# Cluster-tilting theory

Lead Research Organisation: University of Leeds
Department Name: Pure Mathematics

### Abstract

ManifoldsA manifold is a mathematical space which is locally like the usual coordinate space of some fixed dimension, known as Euclidean space. The earth itself is an example of a two-dimensional manifold: maps exploit the fact that, if you don't go too far from a given point, the earth behaves pretty much like a plane. The earth could be covered entirely by small maps making use of this local planar structure, although an accurate map of the entire world must be a globe. A one-dimensional manifold, such as a circle, looks locally like a line, and there are manifolds of higher dimension as well. Manifolds play an important role in mathematics because there are so many examples and because they can be studied using the properties of Euclidean space. SymmetriesThe symmetries of an object are the transformations which leave it looking exactly as it was before, such as a rotation of a square through a quarter of a revolution. Two symmetries can always be composed to give a third. This, together with other properties, gives the set of symmetries of any object the structure of a group. While there are only 8 symmetries of a square, the circle has infinitely many symmetries, forming a one-dimensional manifold. The manifold and group structures are compatible, and such an object is known as a Lie group. Lie groups often arise as certain symmetry groups known as gauge groups in physics as well.Lie groupsIn the 1870s, Sophus Lie studied Lie groups by linearizing them, to obtain simpler spaces known as Lie algebras. Lie algebras can be studied using more elementary mathematical notions from linear algebra, which is the study of vector spaces (such as Euclidean spaces) and their linear transformations. This and other developments led to the modern subject of Lie theory in which Lie algebras are studied via their representation theory: each element of the Lie algebra is represented by a transformation of a vector space. Such transformations can be represented more concretely via matrices, which are rectangular arrays of numbers, provided a basis is chosen (essentially a choice of coordinates).Quantum groupsIn 1985, deformed versions of the classical Lie algebras, or quantum groups, were introduced by Drinfeld and Jimbo, and this led to a revolution in the way that Lie algebras were studied. In particular, Kashiwara and Lusztig introduced the canonical basis in 1990, with beautiful properties. It simultaneously gives rise to bases for all representations for the Lie algebra (of a certain kind). The canonical basis has applications to other fields as well, such as the representation theory of affine Hecke algebras.The projectAttempts to describe the canonical basis explicitly have led to a lot of interesting mathematics, including the cluster algebras of Fomin and Zelevinsky, introduced to model its multiplicative properties. Cluster categories and cluster-tilting theory were introduced in order to understand cluster algebras. These objects were defined using representations of quivers: a quiver is a graph with oriented edges and a representation is a vector space for each vertex and a transformation for each edge. As well as giving insight into the canonical basis, cluster-tilting theory has rapidly become a powerful tool for the study of representations of quivers.This model of the canonical basis is still incomplete and work on the proposed project will help towards a better picture via quantum cluster algebras. The project will also work on developing cluster-tilting theory from the perspective of representations of quivers by describing its combinatorial properties and generalising it to a wider context in several directions. The project will also develop stronger connections to the diagram algebras arising in statistical mechanics.

### ORCID iD

Bethany Rose Marsh (Principal Investigator)  http://orcid.org/0000-0002-4268-8937

### Publications

10 25 50
Bakke Buan A (2013) From triangulated categories to module categories via localisation II: calculus of fractions in Journal of the London Mathematical Society

Barot M (2014) Reflection group presentations arising from cluster algebras in Transactions of the American Mathematical Society

Baur K (2012) Categorification of a frieze pattern determinant in Journal of Combinatorial Theory, Series A

Baur K (2016) Dimer models and cluster categories of Grassmannians in Proceedings of the London Mathematical Society

Baur K (2013) Torsion Pairs and Rigid Objects in Tubes in Algebras and Representation Theory

Baur K (2012) A geometric model of tube categories in Journal of Algebra

Buan A (2012) From triangulated categories to module categories via localisation in Transactions of the American Mathematical Society

Buan A (2012) From triangulated categories to module categories via localization II: calculus of fractions in Journal of the London Mathematical Society

Buan A (2009) Cluster structures from 2-Calabi-Yau categories with loops in Mathematische Zeitschrift

Fordy A (2010) Cluster mutation-periodic quivers and associated Laurent sequences in Journal of Algebraic Combinatorics

Description Cluster algebras are mathematical objects that were first defined around 2000 by S. Fomin and A. Zelevinsky, and were introduced to study the canonical basis of a quantized enveloping algebra. Quantized enveloping algebras can be regarded as deformed versions of the classical Lie algebras which arose in the context of continuous symmetries early in the twentieth century. Their canonical bases were introduced around 1990 in order to describe them more explicitly.

