# Lectures

## Anne Dranowski

**Crystals and Preprojective Algebra Modules**

Underlying every irreducible representation of a semisimple simply-laced Lie algebra is a colored directed graph called its crystal. Crystals can be realized combinatorially, geometrically, and categorically. For example, components of Springer fibres yield a geometric realization, which is compatible with the combinatorial realization given by semistandard Young tableaux. In joint work with B. Elek, J. Kamnitzer and C. Morton-Ferguson, we generalize the latter well-known type A specific combinatorial realization. In place of tableaux we work with reverse plane partitions, and to establish the crystal structure, we relate these to a space of modules for the preprojective algebra. In these lectures we will begin by reviewing the classical results on tableaux and Springer fibres, as well as the notion of a g-crystal. Next we will cover the combinatorics necessary for working with reverse plane partition (Stembridge). We will then discuss the crystal structures on components of quiver Grassmannians and cores of quiver varieties (Nakajima, Savage and Tingley). Finally, we will discuss the crystal structure on reverse plane partitions, as well as future directions, including possible connections to recent work of Garver, Patrias, and Thomas, as well as Baumann, Knutson, and Kamnitzer.

**References**

- Berenstein, A. & Kazhdan, D. (2007). Lecture Notes on Geometric Crystals and their Combinatorial Analogues. Mathematical Society of Japan Memoirs Vol. 17, 1-9.
- Bjorner, A., & Brenti, F. (2006). Combinatorics of Coxeter groups (Vol. 231). Springer Science & Business Media.
- Hong, J., & Kang, S. J. (2002). Introduction to quantum groups and crystal bases (Vol. 42). American Mathematical Soc..
- Nakajima, H. (2001). Quiver varieties and tensor products. Inventiones mathematicae, 146(2), 399-449.
- Ram, A. (2015). The glass bead game. BrisScience series at The EDGE.
- Ram, A., & Kleshchev, A. S. (2010). Homogeneous representations of Khovanov-Lauda algebras. Journal of the European Mathematical Society, 12(5), 1293-1306.
- Saito, Y. (2002). Crystal bases and quiver varieties. Mathematische Annalen, 324(4), 675-688.
- Savage, A. (2010). Lectures on geometric realizations of crystals. arXiv preprint arXiv:1003.5019.
- Savage, A., & Tingley, P. (2011). Quiver grassmannians, quiver varieties and the preprojective algebra. Pacific journal of mathematics, 251(2), 393-429.
- Stembridge, J. R. (2001). Minuscule elements of Weyl groups. Journal of Algebra, 235(2), 722-743.
- Tingley, P. (2008). Three combinatorial models for $\widehat{\mathrm{sl}}_n$ crystals, with applications to cylindric plane partitions. International Mathematics Research Notices, 2008.

## Bernard Leclerc

**Generalized preprojective algebras**

Classical preprojective algebras were introduced by Gelfand-Ponomarev and further studied by Dlab-Ringel and many others. Their connection with Lie theory was discovered by Lusztig, who realized the enveloping algebra of the positive part of a symmetric Kac-Moody Lie algebra as a convolution algebra of constructible functions on module varieties of a preprojective algebra. In joint work with Geiss and Schröer, we used Lusztig’s construction to obtain categorifications of the cluster algebra structure on coordinate rings of certain finite-dimensional unipotent subgroups of symmetric Kac-Moody groups labelled by Weyl group elements. These results require the generalized Cartan matrix to be symmetric. So the question arises of a possible generalization to the symmetrizable case. This motivated a series of joint works with Geiss and Schröer on a new class of 1-Iwanaga-Gorenstein algebras associated with symmetrizable generalized Cartan matrices, and their generalized preprojective algebras. In these talks I will give a survey of this construction and its application to crystal bases, and I will explain how it relates to joint work with Hernandez on representations of quantum affine algebras. Finally if time allows, I will discuss some recent developments by Murakami and Fujita-Murakami.

