# Cluster algebras, representation theory, and Poisson geometry (11w5137)

Arriving in Banff, Alberta Sunday, September 4 and departing Friday September 9, 2011

## Organizers

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

Christof Geiss (Universidad Nacional Autónoma de México)

Michael Shapiro (Michigan State University)

Hugh Thomas (University of New Brunswick)

## Objectives

Origins of Cluster Algebras

Cluster algebras where conceived in spring 2000 by S. Fomin and A. Zelevinsky [FZ02a] as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. They are constructively defined commutative algebras with a distinguished set of generators (cluster variables) grouped into overlapping subsets (clusters) of fixed cardinality. They have the unusual feature that both the generators and the relations among them are not given from the outset, but are produced by an elementary iterative process of seed mutation. This procedure appears counter-intuitive at first, but it seems to encode a somehow universal phenomenon which might explain the explosive development of this topic. Indeed, by now close connections to Poisson geometry, Teichmueller theory, representation theory of finite dimensional associative algebras and Lie theory and Coxeter groups have been discovered. The theory of cluster algebras was further developed in the subsequent papers [FZ02b, FZ03, BFZ05, BZ05, FZ07, DWZ08, DWZ09]. Remarkably, in the last two paper of this series super potentials borrowed from mathematical physics play a prominent role.

Much of this research has been conducted by first rate mathematicians like A. Goncharov, B. Keller, M. Kontsevich, B. Leclerc, H. Nakajima, I. Reiten, C.M. Ringel.

Teichmueller Theory

Fock and Goncharov introduced in the seminal paper [FG06b] higher Teichmueller theory as a fusion of the classical theory with representation theory by an algebraic geometry approach, opening completely new perspectives. Somehow Gekhtman, Shapiro and Vainshtein put forth in [GSV03] and [GSV05] a similar program. Basic ingredients of the emerging theory include explicit coordinate descriptions of the the appropriate decorated Teichmueller space (an idea which goes back to W. Thurston [T80] and R. Penner [P87]), and the concept of of total positivity in Lie theory which originated in the work of G. Lusztig [L94, L98]. In [FG06b, FG03] it was shown that the A and X versions of the Teichmueller and lamination spaces can be obtained as the positive real and tropical points of certain cluster A- and X-varieties, the putative spectra of cluster algebras.

These ideas motivated S. Fomin, M. Shapiro and D. Thurston to develop in [FST09] systematically the theory of cluster algebras associated to triangulated surfaces. This last paper sparked even before its publication quite some research into various descriptions. For example Labardini [L09] started to associate to the cluster algebras from that paper quiver potentials. The combinatorial research of several authors on the positivity of cluster expansions for this class of algebras culminated recently in a complete answer [MSW09]. And, based on [L09] in [ABCP09] the connection to the representation theory of (tame) gentle algebras was made.

Quantum Teichmueller spaces were constructed independently by Chekhov and Fock [CF99] and by Kashaev [K98], they are also linked to cluster algebras and varieties [G07, FG09]. In [T04] a conjecture of Verlinde is further discussed, namely that the mapping class group acts on the quantum Teichmueller space in the same way as on conformal blocks of the Liouville conformal field theory.

The relation between Poisson geometry and cluster structures in double Bruhat cells (for Lie groups and their flag varieties) were further investigated by K. Brown, K. Goodearl and M. Yakimov [BGY06,WY07] and by Kogan and Zelevinsky [KZ02].

As a result of these developments, cluster theory becomes enriched by new examples as well as new features such as duality (see also [FZ07]), Poisson structure and quantization (see also [GSV03,FG06a]) and relations to algebraic $K$-theory and the dilogarithm.

References

[ABCP09] I. Assem, Th. Bruestle, G. Charbonneau-Jodoin, and P.G. Plamondon, Gentle algebras arising from surface triangulations, 37 pages, 2009, arXiv:0903.3347v2 [math.RT]

[BFZ05] A. Berenstein, S. Fomin, and A. Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), no. 1, 1-52.

[BZ05] A. Berenstein and A. Zelevinsky, Quantum cluster algebras, Adv. Math. 195 (2005), no. 2, 405-455.

[BGY06] K. A. Brown, K. R. Goodearl, and M. Yakimov, Poisson structures on affine spaces and flag varieties. I. Matrix affine Poisson space, Adv. Math. 206 (2006), no. 2, 567-629.