Cluster algebras have had a wide impact in mathematics since then, because they encode natural ideas present in a variety of contexts. At the heart of a cluster algebra is the fundamental notion of mutation of a quiver, or directed network. The mutation relies on a choice of vertex in the quiver, and has only a local effect: arrows incident with the chosen vertex and adjacent ones are changed. This mutation rule and related combinatorics has particular relevance to the representation theory of finite-dimensional algebras and algebraic category theory. Essentially, a category is a collection of objects and relationships between them, a general notion allowing abstraction of mathematical theories.

In [BMV], we gave axioms for a category to satisfy in order for it to be modelled by the cluster theory described above. This generalized existing results by allowing loops in the quivers: in such cases the quivers were replaced by appropriate skew-symmetric matrices.

The article [FM] classified those quivers which, after mutation, coincide with the original quiver, as well as constructing quivers with higher periodic properties. This allowed the construction of discrete systems (recurrences) with good integrability properties - giving rise to interdisciplinary results.

A key problem in representation theory is to understand the module category of an algebra, which explains how it can be represented by matrices of numbers. A quiver as above has an associated algebra, the path algebra, and its cluster category can be regarded as an extension of the module category. Early results in cluster theory showed how to obtain module categories via additive quotients. The articles [BuMa1] and [BuMa2] provided a new method for obtaining module categories from cluster categories: via localization, or specifying that some pairs of objects are the same, and connected the theory with notions of generalized abelian categories.

Mutation of quivers is generally modelled in an associated cluster category via mutation of a certain class of objects, known as cluster-tilting objects: the quivers can be read off from the objects via a certain construction. The aim of [MP] was to focus on a larger class of objects in the cluster category (the rigid objects). We showed that mutation of this larger class could be modelled using a more general class of quivers, in which each arrow is given a certain "colour" or label. We showed that if the cluster category was associated to a surface with boundary and marked points (such as a sphere, disk or torus), then the coloured quiver could also be read off from the geometry. We gave, in some contexts, new mutation rules for coloured quivers.

In more detail, a cluster algebra is defined by starting with an initial collection of rational functions (quotients of polynomials), or cluster, together with the quiver mentioned above, with vertices labelled by the functions in the cluster. The quiver and cluster together are known as a seed, and are mutated together to form a new seed; each seed can be regarded as a kind of chart, or map, of an associated space. The Laurent Phenomenon of Fomin and Zelevinsky states that, even after many mutations, the functions in the seeds obtained have good expressions in terms of the original cluster chosen. Important examples of cluster algebras arise from Grassmannians which model the way one space can be embedded into another.
The article [MS] gives beautiful descriptions of such expressions arising from matchings, or dimers, on planar graphs describing the Grassmannian cluster algebra structure known as plabic graphs.

The Grassmannian theme is continued in the article [BKM], which builds on the article [JKS] to give a categorical model for the cluster algebra structure on the Grassmannian discussed above. The main result is that the quivers describing this structure (in a key special case) can be derived from the categorical model. The paper goes a step further and describes, for each such quiver, an algebra arising from the categorical model. The quiver can be embedded into a disk in such a way that the complement is a union of polygons, each bounded by a cycle in the quiver (such an embedding is known as a dimer model, and plays a role in mathematical physics as well as mathematics). The relations on the algebra (part of its description) can then be naturally read off the quiver together with its embedding. These results (together with [MS] above and other papers) open up some interesting questions: can such algebras be described for arbitrary surfaces, rather than just for disks? What do the corresponding cluster algebras model?

