# Stability Conditions and Representation Theory of Finite-Dimensional Algebras (18w5178)

Arriving in Oaxaca, Mexico Sunday, October 28 and departing Friday November 2, 2018

## Organizers

Thomas Brüstle (Bishop's University and Université de Sherbrooke)

José Antonio de la Peña (Centro de Investigación en Matemáticas)

David Pauksztello (University of Manchester)

David Ploog (Universität Hannover)

## Objectives

1. Semi-invariant theory

The work of Schofield [44] and King [31] introduced the notion of (semi-)stability to quiver representations, in the latter case with the two-fold motivation of using moduli spaces of semistable representations to organise the representation theory of wild algebras, and secondly to connect the moduli spaces occurring in the representation-theoretic setting with those occurring in geometry, which are related via derived equivalence. Indeed, such an approach has borne fruit in classification problems for weighted projective lines and non- commutative curves studied in [21, 35, 37]. This is now a highly active area with a large number of recent publications.

For example, the study of rings of quiver semi-invariants by Derksen and Weyman has been expanded to the context of cluster algebras in [16] and [26]. The work of Igusa, Orr, Todorov and Weyman shows that walls in the semi-invariant picture correspond to the c-vectors in cluster theory. These c-vectors are also studied in quantum field theory (by Cecotti, Cordova, Vafa and many others), where they are interpreted as charges of BPS particles. It turns out that maximal paths in the semi-invariant picture, oriented in positive direction with respect to an inherently given bilinear form, give rise to a complete sequence of charges, called spectrum of a BPS particle in quantum field theory. Moreover, the same sequences of c-vectors correspond to maximal green sequences, which are studied by Reineke to find combinatorial DT-invariants. One objective will be to use methods from representation theory to control which c-vectors can occur, thus controlling the existence of BPS particles.

Another aspect of this workshop will be to investigate the connections between the semi-invariant theory viewpoint and Bridgeland stability conditions. For example, the work of Ingalls-Thomas and Ingalls-Paquette-Thomas in [27, 28] identifies which extension-closed abelian subcategories may occur as semistable subcategories. In Bridgeland stability con- ditions, the semistable subcategories are the slices of the slicings, suggesting possible con- nections between the work [27, 28] on classifications of semistable subcategories and classi- fications of slicings in [22, 30], which in turn can lead to classifications of t-structures and a detailed understanding of the wall-and-chamber structure of the stability manifold.

2. Bridgeland stability spaces in representation theory

Bridgeland stability conditions are now well established in algebraic geometry. Unfortunately, computations with stability conditions, particularly in the geometric setting are notoriously difficult. For example, while classical, Mumford-style stable sheaves on varieties abound, it is extremely hard to even construct single Bridgeland stability conditions on derived categories of higher-dimensional varieties; see for example the work of Bayer-Macrı-Toda [6] in this direction. Naturally, it is even harder to say something about the whole stability space, which is only fully understood in sporadic cases [5, 9, 18, 38, 40, 45]. For instance, among experts, it is widely believed that if the Bridgeland stability manifold is non- empty then it is contractible, but this is known in very few cases. Such a statement would have strong implications in algebraic geometry, where it would be useful in the computation of autoequivalence groups, for example.

Algebraic examples are often more tractable: for instance, in Macrı’s approach to stability conditions on curves in [38] and Bayer and Macrı’s study of the space of stability conditions for the local projective plane in [5], exceptional collections provide a tool to understand the ‘algebraic’ part of the stability manifold and obtain simple-connectedness results. This has led some authors, for example Dimitrov, Haiden, Katzarkov and Kontsevich, to look at more tractable examples in the representation theory of finite-dimensional algebras [17].