[CF99] L. O. Chekhov and V. V. Fock, Quantum Teichmueller spaces, Teoret. Mat. Fiz. 120 (1999), no. 3, 511-528, arXiv: math/9908165 [math.QA].

[DWZ08] H. Derksen, J. Weyman, and A. Zelevinsky, Quivers with potentials and their representations. I. Mutations, Selecta Math. (N.S.) 14 (2008), no. 1, 59-119.

[DWZ09] ----, Quivers with potentials and their representations II: Applications to cluster algebras, 44 pages, 2009, arXiv:0904.0676v1 [math.RA].

[FG03] V. Fock and A Goncharov, Cluster ensembles, quantization and the dilogarithm, 69 pages, arXiv:math/0311245 [math.AG].

[FG06a] ----, Cluster X-varieties, amalgamation and Poisson-Lie groups, Algebraic geometry and number theory, Progr. Math., vol. 253, Birkhaeuser Boston, Boston, MA, 2006, Volume dedicated to V. Drinfeld, pp. 27-68.

[FG06b] ----, Moduli spaces of local systems and higher Teichmueller theory, Publ. Math. Inst. Hautes Etudes Sci. (2006), no. 103, 1-211.

[FG09] ----, The quantum dilogarithm and representations of quantum cluster varieties, Invent. Math. 175 (2009), no. 2, 223-286.

[FST09] S. Fomin, M. Shapiro, and D. Thurston, Cluster algebras and triangulated surfaces. Part I: Cluster complexes., Acta Math. 201 (2008), no. 1, 83-146.

[FZ02a] S. Fomin and A. Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497-529 (electronic).

[FZ02b] ----, The Laurent phenomenon, Adv. in Appl. Math. 28 (2002), no. 2, 119-144.

[FZ03] ----, Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), no. 1, 63-121.

[FZ07] ----, Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007), no. 1, 112-164.

[GSV03] M. Gekhtman, M. Shapiro, and A. Vainshtein, Cluster algebras and Poisson geometry, Mosc. Math. J. 3 (2003), no. 3, 899-934, 1199, Dedicated to Vladimir Igorevich Arnold on the occasion of his 65th birthday.

[GSV05] ----, Cluster algebras and Weil-Petersson forms, Duke Math. J. 127 (2005), no. 2, 291-311.

[G07] A. Goncharov, Pentagon relation for the quantum dilogarithm and quantized M(0,5)cyc, to appear in Progress in Mathematics volume (Birkhauser) dedicated to the memory of Alexander Reznikov, arXiv/0704.405v2 [math.QA].

[K98] R. M. Kashaev, Quantization of Teichmueller spaces and the quantum dilogarithm, Lett. Math. Phys. 43 (1998), no. 2, 105-115, arXiv:math/9706018 [math.QA].

[KZ02] M. Kogan and A. Zelevinsky, On symplectic leaves and integrable systems in standard complex semisimple Poisson-Lie groups, Int. Math. Res. Not. (2002), no. 32, 1685-1702.

[L09] D. Labardini-Fragoso, Quivers with potentials associated to triangulated surfaces, Proc. Lond. Math. Soc. (3) 98 (2009), no. 3, 797-839, arXiv:0803.1328v3.

[L94] G. Lusztig, Total positivity in reductive groups, Lie theory and geometry, Progr. Math., vol. 123, Birkhaeuser Boston, Boston, MA, 1994, pp. 531-568.

[L98] ----, Introduction to total positivity, Positivity in Lie theory: open problems, de Gruyter Exp. Math., vol. 26, de Gruyter, Berlin, 1998, pp. 133-145.

[MSW] G. Musiker, R. Schiffler, and L. Williams, Positivity for cluster algebras from surfaces, 67 pages, 2009, arXiv:0906.0748v1 [math.CO].

[P87] R. C. Penner, The decorated Teichmueller space of punctured surfaces, Comm. Math. Phys. 113 (1987), no. 2, 299-339.

[T04] J. Teschner, On the relation between quantum Liouville theory and the quantized Teichmueller spaces, Proceedings of 6th International Workshop on Conformal Field Theory and Integrable Models, vol. 19, 2004, pp. 459-477.

[T80] W. Thurston, The geometry and topology of three-manifolds, Princeton University notes, http://www.msri.org/publications/books/gt3m.

