Osaka Central Advanced Mathematical Institute (OCAMI)
Osaka Metropolitan University
MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics

OCAMI Reports Vol. 1 (2022)
doi: 10.24544/ocu.20220521-001

Preprojective Algebras andCalabi-Yau Algebras
Organized by
Yuya Mizuno
Haruhisa Enomoto
Yoshiyuki Kimura
Masato Okado
Toshiya Yurikusa
March 1–4, 2022
Abstract
This workshop held March 1–4, 2022 to conduct international researchexchanges on Preprojective Algebras and Calabi-Yau Algebras.

2020 Mathematics Subject Classiﬁcation.
16G10, 16G20
Key words and Phrases.
Preprojective Algebras, Calabi-Yau Algebras, cluster algebras,cluster theory, crystals, categoriﬁcation© 2022 OCAMI.
OCAMI. Preprojective Algebras and Calabi-Yau Algebras. OCAMI Reports. Vol. 1, Osaka Central
Advanced Mathematical Institute, Osaka Metropolitan University. 2022, 93pp.
doi: 10.24544/ocu.20220521-001

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OCAMI Reports Vol. 1 (2022)

PrefaceThis volume of OCAMI Reports summarizes the workshop Preprojective Algebras
and Calabi-Yau Algebras held from March 1 to March 4 in 2022 online by Zoom
(because of the COVID-19 pandemic). This workshop was supported by Osaka city
University, Advanced Mathematical Institute MEXT Joint Usage/Research Center
on Mathematics and Theoretical Physics.
Preprojective algebras are algebras determined by a quiver, which are not only
important in their representation theory but are also fundamental objects used in theLusztig and Nakajima quiver varieties, and they have a deep connection between
representation theory of Lie algebras and quantum groups. The purpose of thisworkshop is to bring together researchers focusing on the representation theory of
preprojective algebras and Calabi-Yau algebras and the connection between related
combinatorics, geometry and integrable systems.
We have three lecturers : Professor Anne Dranowski gave a series of lectures
about Crystals and Preprojective Algebra Modules. Professor Matthew Pressland
gave a series of lectures about Dimer models: consistency, Calabi-Yau properties andcategoriﬁcation. Professor Bernard Leclerc gave a series of lectures about Generalized
preprojective algebras. Moreover, we have two Speakers : Professor Akishi Ikeda gave
a talk about Calabi-Yau algebras and canonical bundles. Professor Kota Murakamigave a talk about Deformed Cartan matrices and generalized preprojective algebras.
All speakers gave talks on recent developments, which promoted the fruitful
discussions for especially young researchers. The organizers are convinced that discussions will lead to future research projects and a discovery of a new relation.

Preprojective Algebras and Calabi-Yau Algebrasiii

Organizers
Yuya Mizuno
Faculty of Liberal Arts, Sciences and Global Education, Osaka Metropolitan University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan
Email address: yuya.mizuno@omu.ac.jp
Haruhisa Enomoto
Graduate School of Science, Osaka Metropolitan University, 1-1 Gakuen-cho, Nakaku, Sakai, Osaka 599-8531, Japan
Email address: the35883@osakafu-u.ac.jp
Yoshiyuki Kimura
Faculty of Liberal Arts, Sciences and Global Education, Osaka Metropolitan University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan
Email address: ysykimura@omu.ac.jp
Masato Okado
Graduate School of Science, Osaka Metropolitan University, 3-3-138 Sugimoto, Sumiyoshiku, Osaka, 558-8585, Japan
Email address: okado@omu.ac.jp
Toshiya Yurikusa
Mathematical Institute, Tohoku University, Aoba-ku, Sendai, 980-8578, Japan
Email address: toshiya.yurikusa.d8@tohoku.ac.jp

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OCAMI Reports Vol. 1 (2022)

Contents
Matthew Pressland
Dimer models: consistency, Calabi – Yau properties and categorification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Anne Dranowski
Crystals and Preprojective Algebra Modules

8. . . . . . . . . . . . . .