A reflection in the plane fixes the points on the line of reflection, while a reflection in three-dimensional space fixes the points on a plane of reflection, such as mirror. Reflections can be defined similarly in higher dimensions. Two reflections can be combined to give a new transformation of space; this procedure can be iterated to produce new transformations, in a way analogous to the creation of multiple images when two mirrors are placed together. In special circumstances, only finitely many new transformations are produced, such as when two mirrors are placed together at an angle which is an integer fraction of a full rotation. Classically, such reflection groups have been fully classified. In [BaMa], we give new combinatorial descriptions, or presentations, of reflection groups using the theory of cluster algebras described above. We extended this to new descriptions of the corresponding Artin braid groups in [GM], explaining how they could be understood geometrically using triangulated surfaces. We showed that these presentations could be interpreted categorically, with the generators acting as spherical twists on derived categories of Ginzburg differential-graded algebras, which play an important role in cluster theory.

In a cluster algebra, each cluster variable can be expressed in terms of the cluster variables in a fixed initial cluster as a fraction: the division of a polynomial by a monomial element; the exponents in the monomial give a labelling of the cluster variable by an integer vector known as its denominator vector. It was originally expected that such vectors would always have a representation-theoretic interpretation. In [MR], we answer a question posed by T. Nakanishi, giving an example of a cluster algebra with a denominator vector which is not described representation-theoretically in the expected way. However, we are able to give a new way of doing this. In order to do this, we give a lot of useful information about the corresponding representation theory: a classification of certain kinds of modules (rigid and Schurian) over the associated cluster-tilted algebra.

[BaMa] M. Barot and R. J. Marsh. Reflection group presentations arising from cluster algebras. Trans. Amer. Math. Soc. 367 (2015), 1945-1967.
[BKM] K. Baur, A. D. King and R. J. Marsh, Dimer models and cluster categories of Grassmannians. Proc. Lond. Math. Soc. (3) 113 (2016), no. 2, 213-260.
[BuMa1] A. B. Buan and R. J. Marsh. From triangulated categories to module categories via localisation. Trans. Amer. Math. Soc. 365, no. 6, 2845-2861 (2012). [BuMa2] A. B. Buan and R. J. Marsh. From triangulated categories to module categories via localisation II: Calculus of fractions. J. London Math. Soc. (2012) 86 (1): 152-170.
[BMV] A. B. Buan, R. J. Marsh and D. F. Vatne. Cluster structures from 2-Calabi-Yau categories with loops. Math. Zeit. 265 no. 4 (2010), 951-970.
[FM] A. P. Fordy and R. J. Marsh. Cluster mutation-periodic quivers and associated Laurent sequences. J. Alg. Combin. 34, no. 1 (2011), 19-66.
[GM] R. J. Marsh and J. Grant. Braid groups and quiver mutation. Pacific Journal of Mathematics 290-1 (2017), 77-116.
[JKS] B. T. Jensen, A. D. King and X. Su, A categorification of Grassmannian cluster algebras. Proc. Lond. Math. Soc. (3) 113 (2016), no. 2, 185-212.
[MP] R. J. Marsh and Y. Palu. Coloured quivers for rigid objects and partial triangulations: The unpunctured case. Proc. London Math. Soc. (2013), 108, no. 2 (2013), 411-440.
[MR] R. J. Marsh and I. Reiten, Rigid and Schurian modules over cluster-tilted algebras of tame type. Math. Zeitschrift 284 (2016), no. 3-4, 643-682.
[MS] R. J. Marsh and J. S. Scott. Twists of Plücker coordinates as dimer partition functions. Comm. Math. Phys. 341 Issue 3 (2016), 821-884.
Exploitation Route The presentations in output [BarMa] have been used in [FeTu1] to construct hyperbolic manifolds of finite volume with large symmetry groups. This approach has been generalized further to affine types and quivers (cluster algebras) associated to unpunctured surfaces and orbifolds. Relations generalising those appearing in [BarMa] have been shown to hold in Weyl groups in a very general context in [Se]. The presentations in [BarMa] have been generalized to affine and and quotients of Coxeter groups associated to surfaces and orbifolds in [FeTu2]. The presentations of Weyl groups in [BarMa] were lifted to give new presentations of Artin braid groups in [HHLP], including the non-simply laced case. In the simply-laced case, modified versions of lifted presentations were given in [GM] (see Key Findings), with topological and categorical interpretations.