[WY07] Ben Webster and Milen Yakimov, A Deodhar-type stratification on the double flag variety, Transform. Groups 12 (2007), no. 4, 769-785.

Cluster algebras where conceived in spring 2000 by S. Fomin and A. Zelevinsky [FZ02a] as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. They are constructively defined commutative algebras with a distinguished set of generators (cluster variables) grouped into overlapping subsets (clusters) of fixed cardinality. They have the unusual feature that both the generators and the relations among them are not given from the outset, but are produced by an elementary iterative process of seed mutation. This procedure appears counter-intuitive at first, but it seems to encode a somehow universal phenomenon which might explain the explosive development of this topic. Indeed, by now close connections to Poisson geometry, Teichmueller theory, representation theory of finite dimensional associative algebras and Lie theory and Coxeter groups have been discovered. The theory of cluster algebras was further developed in the subsequent papers [FZ02b, FZ03, BFZ05, BZ05, FZ07, DWZ08, DWZ09]. Remarkably, in the last two paper of this series super potentials borrowed from mathematical physics play a prominent role.

Much of this research has been conducted by first rate mathematicians like A. Goncharov, B. Keller, M. Kontsevich, B. Leclerc, H. Nakajima, I. Reiten, C.M. Ringel.

Teichmueller Theory

Fock and Goncharov introduced in the seminal paper [FG06b] higher Teichmueller theory as a fusion of the classical theory with representation theory by an algebraic geometry approach, opening completely new perspectives. Somehow Gekhtman, Shapiro and Vainshtein put forth in [GSV03] and [GSV05] a similar program. Basic ingredients of the emerging theory include explicit coordinate descriptions of the the appropriate decorated Teichmueller space (an idea which goes back to W. Thurston [T80] and R. Penner [P87]), and the concept of of total positivity in Lie theory which originated in the work of G. Lusztig [L94, L98]. In [FG06b, FG03] it was shown that the A and X versions of the Teichmueller and lamination spaces can be obtained as the positive real and tropical points of certain cluster A- and X-varieties, the putative spectra of cluster algebras.

These ideas motivated S. Fomin, M. Shapiro and D. Thurston to develop in [FST09] systematically the theory of cluster algebras associated to triangulated surfaces. This last paper sparked even before its publication quite some research into various descriptions. For example Labardini [L09] started to associate to the cluster algebras from that paper quiver potentials. The combinatorial research of several authors on the positivity of cluster expansions for this class of algebras culminated recently in a complete answer [MSW09]. And, based on [L09] in [ABCP09] the connection to the representation theory of (tame) gentle algebras was made.

Quantum Teichmueller spaces were constructed independently by Chekhov and Fock [CF99] and by Kashaev [K98], they are also linked to cluster algebras and varieties [G07, FG09]. In [T04] a conjecture of Verlinde is further discussed, namely that the mapping class group acts on the quantum Teichmueller space in the same way as on conformal blocks of the Liouville conformal field theory.

The relation between Poisson geometry and cluster structures in double Bruhat cells (for Lie groups and their flag varieties) were further investigated by K. Brown, K. Goodearl and M. Yakimov [BGY06,WY07] and by Kogan and Zelevinsky [KZ02].

As a result of these developments, cluster theory becomes enriched by new examples as well as new features such as duality (see also [FZ07]), Poisson structure and quantization (see also [GSV03,FG06a]) and relations to algebraic $K$-theory and the dilogarithm.

References

[ABCP09] I. Assem, Th. Bruestle, G. Charbonneau-Jodoin, and P.G. Plamondon, Gentle algebras arising from surface triangulations, 37 pages, 2009, arXiv:0903.3347v2 [math.RT]

[BFZ05] A. Berenstein, S. Fomin, and A. Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), no. 1, 1-52.

[BZ05] A. Berenstein and A. Zelevinsky, Quantum cluster algebras, Adv. Math. 195 (2005), no. 2, 405-455.

[BGY06] K. A. Brown, K. R. Goodearl, and M. Yakimov, Poisson structures on affine spaces and flag varieties. I. Matrix affine Poisson space, Adv. Math. 206 (2006), no. 2, 567-629.

[CF99] L. O. Chekhov and V. V. Fock, Quantum Teichmueller spaces, Teoret. Mat. Fiz. 120 (1999), no. 3, 511-528, arXiv: math/9908165 [math.QA].