Bernard Leclerc
Generalized preprojective algebras . . . . . . . . . . . . . . . . . . . . 45
Akishi Ikeda
Calabi-Yau algebras and canonical bundles . . . . . . . . . . . . . . . 68
Kota Murakami
Deformed Cartan matrices and generalized preprojective algebras. . 81Preprojective Algebras and Calabi-Yau Algebras

Dimer models: consistency, Calabi–Yau
properties and categorification
Matthew Pressland
Abstract. A dimer model is a bipartite graph drawn in a surface. First
introduced in the context of statistical mechanics, dimer models became asignificant topic in string theory around fifteen years ago. In mathematics,
a key development at this time was the study of consistency conditions, andthe use of dimer models on the torus to construct non-commutative crepant
resolutions of 3-dimensional Gorenstein singularities. A key property of aconsistent dimer model is that its associated non-commutative dimer algebra is3-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 thecluster structure on the Grassmannian and its positroid strata. In this lecture
series, I will survey these ideas.
School of Mathematics and Statistics, University of Glasgow, Glasgow, UK
Email address: matthew.pressland@glasgow.ac.uk

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DIMER MODELS: CONSISTENCY, CALABI–YAU PROPERTIES AND
CATEGORIFICATION
MATTHEW PRESSLAND

1. Dimer models
A dimer model D consists of a finite bipartite graph drawn in the interior of an oriented
topological surface Σ, together with a finite collection of half-edges which connect nodes of thisgraph to distinct points on the boundary ∂Σ. We require that Σ◦ \ D is a union of open discs, eachof which is called a face of D. Dimer models have found many applications in mathematics and inphysics, relating to statistical mechanics [22, 34], string theory [12, 18], non-commutative geometry
and singularity theory [3, 6, 9], and most recently to the theory of cluster algebras and categories[1, 8, 14, 24, 26, 27, 32]. We are particularly interested in the case that either Σ is the torus (which
is a closed surface, so that D is an honest bipartite graph, without half-edges) or that Σ is the disc.
To a dimer model D we may associate the dual quiver Q = QD , with vertex set Q0 given by thefaces of D, and arrow set Q1 given by the edges (and half-edges) of D. An arrow in Q1 connects
the two faces incident along the corresponding edge, and is oriented so that the black node of thisedge is on the left of the arrow (or, if the edge is a half-edge with only a white node, this whitenode is on the right of the arrow). Due to its embedding in Σ, the quiver Q also has a natural set
Q2 of faces, distinguished cycles in Q bounding components of Σ \ Q, which are in bijection with
the nodes of D. The bipartite structure of D is reflected in the orientations of these distinguishedcycles, which can be either positive or negative, with two faces sharing an arrow having opposite
orientations. Positive quiver faces correspond to black nodes of D, while negative ones correspondto white nodes.
Definition 1. The dimer algebra AD of a dimer model D is the complete path algebra of QD
modulo the closed ideal generated by relations p − q, where p, q : i → i both bound a face, and−+−p+
a − pa for a ∈ Q1 \ F1 , where apa and apa are, respectively, the boundaries of the positively andnegatively oriented faces incident with a.
∗

∗
∗

•

•

∗

∗

Figure 1. A dimer model in the disc, together with its dual quiver. Frozen
vertices of this quiver are indicated by asterisks, and frozen arrows are dashed.
In the case that D is connected (which is automatic if Σ is closed), the relations of the formp − q are redundant, and so AD is an instance of a frozen Jacobian algebra. Here we treat Q as an
ice quiver, in the sense of [1, 29, 30] (so there are frozen arrows as well as frozen vertices, and not
Date: April 29, 2022.
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MATTHEW PRESSLAND

every arrow between frozen vertices has to be frozen), by taking the frozen vertices to be faces of D
incident with ∂Σ and the frozen arrows to be the half-edges of D. To define the frozen Jacobianalgebra we also put a potential on Q, which is the sum of positively oriented faces minus the sum of−negatively oriented faces; thus each relation p+
a − pa is the derivative of this potential with respect
to the unfrozen arrow a. Note that the frozen arrows, which correspond to half-edges, lie in theboundary of only one face, and so do not contribute relations.
2. Consistency and the Calabi–Yau condition
A dimer model D has a collection of zig-zag paths, walks along the edges which turn maximallyto the right at black nodes and maximally to the left at white nodes. Thus each edge of the dimer
model may be treated as the transverse intersection of two zig-zag paths, which traverse this edgein opposite directions. Each zig-zag path is either a closed loop in the interior of Σ, or a pathwith both endpoints on ∂Σ. In the following definition, the zig-zag paths starting and ending on ahalf-edge of D are taken to cross transversely at this half edge if they are distinct (cf. Figure 2),
but not if they are the same: in this second case we consider the zig-zag path to be a closed loop at
the boundary point on the half-edge.
e be its lift to the universal cover of Σ; note that
Definition 2. Let D be a dimer model, and let De lift those of D. We say that D is consistent if
the zig-zag paths of D(a) its zig-zag paths have no transverse self-intersections, ande intersect twice, they are oriented in opposite directions between(b) if two zig-zag paths on D
these two intersection points.