The article [Fr] mentions the twist map on the Grassmannian considered in [MaSc] as an important example of a quasi-automorphism of the Grassmannian cluster algebra, and indicates that much of the set-up in [Fr] was developed with the Grassmannian application in mind, to be investigated in a forthcoming paper.
The article [MuSp] introduces a twist map for positroid varieties, mentioning [MaSc] as the "most important precedent" for the paper (which introduces a twist map for the Grassmannian itself). The article [FoSh] exploits the explicit maximal green sequences given in [MaSc] to show that quivers associated to reduced plabic graphs admit a green-to-red sequence of mutations. This, in turn, implies that the "Enough Global Monomials" property holds for the corresponding cluster algebras.

The surface perspective for Brauer graph algebras developed in [MaSch] was used in the article [Sch] studying the relationship between gentle algebras and Brauer graph algebras and played a role in [GS] which introduced the notion of a Brauer configuration algebra, a generalisation of a Brauer graph algebra. The article [AAC] classifies the two-term tilting complexes over a Brauer graph algebra, using this to determine which Brauer graph algebras are tilting-discrete.

The article [FM] constructs an interesting class of integer recurrences with Laurent polynomial properties that are interesting from the point of view of discrete Integrable systems. For example, the integrability and other properties of these have been studied in [F] in the context of Poisson structures, Liouville's equation and Baecklund transformations. They have also been studied in [FH1] in terms of their symplectic structure, and in [FH2]. A particularly interesting direction, informed by [FM], is to consider recurrences associated to Laurent Phenomenon algebras [HW]. The higher period quivers defined by [FM] remain unclassified. The article [FM] is cited in the article [DMSS] which uses algebraic geometric techniques to study the cluster algebra positivity conjecture (concerning coefficients in the Laurent expansion of cluster variables in terms of a fixed initial cluster) in the quantum case. In particular, this is shown for the Somos-5 quiver appearing in [FM] via a consideration of gradings on this quiver. Section 2 of [JMZ] introduces and explains definitions and results from [FM]. The paper then studies the Gale- Robinson case (which was highlighted in [FM]) in detail, linking the quiver from [FM] with a brane tiling (a mathematical physics notion) and pinecone graphs of [BPW] (from combinatorics) to give perfect matching formulas for the terms in the recurrence. In [EF] a relationship between Somos-4 sequences and a del Pezzo surface is discussed, in terms of pyramid partition functions in a mathematical physics context. It is pointed out that [FM] also contains such a relationship (the Somos 4 sequence arises from a period 1 quiver in [FM], which also can be attached to the same del Pezzo surface). The article [CrSo] studies symplectic maps arising from reductions of cluster iteration maps arising from quivers which are periodic in the sense of [FM], which provides an important setting for this paper. The paper [CMS] builds on [CrSo], studying the dynamics of the iteration maps of two particular 2-periodic quivers. Cluster variables associated to cluster algebras of a number of mutation-periodic quivers (in the sense of [FM], or a generalisation) are shown to satisfy q-Painleve equations in [Nao]. The article [HI] studies T- and Y-systems associated to periodic quivers in the sense of [FM]. The periodicity approach to Laurent sequences using cluster algebras in [FM] has been generalised to the case of Laurent phenomenon algebras (in the sense of Lam-Pylyavskyy [LP]) in [ACH]. A generalisation of the approach in [FM] also appears in [GP] in the context of the pentagram map. The recurrences introduced in [FM] play an important role in the study [HKTW] of integrability in the context of the Hirota-Miwa equation (or octahedron recurrence). The article [CMS] studies cluster maps, which arise from the periodic quivers introduced in [FM]; it considers related Poisson structures and the dynamics of cluster maps. The article [FJ] considers the periodic quivers introduced in [FM], studying the corresponding quantum cluster algebras and their relationship to quantized Weyl algebras. The article [OT] introduces and studies new families of integer sequences based on the sequences that were introduced in [FM] using the periodic quivers.

The article [BMV] shows how cluster combinatorics can be modelled by a category even when there are loops in the quiver; it has been taken forward by [ZZ] which shows how to obtain cluster variables in this case, giving a stronger categorification. This suggests that cluster categorifications may exist in a wider context. [BMV] has motivated the study of algebras associated to cluster tubes; a start was made in [V,XL1,XL2]. [BMV] was part of the motivation for the study of maximal rigid objects without loops in [XO1,XO2]. Results from [BMV] are used for the development of examples in [CZZ]. The cluster categorification in type B is related to cluster algebra gradings in [G]. The article [BMV] is cited in [LX]. [BMV] forms part of the motivation for [BPR], since it indicates a source of maximal rigid objects which are not cluster-tilting objects (in cluster tubes). The perspective of studying 2-Calabi-Yau categories with cluster structures with loops developed in [BMV] formed part of the motivation for [GLS]. This theory is also used in [Ch] to study certain 2-Calabi-Yau categories related to derived categories.