In representation theory, much less has been done on spaces of Bridgeland stability condi- tions. In the work of Broomhead, Pauksztello and Ploog [10] and Qiu and Woolf [42], the contractibility of the stability manifold was established for the family of finite-dimensional algebras that have discrete derived category – in this context the whole stability manifold is ‘algebraic’. In [40], Okada shows that the stability manifold of Db(kA1) is C2. Building on the work of Macrı [38], it is shown in [18] that the space of stability conditions of Db(kA2) is contractible. However, no other representation-theoretic examples are known.

Therefore, one objective is to understand further examples of spaces of stability conditions in representation theory. It is natural first to look at tame hereditary examples and attempt to extend the methods of [18] and [38] to all tame hereditary algebras; it is expected that type A should follow naturally from [18], but the situation for type D may be more difficult. A second approach is to look for silting-discrete algebras. The notion of silting-discreteness was introduced by Aihara in [2], and silting-discreteness was a key ingredient in the contractibility proof in [10]. In unpublished work by Grant and Iyama, examples of symmetric algebras with disconnected silting quiver, a notion due to Aihara and Iyama in [3], have been identified. This presents a natural place to search for potential counterexamples to the belief that the stability manifold is contractible whenever it is non-empty.

3. Cluster theory, mutation, wall-crossing and scattering diagrams

The semi-invariant picture of quiver representations has re-appeared in mathematical physics and mirror symmetry as scattering diagrams in [23] and in Kontsevich and Soibelman’s study of wall crossing in the context of Donaldson-Thomas invariants in integrable systems and mirror symmetry [34]. It is now well known that there are many connections between different types of mutation, for example, cluster mutation, silting mutation and mutation of exceptional collections [3, 11, 14, 29]. In [32, 41], King-Qiu and Qiu observed that wall crossing in the stability manifold of Db(kQ) for Q Dynkin was closely related to the cluster exchange graph. The compatibility of mutation of simple objects of hearts, tilting of bounded t-structures and mutation silting objects exhibited in [33] (also Woolf [45]) pro- vides the theoretical tools to examine this relationship more deeply. Thus, we aim to explore the connection between cluster combinatorics and stability conditions/wall crossing in representation theory.

Several authors have studied the categorification of the cluster structure for cluster algebras stemming from marked surfaces [4, 12, 13]. Triangulated surfaces with no punctures give rise to gentle algebras [4] and the structure of (part of) their stability space, i.e. the wall and chamber structure, can be studied from the viewpoint of support-τ-tilting theory [1, 15]. In fact, a similar approach seems to work more generally for all surfaces with non- empty boundary [36]. A very similar construction, perhaps even the same, has been studied independently by Haiden-Katzarkov-Kontsevich in [24], where the objects in the Fukaya category of a surface on the ‘A side’ of homological mirror symmetry are classified by similar combinatorial methods in order to study its space of stability conditions.

In summary, it seems that cluster algebras and cluster theory provide a common intersection of interests between the mathematical physics/algebraic geometry community that are interested in stability spaces and the representation theorists. The theory in both fields is developing rapidly, but some results are not necessarily efficiently communicated between the communities. Therefore, an objective of this workshop will be to provide a forum for this communication between these communities.

References

[1] T. Adachi, O. Iyama, I. Reiten, τ-tiling theory, Compos. Math. 150 (2014), 415–452.

[2] T. Aihara, Tilting-connected symmetric algebras, Algebr. Represent. Theory 16 (2013), 873–894.

[3] T. Aihara, O. Iyama, Silting mutation in triangulated categories, J. London Math. Soc. 85 (2012), 633–668.

[4] I. Assem, T. Brustle, G. Charbonneau-Jodoin, P-G. Plamondon, Gentle algebras arising from surface triangulations, Algebra Number Theory 4 (2010), 201–229.

[5] A. Bayer, E. Macri, The space of stability conditions on the local projective plane, Duke Math. J. 160 (2011), 263–322.

[6] A. Bayer, E. Macri, Y. Toda, Bridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities, J. Algebraic Geom. 23 (2014), 117–163.

[7] A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Ast ́erisque 100 (1982).