Figure 2. The zig-zag paths of the dimer model in Figure 1, lifted off of the edges
for readability. We see from these paths that this dimer model is consistent; whilee = D intersect twice, they are oriented in opposite
some pairs of zig-zag paths on D
directions between the crossings.
Consistent dimer models will turn out to have Calabi–Yau dimer algebras, in the sense of thefollowing definition.
Definition 3 ([29]). Let A be a Noetherian algebra with enveloping algebra Aε = A ⊗ Aop , andlet e ∈ A be an idempotent element, i.e. e2 = e. Write Deb (A) = {X ∈ Db (A) : eX = 0} andΩA = RHomAε (A, Aε ). Say that (A, e) is (bimodule) internally d-Calabi–Yau if(a) gl. dim A 6 d and Aε A has a finite resolution by finitely-generated projective Aε -modules;(b) there is a triangle
Σ−d A

ΩA

X

Σ1−d Ain D(Aε ) such that RHomA (X, M ) = 0 = RHomAop (X, N ) for any M ∈ Deb (A) andN ∈ Deb (Aop ).
We say A is (bimodule) d-Calabi–Yau if we may choose X = 0 in (b), so that A ∼= Σd ΩA in D(Aε ).
In this case the condition on the global dimension in (a) is redundant, and (A, 0) is internallyd-Calabi–Yau.

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When (A, e) is internally d-Calabi–Yau, we may show using a result of Keller [23, Lem. 4.1] that
there is a bifunctorial isomorphism∗
ExtiA (M, N ) = Extd−i
A (N, M )for any M ∈ Mod A and any finite-dimensional N ∈ mod A/AeA, where (–)∗ denotes duality overthe ground field. Thus we recover the traditional Calabi–Yau duality formula, at least under certain
conditions on the A-modules involved.
Theorem
4. Let D be a consistent dimer model on a surface Σ, with dimer algebra A. LetP
e = i∈F0 ei be the sum of the vertex idempotents at frozen vertices (so e = 0 if Σ is closed). Then
(A, e) is internally 3-Calabi–Yau if(a) Σ is a closed surface with positive genus [9] (see also [6] for the case that Σ is a torus), or(b) if Σ is a disc and D is connected [28].
We remark that if Σ is closed with genus at least 2, then the dimer algebra of a consistent dimer
model on Σ is not Noetherian [4], and so the definition of internally Calabi–Yau algebra does notstrictly apply: we use instead the more general definition (with more technical assumptions) as
originally stated in [29].
3. Categorification
In general, internally Calabi–Yau algebras may be used to define stably Calabi–Yau Frobenius
categories with cluster-tilting objects. Here a Frobenius category is an exact category E withenough projective and injective objects, which coincide; a prototypical example is the module
category of a finite-dimensional selfinjective algebra. The stable category E of E is the quotient bymorphisms factoring over a projective object, and is canonically triangulated [19]. We say E isstably d-Calabi–Yau if E is d-Calabi–Yau, i.e. if Σd is a Serre functor on this triangulated category.
An object T ∈ E is d-cluster-tilting if
add T = {X ∈ E : ExtiE (T, X) = 0 for i = 1, . . . , d − 1}
= {X ∈ E : ExtiE (X, T ) = 0 for i = 1, . . . , d − 1}.
The second equality is automatic when E is stably d-Calabi–Yau. It is immediate from this definition
that add T must contain all projective-injective objects in E, and that a 1-cluster-tilting object
is the same as an additive generator, i.e. an object T such that add T = E. Frobenius categories
with these properties (for d = 3) play an important role in the categorification of cluster algebras
with frozen variables [10, 13, 15, 21, 28]. Further examples, with d = 2, are given by categories
of Cohen–Macaulay modules over Kleinian singularities, as appearing in the classical McKay
correspondence; see [16, 25].
Theorem 5 ([29]). Let A be a Noetherian algebra, and let e ∈ A be an idempotent such that A/AeA
is finite-dimensional and (A, e) is internally d-Calabi–Yau. Write B = eAe. Then(a) B is Iwanaga–Gorenstein (meaning that B is Noetherian and has finite injective dimension
as both a left and a right B-module), so the category
GP(B) = {X ∈ mod B : ExtiB (X, B) = 0 for all i > 0}of Gorenstein projective B-modules is Frobenius,(b) the stable category GP(B) is (d − 1)-Calabi–Yau,(c) eA ∈ GP(B) is (d − 1)-cluster-tilting in this category, and(d) the natural maps A → EndB (eA)op and A/AeA → EndB (eA)op (the latter denoting theendomorphism algebra of eA in GP(B)) are isomorphisms.
This theorem may be applied in the case that A is a dimer algebra from a consistent connected
dimer model on the torus or on the disc; the necessary Calabi–Yau property is the result ofTheorem 4. In the case of the disc we take e to be the boundary idempotent, whereas in the caseof the torus e must be non-zero so that A/AeA is finite-dimensional. While dimer algebras on
higher genus closed surfaces are Calabi–Yau, they are not Noetherian. The categories and algebras