Output [JP] is used in [HJ1] to give a categorical interpretation of infinite frieze patterns of integers. Ideas from [JP] are used across [HJ2] in order to continue the study of friezes and to develop a modified Caldero-Chapoton map depending on a rigid object.

The geometric model in [BaMa1] is used to give a geometric description of torsion pairs and maximal rigid objects in tubes in [BBM]. It is used in [HJR] in order to give a geometric description of torsion pairs in cluster tubes. It formed part of the motivation for the definition of asymptotic arcs in [BPT].

Work in [BuMa1,BuMa2] has been put in a wider context and generalized in [Bel, Nak1, Liu], leading to new categorical notions such as twin torsion pairs. The articles [BuMa1,BuMa2] also formed a key
motivation for [MaPa], which considered the localisations in these papers under mutation.
The article [Nak1] was strongly motivated by the article [BuMa2].

The article [Nak2], put an equivalence found in [MaPa] in the context of cluster categories into a wider
triangulated category context (i.e. generalizing it), showing that it could be interpreted as an equivalence of hearts.

Output [M] has been used in courses/working seminars at the University of Michigan, the University of Oxford and the Institut Henri Poincare.

A result from [BBM] is used in [CSP] in the study of torsion pairs in triangulated categories. Results from [BBM] were recovered and refined in the context of tilting sheaves over weighted noncommutative regular projective curves in [AK].

The article [MM] forms part of the geometrical motivation for [DVM].

The article [MaSch] uses geometrical and homological ideas from [BKM] to describe a relationship between Brauer graph algebras and cluster algebras. The properties of the cluster categorification of the Grassmannian developed in [BKM] (and [JKS]) formed a key set of examples of the internally Calabi-Yau categories introduced in [Pr]. This cluster categorification is also used to study friezes of integers in [BFGST].
The article [BauMa] investigating the combinatorics of triangulations in the context of the Grassmannian cluster algebra was motivated by study of the Scott map in [BKM].

The arc-cutting approach to geometric cluster algebras developed in [MaPa2] was used in [HL] to further study geometric cluster algebras. The geometric interpretation of Iyama-Yoshino reduction given in [MaPa2] is used to good effect in [CS] to interpret extensions in a geometric cluster category and thence to give results concerning modules over the gentle algebras which arise from surfaces as Jacobian algebras relating to the cluster structure. The article [MaPa2] focuses on unpunctured surfaces, but the article [QZ] considers the case of a punctured surface, describing the corresponding cluster category; the geometric interpretation of Iyama-Yoshino reduction in [MaPa2] is a key ingredient in this study.

The article [Gr] is used in [GrMa] to show that certain monomials of functors can be represented as single periodic twists, giving some insight into the results in the paper.