**References**

- Ringel, The preprojective algebra of a quiver (1996)
- Lusztig, Semicanonical bases arising from enveloping algebras (2000)
- Geiss, Leclerc, Schröer, Cluster algebras in algebraic Lie theory (2013)
- Hernandez, Leclerc, A cluster algebra approach to q-characters of Kirillov-Reshetikhin modules (2016)
- Geiss, Leclerc, Schröer, Quivers with relations for symmetrizable Cartan matrices, I: Foundations (2017), IV: Crystal graphs and semicanonical functions (2018)
- Murakami, PBW parametrizations and generalized preprojective algebras (2020) Fujita, Murakami, Deformed Cartan matrices and generalized preprojective algebras of finite type (2021)

## Matthew Pressland

**Dimer models: consistency, Calabi–Yau properties and categorification**

A dimer model is a bipartite graph drawn in a surface. First introduced in the context of statistical mechanics, dimer models became a significant topic in string theory around fifteen years ago. In mathematics, a key development at this time was the study of consistency conditions, and the use of dimer models on the torus to construct non-commutative crepant resolutions of 3-dimensional Gorenstein singularities. A key property of a consistent dimer model is that its associated non-commutative dimer algebra is 3-Calabi–Yau. More recently, dimer models have reappeared, now on surfaces with boundary and sometimes called plabic graphs or Postnikov diagrams, in the context of categorifying cluster algebras with coefficients, notably the cluster structure on the Grassmannian and its positroid strata. In this lecture series, I will survey these ideas.

**Main References**

- K. Baur, A.D. King, B.R. Marsh: Dimer models and cluster categories of Grassmannians, Proc. London Math. Soc. 113 (2016), pp. 213–260.
- N. Broomhead: Dimer models and Calabi–Yau algebras, Mem. Amer. Math. Soc. 215 (2012)
- İ. Çanakçı, A. King, M. Pressland: Perfect matchings, dimer partition functions and cluster characters, arXiv:2016.15924 [math.RT]
- B.T. Jensen, A.D. King, X. Su: A categorification of Grassmannian cluster algebras, Proc. London Math. Soc. 113 (2016), pp. 185–212
- M. Pressland: Calabi–Yau properties of Postnikov diagrams, arXiv:1912.12475 [math.RT]

**Additional References**

- R. Bocklandt: A dimer ABC, Bull. Lond. Math. Soc. 48 (2016), pp. 387–451
- S. Franco: Bipartite field theories: from D-brane probes to scattering amplitudes, J. High Energy Phys., JHEP11(2012)141
- S. Franco, A. Hanany, D. Vegh, B. Wecht, K.D. Kennaway: Brane dimers and quiver gauge theories, J. High Energy Phys., JHEP01(2006)096
- P. Galashin, T. Lam: Positroid varieties and cluster algebras, Ann. Sci. Éc. Norm. Supér., to appear (arXiv:1906.03501 [math.CO])
- M. Pressland: Internally Calabi–Yau algebras and cluster-tilting objects, Math. Z. 287 (2017), pp. 555–585
- J.S. Scott: Grassmannians and cluster algebras, Proc. London Math. Soc. 92 (2006), pp. 345–380.

# Speakers

## Akishi Ikeda

**Calabi-Yau algebras and canonical bundles**

In this talk, we explore common features of Calabi-Yau algebras in representation theory and Calabi-Yau manifolds in geometry. First we see the role of inverse dualizing complexes in both sides and recall why Calabi-Yau algebras are called “Calabi-Yau” algebras. Next we see that the Calabi-Yau completions (derived preprojective algebras) can be interpreted asthe total spaces of canonical bundles of smooth varieties. If time permits, I explain the relationship between double gradings of Calabi-Yau completions and weights of torus actions on canonical bundles. This is a joint work with Yu Qiu.

## Kota Murakami

**Deformed Cartan matrices and generalized preprojective algebras**

E.Frenkel-Reshetikhin introduced a kind of quantization of the Cartan matrix, called the $(q, t)$-deformed Cartan matrix, which characterizes some aspects of the representation theory of affine quantum groups and deformed W-algebras with their specializations. In this talk, we will interpret the $(q, t)$-deformed Cartan matrix as an invariant from the representation theory of a bigraded version of the generalized preprojective algebra introduced by Geiss-Leclerc-Schröer. In particular, we will interpret several numerical properties of the $(q, t)$-deformed Cartan matrix as homological properties from the representation theory of the generalized preprojective algebra. This talk is based on a joint work with Ryo Fujita.