[DWZ08] H. Derksen, J. Weyman, and A. Zelevinsky, Quivers with potentials and their representations. I. Mutations, Selecta Math. (N.S.) 14 (2008), no. 1, 59-119.

[DWZ09] ----, Quivers with potentials and their representations II: Applications to cluster algebras, 44 pages, 2009, arXiv:0904.0676v1 [math.RA].

[FG03] V. Fock and A Goncharov, Cluster ensembles, quantization and the dilogarithm, 69 pages, arXiv:math/0311245 [math.AG].

[FG06a] ----, Cluster X-varieties, amalgamation and Poisson-Lie groups, Algebraic geometry and number theory, Progr. Math., vol. 253, Birkhaeuser Boston, Boston, MA, 2006, Volume dedicated to V. Drinfeld, pp. 27-68.

[FG06b] ----, Moduli spaces of local systems and higher Teichmueller theory, Publ. Math. Inst. Hautes Etudes Sci. (2006), no. 103, 1-211.

[FG09] ----, The quantum dilogarithm and representations of quantum cluster varieties, Invent. Math. 175 (2009), no. 2, 223-286.

[FST09] S. Fomin, M. Shapiro, and D. Thurston, Cluster algebras and triangulated surfaces. Part I: Cluster complexes., Acta Math. 201 (2008), no. 1, 83-146.

[FZ02a] S. Fomin and A. Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497-529 (electronic).

[FZ02b] ----, The Laurent phenomenon, Adv. in Appl. Math. 28 (2002), no. 2, 119-144.

[FZ03] ----, Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), no. 1, 63-121.

[FZ07] ----, Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007), no. 1, 112-164.

[GSV03] M. Gekhtman, M. Shapiro, and A. Vainshtein, Cluster algebras and Poisson geometry, Mosc. Math. J. 3 (2003), no. 3, 899-934, 1199, Dedicated to Vladimir Igorevich Arnold on the occasion of his 65th birthday.

[GSV05] ----, Cluster algebras and Weil-Petersson forms, Duke Math. J. 127 (2005), no. 2, 291-311.

[G07] A. Goncharov, Pentagon relation for the quantum dilogarithm and quantized M(0,5)cyc, to appear in Progress in Mathematics volume (Birkhauser) dedicated to the memory of Alexander Reznikov, arXiv/0704.405v2 [math.QA].

[K98] R. M. Kashaev, Quantization of Teichmueller spaces and the quantum dilogarithm, Lett. Math. Phys. 43 (1998), no. 2, 105-115, arXiv:math/9706018 [math.QA].

[KZ02] M. Kogan and A. Zelevinsky, On symplectic leaves and integrable systems in standard complex semisimple Poisson-Lie groups, Int. Math. Res. Not. (2002), no. 32, 1685-1702.

[L09] D. Labardini-Fragoso, Quivers with potentials associated to triangulated surfaces, Proc. Lond. Math. Soc. (3) 98 (2009), no. 3, 797-839, arXiv:0803.1328v3.

[L94] G. Lusztig, Total positivity in reductive groups, Lie theory and geometry, Progr. Math., vol. 123, Birkhaeuser Boston, Boston, MA, 1994, pp. 531-568.

[L98] ----, Introduction to total positivity, Positivity in Lie theory: open problems, de Gruyter Exp. Math., vol. 26, de Gruyter, Berlin, 1998, pp. 133-145.

[MSW] G. Musiker, R. Schiffler, and L. Williams, Positivity for cluster algebras from surfaces, 67 pages, 2009, arXiv:0906.0748v1 [math.CO].

[P87] R. C. Penner, The decorated Teichmueller space of punctured surfaces, Comm. Math. Phys. 113 (1987), no. 2, 299-339.

[T04] J. Teschner, On the relation between quantum Liouville theory and the quantized Teichmueller spaces, Proceedings of 6th International Workshop on Conformal Field Theory and Integrable Models, vol. 19, 2004, pp. 459-477.

[T80] W. Thurston, The geometry and topology of three-manifolds, Princeton University notes, http://www.msri.org/publications/books/gt3m.

[WY07] Ben Webster and Milen Yakimov, A Deodhar-type stratification on the double flag variety, Transform. Groups 12 (2007), no. 4, 769-785.