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MATTHEW PRESSLAND

appearing in Theorem 5 have useful applications, to (non-commutative) algebraic geometry in thecase of dimers on the torus, and to cluster algebras [11] in the case of dimers on the disc.
3.1. The torus. For D a consistent dimer model on the torus, choose any vertex 0 ∈ Q0 , and set
e = e0 . Then B = eAe coincides with the centre of A (in particular, it is commutative), and is a
Gorenstein toric 3-fold singularity [6]. Moreover [17, 33], every such singularity appears in this way.
In this case GP(B) = CM(B) is the category of Cohen–Macaulay modules over this Gorenstein ring,and the stable category GP(B) is the singularity category of B [7]. Furthermore, A ∼= EndB (eA)op
is a non-commutative crepant resolution of Spec B, and hence [2, 5] Db (A) ' Db (coh X) where
X → Spec(B) is a crepant resolution and coh X denotes the category of coherent sheaves on X.
Modifying the dimer model by a local move known as Seiberg duality corresponds to modifying
this crepant resolution via a flop, and modifying the cluster-tilting object eA via mutation, cf. [35].
This is a 3-dimensional analogue of the classical McKay correspondence in dimension 2. Inthat setting, one considers a Kleinian singularity B, i.e. the ring of functions on the quotient of
C2 by a finite subgroup G of SL(2, C) (localised at the origin). Such subgroups are classified by
Dynkin diagrams, and if we take A to be the preprojective algebra of the corresponding extended(or affine) diagram, then A is 2-Calabi–Yau. In particular, (A, e) is internally Calabi–Yau when eis the vertex idempotent at the extending vertex. In this case eAe = B (again isomorphic to thecentre of A), and A/AeA is the preprojective algebra of the unextended Dynkin diagram, which isfinite-dimensional. Again GP(B) = CM(B) is the category of Cohen–Macaulay modules for thecommutative ring B and Db (A) ' Db (coh X) for X the (unique) crepant resolution of the Kleinian
singularity. It follows from Theorem 5 that GP(B) = CM(B) has a 1-cluster-tilting object; this isequivalent to the fact that CM(B) has finitely many indecomposable objects up to isomorphism, as
originally proved by Herzog [20].
3.2. The disc. When D is a connected consistent dimer model on the disc, we take e to be theboundary idempotent of A. Since D typically has multiple boundary faces, this means that B is
not commutative. However, it is shown in [28] that in this case the category GP(B) categorifies
the cluster algebra structure on an open positroid variety in the Grassmannian, as defined by
Galashin–Lam [14], and by Serhiyenko–Sherman-Bennett–Williams [32] in some special cases. This
generalises work of Jensen–King–Su [21], who provide a categorification for the unique dense
positroid variety, whose cluster structure is due to Scott [31], and of Baur–King–Marsh [1], relating
this categorification to dimer models.
More precisely, a dimer model D in the disc with n half-edges determines a permutation π of this(cyclically ordered) set of half-edges, by the condition that the zig-zag path starting at half-edge eends at half-edge π(e). If D is connected, then this permutation has no fixed points; otherwise weshould ‘decorate’ it by marking each fixed point as either positive or negative depending on theorientation of the zig-zag path starting and ending at this point. As explained by Postnikov [27],this permutation (with its decoration if it has fixed points) determines an open subvariety of theGrassmannian Grnk of k-dimensional subspaces of Cn , called an open positroid variety. Here n issimply the number of half-edges of D, while k is a little harder to describe, but is still determined
combinatorially by D or π. It is shown in [14, 31, 32] how to use D to construct the initial seed ofa cluster algebra structure on the coordinate ring of this variety, the quiver of this seed being QD .
It is this structure which we categorify via GP(B) when D is connected. Modifying D via Seiberg
duality corresponds to mutating this initial seed.
In the case that π(e) is always exactly k clockwise steps from e around the boundary of thedisc (as in Figure 2, where k = 3), the corresponding positroid variety is dense in Grnk , and so theresults of Scott [31], Jensen–King–Su [21] and Baur–King–Marsh [1] apply. One consequence ofBaur–King–Marsh’s work is that in this case the categorification GP(B) described via the dimer
model coincides with Jensen–King–Su’s (who defined B directly via a quiver with relations in this∼special case). The isomorphism A → EndB (eA)op appearing in Theorem 5 also recovers a result ofBaur–King–Marsh for the dense positroid variety, as well as extending it to positroid varieties formore general permutations.