[AAC] T. Adachi, T. Aihara, and A. Chan, Classification of two-term tilting complexes over Brauer graph algebras. Math. Z. 290 (2018), no. 1-2, 1-36.
[ACH] J. Alman, C. Cuenca and J. Huang, Laurent phenomenon sequences. J. Algebraic Combin. 43 (2016), no. 3, 589-633.
[AK] L. Angeleri Hügel and D. Kussin, Large tilting sheaves over weighted noncommutative regular projective curves. Doc. Math. 22 (2017), 67-134.
[BauMa] K. Baur and P. Martin, The fibres of the Scott map on polygon tilings are the flip equivalence classes. Monatsh. Math. 187 (2018), no. 3, 385-424.
[BarMa] M. Barot and R. J. Marsh. Reflection group presentations arising from cluster algebras. Trans. Amer. Math. Soc. 367 (2015), 1945-1967.
[BFGST] K. Baur, E. Faber, S. Gratz, K. Serhiyenko and G. Todorov, Mutation of friezes.
Bull. Sci. Math. 142 (2018), 1-48.
[BaMa1] K. Baur and R. J. Marsh. A geometric model of tube categories. J. Algebra 362 (2012), 178--191.
[BPT] K. Baur, M. J. Parsons and M. Tschabold, Infinite friezes. European J. Combin. 54 (2016), 220-237.
[Bel] A. Beligiannis, Rigid objects, triangulated subfactors and abelian localizations. Math. Z. 274 (2013), no. 3-4, 841-883.
[BBM] K. Baur, A. B. Buan and R. J. Marsh, Torsion pairs and rigid objects in tubes. Algebr. Represent. Theory 17 (2014), no. 2, 565-591.
[BKM] K. Baur, A. King and R. J. Marsh, Dimer models and cluster categories of Grassmannians. Proc. Lond. Math. Soc. (3) 113 (2016), no. 2, 213-260.
[BPW] M. Bousquet-Melou, J. Propp and J. West, Elect. J. Combin. 16, no. 1, Research Paper 125 (2009), 1-37.
[BuMa1] A. B. Buan and R. J. Marsh. From triangulated categories to module categories via localisation. Trans. Amer. Math. Soc. 365 (2013), no. 6, 2845-2861.
[BuMa2] A. B. Buan and R. J. Marsh. From triangulated categories to module categories via localisation II: Calculus of fractions. J. London Math. Soc. (2012) 86 (1): 152-170.
[BMV] A. B. Buan, R. J. Marsh and D. F. Vatne. Cluster structures from 2-Calabi-Yau categories with loops. Math. Zeit. 265 no. 4 (2010), pages 951-970.
[BPR] A. B. Buan, Y. Palu and I. Reiten, Algebras of finite representation type arising from maximal rigid objects. J. Algebra 446 (2016), 426-449.
[Ch] H. M. Chang, Cluster structures in 2-Calabi-Yau triangulated categories of Dynkin type with maximal rigid objects. Acta Math. Sin. (Engl. Ser.) 33 (2017), no. 12, 1693-1704.
[CMS] I. Cruz, H. Mena-Matos and M. Esmerelda Sousa-Dias, Dynamics of the birational maps arising from F0 and dP3 quivers. J. Math. Anal. Appl. 431 (2015), no. 2, 903-918.
[CrSo] I. Cruz and M. E. Sousa-Dias, Reduction of cluster iteration maps. J. Geom. Mech. 6 (2014), no. 3, 297-318.
[CMS] I. Cruz, Inês, H. Mena-Matos and M. E. Sousa-Dias, Multiple reductions, foliations and the dynamics of cluster maps. Regul. Chaotic Dyn. 23 (2018), no. 1, 102-119.
[CS] I. Canakci and S. Schroll, Extensions in Jacobian algebras and cluster categories of marked surfaces. Adv. Math. 313 (2017), 1-49.
[CSP] R. Coelho Simoes and D. Pauksztello,Torsion pairs in a triangulated category generated by a spherical object. J. Algebra 448 (2016), 1-47.
[CZZ] W. Chang, J. Zhang and B. Zhu, On support t-tilting modules over endomorphism algebras of rigid objects. Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 9, 1508-1516.
[DMSS] B. Davison, D. Maulik, J. Schuermann, B. Szendroi, Purity for graded potentials and quantum cluster positivity. Compos. Math. 151 (2015), no. 10, 1913-1944.
[DVM] M. De Visscher and P. P. Martin, On Brauer algebra simple modules over the complex field. Trans. Amer. Math. Soc. 369 (2017), no. 3, 1579-1609.