[8] T. Bridgeland, Stability conditions on triangulated categories, Ann. Math. 166 (2007), 317–345.

[9] T. Bridgeland, Y. Qiu, T. Sutherland, Stability conditions and the A2 quiver, arXiv:1406.2566.

[10] N. Broomhead, D. Pauksztello, D. Ploog, Discrete derived categories II: the silting pairs CW complex and the stability manifold, J. London Math. Soc. 93 (2016), 273–300.

[11] Th. Bru ̈stle, D. Yang, Ordered exchange graphs, Advances in representation theory of algebras, 135–193, EMS Ser. Congr. Rep., Eur. Math. Soc., Zu ̈rich, 2013.

[12] Th. Bru ̈stle, Y. Qiu Tagged mapping class groups I: Auslander-Reiten translation, Math. Zeitschrift, Volume 279, Issue 3 (2015), Page 1103–1120.

[13] Th. Bru ̈stle and J.Zhang, On the cluster category of a marked surface, Algebra & Number Theory 5-4 (2011), 529–566. DOI 10.2140/ant.2011.5.529.

[14] A. B. Buan, I. Reiten, H. Thomas, Three kinds of mutation, J. Algebra 339 (2011), 97–113.

[15] L. Demonet, O. Iyama, G. Jasso, τ-tilting-finite algebras, g-vectors and brick-τ-rigid correspondence, arXiv:1503.00285.

[16] H. Derksen, J. Weyman, A. Zelevinsky. Quivers with potentials and their representations I: Mutations., Selecta Mathematica 14.1 (2008): 59–119.

[17] G. Dimitrov, F. Haiden, L. Katzarkov, M. Kontsevich, Dynamical systems and categories, in “The influence of Solomon Lefschetz in geometry and topology”, Contemporary Mathematics 621, 133–170, Amer. Math. Soc., Providence, RI, 2014.

[18] G. Dimitrov, L. Katzarkov, Bridgeland stability conditions on the acyclic triangular quiver, Adv. Math. 288 (2016), 825–886.

[19] M. Douglas, D-branes, categories and N = 1 supersymmetry. Strings, branes and M-theory, J. Math. Phys 42 (2001), 2818–2843.

[20] M. Douglas, Dirichlet branes, homological mirror symmetry, and stability, Proceedings of the Interna- tional Congress of Mathematicians, Vol. III (Beijing, 2002), 395–408, Higher Ed. Press, Beijing, 2002.

[21] W. Geigle, H. Lenzing, A class of weighted projective curves arising in representation theory of finite- dimensional algebras, in “Singularities, representations of algebras, and vector bundles (Lambrecht, 1985)”, Lecture Notes in Mathematics vol. 1273, 265–297, Springer, Berlin, 1987.

[22] A. L. Gorodentsev, S. A. Kuleshov, A. N. Rudakov, t-stabilities and t-structures on triangulated cate- gories, Izv. Math 68 (2004), 749–781.

[23] M. Gross, B. Siebert, From real affine geometry to complex geometry, Annals of mathematics, 174, (2011), 1301–1428.

[24] F. Haiden, L. Katzarkov, M. Kontsevich, Flat surfaces and stability structures, arXiv:1409.8611.

[25] G. Harder, M. S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1974/75), 215–248.

[26] K. Igusa, K. Orr, G. Todorov, J. Weyman, Cluster complexes via semi-invariants, Compositio Mathematica, 145(04), (2009), 1001–1034.

[27] C. Ingalls, H. Thomas, Noncrossing partitions and representations of quivers, Compos. Math. 145 (2009), 1533–1562.

[28] C. Ingalls, C. Paquette, H. Thomas, Semi-stable subcategories for Euclidean quivers, Proc. London Math. Soc. 110 (2015), 805–840.

[29] O. Iyama, Y. Yoshino, Mutation in triangulated categories and rigid Cohen-Macaulay modules, Invent. Math. 172 (2008), 117–168.