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References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]

[26]
[27]
[28]

K. Baur, A. D. King, and B. R. Marsh. Dimer models and cluster categories of Grassmannians. Proc.
Lond. Math. Soc. (3) 113 (2016), pp. 213–260. (arXiv:1309.6524 [math.RT]).
M. van den Bergh. Non-commutative crepant resolutions. In: The legacy of Niels Henrik Abel. Springer,
Berlin, 2004, pp. 749–770. (arXiv:math/0211064 [math.RA]).
R. Bocklandt. A dimer ABC. Bull. Lond. Math. Soc. 48 (2016), pp. 387–451. (arXiv:1510.04242[math.RT]).
R. Bocklandt. Noncommutative mirror symmetry for punctured surfaces. Trans. Amer. Math. Soc.
368 (2016), pp. 429–469. (arXiv:1111.3392 [math.AG]). With an appendix by M. Abouzaid.
T. Bridgeland, A. King, and M. Reid. The McKay correspondence as an equivalence of derived
categories. J. Amer. Math. Soc. 14 (2001), pp. 535–554. (arXiv:math/9908027 [math.AG]).
N. Broomhead. Dimer models and Calabi–Yau algebras. Mem. Amer. Math. Soc. 215 (2012). (arXiv:
0901.4662 [math.AG]).
R.-O. Buchweitz. Maximal Cohen–Macaulay modules and Tate-cohomology over Gorenstein rings.
1987. http://hdl.handle.net/1807/16682.
İ. Çanakçı, A. King, and M. Pressland. Perfect matchings, dimer partition functions and cluster
characters. Preprint (2021). arXiv:2106.15924v2 [math.RT].
B. Davison. Consistency conditions for brane tilings. J. Algebra 338 (2011), pp. 1–23. (arXiv :0812.4185 [math.AG]).
L. Demonet and O. Iyama. Lifting preprojective algebras to orders and categorifying partial flag
varieties. Algebra Number Theory 10 (2016), pp. 1527–1579. (arXiv:1503.02362 [math.RT]).
S. Fomin and A. Zelevinsky. Cluster algebras. I. Foundations. J. Amer. Math. Soc. 15 (2002),
pp. 497–529. (arXiv:math/0104151 [math.RT]).
S. Franco. Bipartite field theories: from D-brane probes to scattering amplitudes. J. High Energy
Phys. (2012), JHEP11(2012)141. (arXiv:1207.0807 [hep-th]).
C. Fu and B. Keller. On cluster algebras with coefficients and 2-Calabi-Yau categories. Trans. Amer.
Math. Soc. 362 (2010), pp. 859–895. (arXiv:0710.3152 [math.RT]).
P. Galashin and T. Lam. Positroid varieties and cluster algebras. To appear in Ann. Sci. École Norm.
Sup. Preprint (2019). arXiv:1906.03501v2 [math.CO].
C. Geiß, B. Leclerc, and J. Schröer. Partial flag varieties and preprojective algebras. Ann. Inst.
Fourier (Grenoble) 58 (2008), pp. 825–876. (arXiv:math/0609138 [math.RT]).
G. Gonzalez-Sprinberg and J.-L. Verdier. Construction géométrique de la correspondance de McKay.
Ann. Sci. École Norm. Sup. (4) 16 (1983), pp. 409–449.
D. R. Gulotta. Properly ordered dimers, R-charges, and an efficient inverse algorithm. J. High Energy
Phys. (2008), JHEP10(2008)014. (arXiv:0807.3012 [hep-th]).
