluster-type mutations for which exchange transformations are given by reciprocal polynomials. In the case of second-order polynomials of the form these transformations are related to triangulations of Riemann surfaces of arbitrary genus with at least one hole/puncture and with an arbitrary number of orbifold points of arbitrary integer orders no. In the second part of the paper, we propose the dual graph description of the corresponding Teichmüller spaces, construct the Poisson algebra of the Teichmüller space coordinates, propose the combinatorial description of the corresponding geodesic functions and find the mapping class group transformations thus providing the complete description of the above Teichmüller spaces. [ABSTRACT FROM AUTHOR]

Q-systems are recursion relations satisfied by the characters of the restrictions of special finite-dimensional modules of quantum affine algebras. They can also be viewed as mutations in certain cluster algebras, which have a natural quantum deformation. In this paper, we explain the relation in the simply laced case between the resulting quantum Q-systems and the graded tensor product of Feigin and Loktev. We prove the graded version of the M=N identities, and write expressions for these as noncommuting evaluated multiresidues of suitable products of solutions of the quantum Q-system. This leads to a simple reformulation of Feigin and Loktev’s fusion coefficients as matrix elements in a representation of the quantum Q-system algebra. [ABSTRACT FROM AUTHOR]

Cluster categories of hereditary algebras have been introduced as orbit categories of their derived categories. Keller has pointed out that for nonhereditary algebras orbit categories need not be triangulated, and he introduced the notion of triangulated hull to overcome this problem. In the more general setup of algebras of global dimension at most 2, cluster categories are defined to be these triangulated hulls of the orbit categories. In this paper, we study the image of the natural functor from the bounded, derived category to the cluster category, that is, we investigate how far the orbit category is from being the cluster category. We show that the cluster combinatorics can be worked with in the orbit category, that is, that it is not necessary to consider the entire cluster category. On the other hand, we show that for wide classes of nonpiecewise hereditary algebras the orbit category is never equal to the cluster category. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the endomorphism algebras of standard cluster tilting objects in the stably 2-Calabi–Yau categories with elements w in Coxeter groups in [4]. They are examples of the 2-Auslander algebras introduced in [12]. Generalizing the work in [9], we show that they are quasihereditary, even strongly quasihereditary in the sense of [18]. We also describe the cluster tilting object giving rise to the Ringel dual and prove that there is a duality between and the category of good modules over the quasihereditary algebra. When w = uv is a reduced word, we show that the 2-Calabi–Yau triangulated category is equivalent to a specific subfactor category of . This is applied to show that a standard cluster tilting object M in and the cluster tilting object Λw ⊕ΩM lie in the same component in the cluster tilting graph. [ABSTRACT FROM PUBLISHER]