[30] P. Jørgensen, D. Pauksztello, Classification of co-slicings and co-t-structures for the Kronecker algebra, J. Pure Appl. Algebra 219 (2015), 569–590.

[31] A. D. King, Moduli of representations of finite dimensional algebras, Quart. J. Math. Oxford 45 (1994), 515–530.

[32] A. D. King, Y. Qiu, Exchange graphs and Ext quivers, Adv. Math. 285 (2015), 1106–1154.

[33] S. Konig, D. Yang, Silting objects, simple-minded collections, t-structures and co-t-structures for finite-dimensional algebras, Doc. Math. 19 (2014), 403–438.

[34] M. Kontsevich, Y. Soibelman, Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and mirror symmetry, Homological Mirror Symmetry and Tropical Geometry. Springer International Publishing, 2014. 197–308.

[35] D. Kussin, Noncommutative curves of genus zero: related to finite dimensional algebras, Mem. Amer. Math. Soc. 201 (2009), x+128.

[36] D. Labardini Fragoso, Quivers with potentials associated to triangulated surfaces, Proc. London Math. Soc. 98 (2009), 797–839.

[37] H. Lenzing, I. Reiten, Hereditary Noetherian categories of positive Euler characteristic, Math. Z. 254 (2006), 133–171.

[38] E. Macri, Stability conditions on curves, Math. Res. Lett. 14 (2007), 657–672.

[39] D. Mumford, Geometric Invariant Theory, Springer-Verlag Berlin (1965).

[40] S. Okada, Stability manifold of P1, J. Algebraic Geom. 15 (2006), 487–505.

[41] Y. Qiu, Stability conditions and quantum dilogarithm identities for Dynkin quivers, Adv. Math. 269 (2015), 220–264.

[42] Y. Qiu, J. Woolf, Contractible stability spaces and faithful braid group actions, arXiv:1407.5986.

[43] A. Rudakov, Stability for an abelian category, J. Algebra 197 (1997), 231–245.

[44] A. Schofield, Semi-invariants of quivers, J. London Mathematical Society (2), no. 3 (1991), 385–395.

[45] J. Woolf, Stability conditions, torsion theories and tilting, J. London Math. Soc. 82 (2010), 663–682.

The work of Schofield [44] and King [31] introduced the notion of (semi-)stability to quiver representations, in the latter case with the two-fold motivation of using moduli spaces of semistable representations to organise the representation theory of wild algebras, and secondly to connect the moduli spaces occurring in the representation-theoretic setting with those occurring in geometry, which are related via derived equivalence. Indeed, such an approach has borne fruit in classification problems for weighted projective lines and non- commutative curves studied in [21, 35, 37]. This is now a highly active area with a large number of recent publications.

For example, the study of rings of quiver semi-invariants by Derksen and Weyman has been expanded to the context of cluster algebras in [16] and [26]. The work of Igusa, Orr, Todorov and Weyman shows that walls in the semi-invariant picture correspond to the c-vectors in cluster theory. These c-vectors are also studied in quantum field theory (by Cecotti, Cordova, Vafa and many others), where they are interpreted as charges of BPS particles. It turns out that maximal paths in the semi-invariant picture, oriented in positive direction with respect to an inherently given bilinear form, give rise to a complete sequence of charges, called spectrum of a BPS particle in quantum field theory. Moreover, the same sequences of c-vectors correspond to maximal green sequences, which are studied by Reineke to find combinatorial DT-invariants. One objective will be to use methods from representation theory to control which c-vectors can occur, thus controlling the existence of BPS particles.

Another aspect of this workshop will be to investigate the connections between the semi-invariant theory viewpoint and Bridgeland stability conditions. For example, the work of Ingalls-Thomas and Ingalls-Paquette-Thomas in [27, 28] identifies which extension-closed abelian subcategories may occur as semistable subcategories. In Bridgeland stability con- ditions, the semistable subcategories are the slices of the slicings, suggesting possible con- nections between the work [27, 28] on classifications of semistable subcategories and classi- fications of slicings in [22, 30], which in turn can lead to classifications of t-structures and a detailed understanding of the wall-and-chamber structure of the stability manifold.