[EF] R. Eager and S. Franco, Colored BPS pyramid partition functions, quivers and cluster trans- formations, Journal of High Energy Physics, 9, Article 038 (2012).
[FeTu1] A. Felikson and P. Tumarkin, Coxeter groups, quiver mutations and geometric manifolds. J. Lond. Math. Soc. (2) 94 (2016), no. 1, 38-60.
[FeTu2] A. Felikson and P. Tumarkin, Coxeter groups and their quotients arising from cluster algebras. Int. Math. Res. Not. IMRN 2016, no. 17, 5135-5186.
[FJ] C. D. Fish and D. A. Jordan, Connected quantized Weyl algebras and quantum cluster algebras. J. Pure Appl. Algebra 222 (2018), no. 8, 2374-2412.
[FoSh] N. Ford and K. Serhiyenko Green-to-red sequences for positroids. J. Combin. Theory Ser. A 159 (2018), 164-182.
[F] A. P. Fordy. Mutation-periodic quivers, integrable maps and associated Poisson algebras. Phil. Trans. R. Soc. A (2011) 369, 1264-1279.
[FH1] A. P. Fordy and A. Hone. Symplectic maps from cluster algebras. Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 091, 12 pages.
[FH2] A. P. Fordy and A. Hone, Discrete integrable systems and Poisson algebras from cluster maps. Comm. Math. Phys. 325 (2014), no. 2, 527-584.
[FM] A. P. Fordy and R. J. Marsh. Cluster mutation-periodic quivers and associated Laurent sequences. J. Alg. Combin. 34, no. 1 (2011), 19-66.
[Fr] C. Fraser, Quasi-homomorphisms of cluster algebras.
Adv. in Appl. Math. 81 (2016), 40-77.
[GLS] C. Geiss, B. Leclerc, J. Schröer, Quivers with relations for symmetrizable Cartan matrices I: Foundations. Invent. Math. 209 (2017), no. 1, 61-158.
[G] J. E. Grabowski, Graded cluster algebras. J. Algebraic Combin. 42 (2015), no. 4, 1111-1134.
[GS] E. L. Green, and S. Schroll, Brauer configuration algebras: a generalization of Brauer graph algebras. Bull. Sci. Math. 141 (2017), no. 6, 539-572.
[GP] M. Glick and P. Pylyavskyy, Y-meshes and generalized pentagram maps. Proc. Lond. Math. Soc. (3) 112 (2016), no. 4, 753-797.
[Gr] J. Grant, Lifts of longest elements to braid groups acting on derived categories. Trans. Amer. Math. Soc. 367 (2015), no. 3, 1631-1669.
[GrMa] J. Grant and R. J. Marsh, Braid groups and quiver mutation. Pacific J. Math. 290 (2017), no. 1, 77-116.
[HHLP] J. Haley, D. Hemminger, A. Landesman and H. Peck, Artin group presentations arising from cluster algebras. Algebr. Represent. Theory 20 (2017), no. 3, 629-653.
[HI] A. N. Hone and R. Inoue, Discrete Painleve equations from Y-systems. Journal of Physics A - Mathematical and Theoretical. Volume 47, Issue 47, Article Number 474007, 2014.
[HJ1] T. Holm and P. Joergensen, SL2-tilings and triangulations of the strip. J. Combin. Theory Ser. A 120 (2013), no. 7, 1817-1834.
[HJ2] T. Holm and P. Joergensen, Generalised friezes and a modified Caldero-Chapoton map depending on a rigid object, II. Bull. Sci. Math. 140 (2016), no. 4, 112-131.
[HJR] T. Holm, P. Joergensen and M. Rubey, Torsion pairs in cluster tubes. J. Algebraic Combin. 39 (2014), no. 3, 587-605.
[HKW] A. N. W. Hone, T. E. Kouloukas and C. Ward. On reductions of the Hirota-Miwa equation. SIGMA Symmetry Integrability Geom. Methods Appl. 13 (2017), Paper No. 057, 17 pp.
[HW] A. N. W. Hone and C. Ward, A family of linearizable recurrences with the Laurent property. Bull. Lond. Math. Soc. 46 (2014), no. 3, 503-516.
[HL] M. Huang, and F. Li, On structure of cluster algebras of geometric type, II: Green's equivalences and paunched surfaces. Pure Appl. Math. Q. 11 (2015), no. 3, 451-490.
[JKS] B. T. Jensen, A. D. King and X. Su, A categorification of Grassmannian cluster algebras. Proc. Lond. Math. Soc. (3) 113(2), 185-212 (2016).