2. Bridgeland stability spaces in representation theory

Bridgeland stability conditions are now well established in algebraic geometry. Unfortunately, computations with stability conditions, particularly in the geometric setting are notoriously difficult. For example, while classical, Mumford-style stable sheaves on varieties abound, it is extremely hard to even construct single Bridgeland stability conditions on derived categories of higher-dimensional varieties; see for example the work of Bayer-Macrı-Toda [6] in this direction. Naturally, it is even harder to say something about the whole stability space, which is only fully understood in sporadic cases [5, 9, 18, 38, 40, 45]. For instance, among experts, it is widely believed that if the Bridgeland stability manifold is non- empty then it is contractible, but this is known in very few cases. Such a statement would have strong implications in algebraic geometry, where it would be useful in the computation of autoequivalence groups, for example.

Algebraic examples are often more tractable: for instance, in Macrı’s approach to stability conditions on curves in [38] and Bayer and Macrı’s study of the space of stability conditions for the local projective plane in [5], exceptional collections provide a tool to understand the ‘algebraic’ part of the stability manifold and obtain simple-connectedness results. This has led some authors, for example Dimitrov, Haiden, Katzarkov and Kontsevich, to look at more tractable examples in the representation theory of finite-dimensional algebras [17].

In representation theory, much less has been done on spaces of Bridgeland stability condi- tions. In the work of Broomhead, Pauksztello and Ploog [10] and Qiu and Woolf [42], the contractibility of the stability manifold was established for the family of finite-dimensional algebras that have discrete derived category – in this context the whole stability manifold is ‘algebraic’. In [40], Okada shows that the stability manifold of Db(kA1) is C2. Building on the work of Macrı [38], it is shown in [18] that the space of stability conditions of Db(kA2) is contractible. However, no other representation-theoretic examples are known.

Therefore, one objective is to understand further examples of spaces of stability conditions in representation theory. It is natural first to look at tame hereditary examples and attempt to extend the methods of [18] and [38] to all tame hereditary algebras; it is expected that type A should follow naturally from [18], but the situation for type D may be more difficult. A second approach is to look for silting-discrete algebras. The notion of silting-discreteness was introduced by Aihara in [2], and silting-discreteness was a key ingredient in the contractibility proof in [10]. In unpublished work by Grant and Iyama, examples of symmetric algebras with disconnected silting quiver, a notion due to Aihara and Iyama in [3], have been identified. This presents a natural place to search for potential counterexamples to the belief that the stability manifold is contractible whenever it is non-empty.

3. Cluster theory, mutation, wall-crossing and scattering diagrams

The semi-invariant picture of quiver representations has re-appeared in mathematical physics and mirror symmetry as scattering diagrams in [23] and in Kontsevich and Soibelman’s study of wall crossing in the context of Donaldson-Thomas invariants in integrable systems and mirror symmetry [34]. It is now well known that there are many connections between different types of mutation, for example, cluster mutation, silting mutation and mutation of exceptional collections [3, 11, 14, 29]. In [32, 41], King-Qiu and Qiu observed that wall crossing in the stability manifold of Db(kQ) for Q Dynkin was closely related to the cluster exchange graph. The compatibility of mutation of simple objects of hearts, tilting of bounded t-structures and mutation silting objects exhibited in [33] (also Woolf [45]) pro- vides the theoretical tools to examine this relationship more deeply. Thus, we aim to explore the connection between cluster combinatorics and stability conditions/wall crossing in representation theory.

Several authors have studied the categorification of the cluster structure for cluster algebras stemming from marked surfaces [4, 12, 13]. Triangulated surfaces with no punctures give rise to gentle algebras [4] and the structure of (part of) their stability space, i.e. the wall and chamber structure, can be studied from the viewpoint of support-τ-tilting theory [1, 15]. In fact, a similar approach seems to work more generally for all surfaces with non- empty boundary [36]. A very similar construction, perhaps even the same, has been studied independently by Haiden-Katzarkov-Kontsevich in [24], where the objects in the Fukaya category of a surface on the ‘A side’ of homological mirror symmetry are classified by similar combinatorial methods in order to study its space of stability conditions.

In summary, it seems that cluster algebras and cluster theory provide a common intersection of interests between the mathematical physics/algebraic geometry community that are interested in stability spaces and the representation theorists. The theory in both fields is developing rapidly, but some results are not necessarily efficiently communicated between the communities. Therefore, an objective of this workshop will be to provide a forum for this communication between these communities.

References

[1] T. Adachi, O. Iyama, I. Reiten, τ-tiling theory, Compos. Math. 150 (2014), 415–452.

[2] T. Aihara, Tilting-connected symmetric algebras, Algebr. Represent. Theory 16 (2013), 873–894.

[3] T. Aihara, O. Iyama, Silting mutation in triangulated categories, J. London Math. Soc. 85 (2012), 633–668.

[4] I. Assem, T. Brustle, G. Charbonneau-Jodoin, P-G. Plamondon, Gentle algebras arising from surface triangulations, Algebra Number Theory 4 (2010), 201–229.

[5] A. Bayer, E. Macri, The space of stability conditions on the local projective plane, Duke Math. J. 160 (2011), 263–322.

[6] A. Bayer, E. Macri, Y. Toda, Bridgeland stability conditions on threefolds I: Bogomolov-Gieseker type inequalities, J. Algebraic Geom. 23 (2014), 117–163.

[7] A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Ast ́erisque 100 (1982).

[8] T. Bridgeland, Stability conditions on triangulated categories, Ann. Math. 166 (2007), 317–345.

[9] T. Bridgeland, Y. Qiu, T. Sutherland, Stability conditions and the A2 quiver, arXiv:1406.2566.

[10] N. Broomhead, D. Pauksztello, D. Ploog, Discrete derived categories II: the silting pairs CW complex and the stability manifold, J. London Math. Soc. 93 (2016), 273–300.

[11] Th. Bru ̈stle, D. Yang, Ordered exchange graphs, Advances in representation theory of algebras, 135–193, EMS Ser. Congr. Rep., Eur. Math. Soc., Zu ̈rich, 2013.

[12] Th. Bru ̈stle, Y. Qiu Tagged mapping class groups I: Auslander-Reiten translation, Math. Zeitschrift, Volume 279, Issue 3 (2015), Page 1103–1120.

[13] Th. Bru ̈stle and J.Zhang, On the cluster category of a marked surface, Algebra & Number Theory 5-4 (2011), 529–566. DOI 10.2140/ant.2011.5.529.

[14] A. B. Buan, I. Reiten, H. Thomas, Three kinds of mutation, J. Algebra 339 (2011), 97–113.

[15] L. Demonet, O. Iyama, G. Jasso, τ-tilting-finite algebras, g-vectors and brick-τ-rigid correspondence, arXiv:1503.00285.

[16] H. Derksen, J. Weyman, A. Zelevinsky. Quivers with potentials and their representations I: Mutations., Selecta Mathematica 14.1 (2008): 59–119.

[17] G. Dimitrov, F. Haiden, L. Katzarkov, M. Kontsevich, Dynamical systems and categories, in “The influence of Solomon Lefschetz in geometry and topology”, Contemporary Mathematics 621, 133–170, Amer. Math. Soc., Providence, RI, 2014.

[18] G. Dimitrov, L. Katzarkov, Bridgeland stability conditions on the acyclic triangular quiver, Adv. Math. 288 (2016), 825–886.

[19] M. Douglas, D-branes, categories and N = 1 supersymmetry. Strings, branes and M-theory, J. Math. Phys 42 (2001), 2818–2843.