[JMZ] I. J. Jeong, G. Musiker, S. Zhang, Gale-Robinson Sequences and Brane Tilings DMTCS Proceedings of the 25th International Conference on Formal Power Series and Al- gebraic Combinatorics (FPSAC 2013, Paris, France), pages 707-718. Available from http://www.dmtcs.org/dmtcs-o js/index.php/proceedings/issue/view/130.
[JP] P. Jorgensen and Y. Palu. A Caldero-Chapoton map for infinite clusters. Trans. Amer. Math. Soc. 365 (2012), 1125-1147.
[LP] T. Lam and P. Pylyavskyy, Laurent phenomenon algebras. Camb. J. Math. 4 (2016), no. 1, 121-162.
[Liu] Y. Liu, Hearts of twin cotorsion pairs on exact categories. J. Algebra 394 (2013), 245-284.
[LX] P. Liu and Y. Xie, On the relation between maximal rigid objects and tau-tilting modules. Colloq. Math. 142 (2016), no. 2, 169-178.
[M] R. J. Marsh, Lecture Notes on Cluster algebras, Zurich Lectures in Advanced Mathematics,European Mathematical Society, 2014. ISBN 978-3-03719-130-9.
[MaPa] R. Marsh and Y. Palu, Nearly Morita equivalences and rigid objects.
Nagoya Mathematical Journal 225, 64-99, 2017.
[MaPa2] R. J. Marsh and Y. Palu, Yann Coloured quivers for rigid objects and partial triangulations: the unpunctured case. Proc. Lond. Math. Soc. (3) 108 (2014), no. 2, 411-440.
[MaSc] R. J. Marsh and J. S. Scott, Twists of Pluecker coordinates as dimer partition functions. Comm. Math. Phys. 341 (2016), no. 3, 821-884.
[MaSch] R. J. Marsh and S. Schroll, The geometry of Brauer graph algebras and cluster mutations. J. Algebra 419 (2014), 141-166.
[MM] R. J. Marsh and P. Martin, Tiling bijections between paths and Brauer diagrams. J. Algebraic Combin. 33 (2011), no. 3, 427-453.
[MuSp] G. Muller and D. E. Speyer, The twist for positroid varieties. Proc. Lond. Math. Soc. (3) 115 (2017), no. 5, 1014-1071.
[Nak1] H. Nakaoka, General heart construction for twin torsion pairs on triangulated categories. J. Algebra 374 (2013), 195-215.
[Nak2] H. Nakaoka, Equivalence of hearts of twin cotorsion pairs on triangulated categories. Comm. Algebra 44 (2016), no. 10, 4302-4326.
[Nao] O. Naoto, Bilinear equations and q-discrete Painleve equations satisfied by variables and coefficients in cluster algebras, J. Physics A - Mathematical and theoretical, Volume: 48, Issue: 35, Article Number: 355201, 2015.
[OT] V. Ovsienko and S. Tabachnikov, Dual numbers, weighted quivers, and extended Somos and Gale-Robinson sequences. Algebr. Represent. Theory 21 (2018), no. 5, 1119-1132.
[Pr] M. Pressland, Internally Calabi-Yau algebras and cluster-tilting objects.Math. Z. 287 (2017), no. 1-2, 555-585.
[Sch] S. Schroll, Trivial extensions of gentle algebras and Brauer graph algebras. J. Algebra 444 (2015), 183-200.
[Se] A. I. Seven, Reflection group relations arising from cluster algebras. Proc. Amer. Math. Soc. 144 (2016), no. 11, 4641-4650.
[V] D. F. Vatne. Endomorphism rings of maximal rigid objects in cluster tubes. Colloq. Math. 123 (2011), no. 1, 63-93.
[XL1] Y. Xie and P. Liu, Lifting to maximal rigid objects in 2-Calabi-Yau triangulated categories. Proc. Amer. Math. Soc. 141 (2013), no. 10, 3361-3367.
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[QZ] Y. Qiu, and Y. Zhou, Cluster categories for marked surfaces: punctured case.Compos. Math. 153 (2017), no. 9, 1779-1819.
[ZZ] Y. Zhou and B. Zhu, Cluster algebras arising from cluster tubes. J. Lond. Math. Soc. (2) 89 (2014), no. 3, 703-723.
Sectors Education,Culture, Heritage, Museums and Collections

URL http://www1.maths.leeds.ac.uk/~marsh/epsrcproject4.html

Description Lecture Notes on Cluster Algebras
Geographic Reach Multiple continents/international
Policy Influence Type Influenced training of practitioners or researchers
URL http://www.ems-ph.org/books/book.php?proj_nr=172

Description ETH
Organisation ETH Zurich
Country Switzerland