[20] M. Douglas, Dirichlet branes, homological mirror symmetry, and stability, Proceedings of the Interna- tional Congress of Mathematicians, Vol. III (Beijing, 2002), 395–408, Higher Ed. Press, Beijing, 2002.

[21] W. Geigle, H. Lenzing, A class of weighted projective curves arising in representation theory of finite- dimensional algebras, in “Singularities, representations of algebras, and vector bundles (Lambrecht, 1985)”, Lecture Notes in Mathematics vol. 1273, 265–297, Springer, Berlin, 1987.

[22] A. L. Gorodentsev, S. A. Kuleshov, A. N. Rudakov, t-stabilities and t-structures on triangulated cate- gories, Izv. Math 68 (2004), 749–781.

[23] M. Gross, B. Siebert, From real affine geometry to complex geometry, Annals of mathematics, 174, (2011), 1301–1428.

[24] F. Haiden, L. Katzarkov, M. Kontsevich, Flat surfaces and stability structures, arXiv:1409.8611.

[25] G. Harder, M. S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1974/75), 215–248.

[26] K. Igusa, K. Orr, G. Todorov, J. Weyman, Cluster complexes via semi-invariants, Compositio Mathematica, 145(04), (2009), 1001–1034.

[27] C. Ingalls, H. Thomas, Noncrossing partitions and representations of quivers, Compos. Math. 145 (2009), 1533–1562.

[28] C. Ingalls, C. Paquette, H. Thomas, Semi-stable subcategories for Euclidean quivers, Proc. London Math. Soc. 110 (2015), 805–840.

[29] O. Iyama, Y. Yoshino, Mutation in triangulated categories and rigid Cohen-Macaulay modules, Invent. Math. 172 (2008), 117–168.

[30] P. Jørgensen, D. Pauksztello, Classification of co-slicings and co-t-structures for the Kronecker algebra, J. Pure Appl. Algebra 219 (2015), 569–590.

[31] A. D. King, Moduli of representations of finite dimensional algebras, Quart. J. Math. Oxford 45 (1994), 515–530.

[32] A. D. King, Y. Qiu, Exchange graphs and Ext quivers, Adv. Math. 285 (2015), 1106–1154.

[33] S. Konig, D. Yang, Silting objects, simple-minded collections, t-structures and co-t-structures for finite-dimensional algebras, Doc. Math. 19 (2014), 403–438.

[34] M. Kontsevich, Y. Soibelman, Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and mirror symmetry, Homological Mirror Symmetry and Tropical Geometry. Springer International Publishing, 2014. 197–308.

[35] D. Kussin, Noncommutative curves of genus zero: related to finite dimensional algebras, Mem. Amer. Math. Soc. 201 (2009), x+128.

[36] D. Labardini Fragoso, Quivers with potentials associated to triangulated surfaces, Proc. London Math. Soc. 98 (2009), 797–839.

[37] H. Lenzing, I. Reiten, Hereditary Noetherian categories of positive Euler characteristic, Math. Z. 254 (2006), 133–171.

[38] E. Macri, Stability conditions on curves, Math. Res. Lett. 14 (2007), 657–672.

[39] D. Mumford, Geometric Invariant Theory, Springer-Verlag Berlin (1965).

[40] S. Okada, Stability manifold of P1, J. Algebraic Geom. 15 (2006), 487–505.

[41] Y. Qiu, Stability conditions and quantum dilogarithm identities for Dynkin quivers, Adv. Math. 269 (2015), 220–264.

[42] Y. Qiu, J. Woolf, Contractible stability spaces and faithful braid group actions, arXiv:1407.5986.

[43] A. Rudakov, Stability for an abelian category, J. Algebra 197 (1997), 231–245.

[44] A. Schofield, Semi-invariants of quivers, J. London Mathematical Society (2), no. 3 (1991), 385–395.

[45] J. Woolf, Stability conditions, torsion theories and tilting, J. London Math. Soc. 82 (2010), 663–682.