A. Hanany and K. D. Kennaway. Dimer models and toric diagrams. Preprint (2005). arXiv:hepth/0503149v2.
D. Happel. Triangulated categories in the representation theory of finite-dimensional algebras. London
Math. Soc. Lecture Note Ser. Vol. 119. Cambridge University Press, 1988.
J. Herzog. Ringe mit nur endlich vielen Isomorphieklassen von maximalen, unzerlegbaren Cohen–
Macaulay-Moduln. Math. Ann. 233 (1978), pp. 21–34.
B. T. Jensen, A. D. King, and X. Su. A categorification of Grassmannian cluster algebras. Proc. Lond.
Math. Soc. (3) 113 (2016), pp. 185–212. (arXiv:1309.7301 [math.RT]).
P. W. Kasteleyn. The statistics of dimers on a lattice: I. The number of dimer arrangements on aquadratic lattice. Phys. A 27 (1961), pp. 1209–1225.
B. Keller. Calabi–Yau triangulated categories. In: Trends in representation theory of algebras andrelated topics. EMS Ser. Congr. Rep. Eur. Math. Soc., Zürich, 2008, pp. 467–489.
B. R. Marsh and J. S. Scott. Twists of Plücker coordinates as dimer partition functions. Comm. Math.
Phys. 341 (2016), pp. 821–884. (arXiv:1309.6630 [math.CO]).
J. McKay. Graphs, singularities, and finite groups. In: The Santa Cruz Conference on Finite Groups(Univ. California, Santa Cruz, Calif., 1979). Vol. 37. Proc. Sympos. Pure Math. Amer. Math. Soc.,
Providence, R.I., 1980, pp. 183–186.
G. Muller and D. E. Speyer. The twist for positroid varieties. Proc. Lond. Math. Soc. (3) 115 (2017),
pp. 1014–1071. (arXiv:1606.08383 [math.CO]).
A. Postnikov. Total positivity, Grassmannians, and networks. Preprint (2006). arXiv:math/0609764v1[math.CO].
M. Pressland. Calabi–Yau properties of Postnikov diagrams. Preprint (2019). arXiv:1912.12475v2[math.RT].

6

[29]
[30]
[31]
[32]

[33]

[34]
[35]

MATTHEW PRESSLAND

M. Pressland. Internally Calabi–Yau algebras and cluster-tilting objects. Math. Z. 287 (2017),
pp. 555–585. (arXiv:1510.06224 [math.RT]).
M. Pressland. Mutation of frozen Jacobian algebras. J. Algebra 546 (2020), pp. 236–273. (arXiv:
1810.01179 [math.RT]).
J. S. Scott. Grassmannians and cluster algebras. Proc. London Math. Soc. (3) 92 (2006), pp. 345–380.
(arXiv:math/0311148 [math.CO]).
K. Serhiyenko, M. Sherman-Bennett, and L. Williams. Cluster structures in Schubert varieties in
the Grassmannian. Proc. Lond. Math. Soc. (3) 119 (2019), pp. 1694–1744. (arXiv:1902.00807[math.CO]).
J. Stienstra. Computation of principal A-determinants through dimer dynamics. In: New developments
in algebraic geometry, integrable systems and mirror symmetry (RIMS, Kyoto, 2008). Vol. 59. Adv.
Stud. Pure Math. Math. Soc. Japan, Tokyo, 2010, pp. 349–369. (arXiv:0901.3681 [math.AG]).
H. N. V. Temperley and M. E. Fisher. Dimer problem in statistical mechanics—an exact result.
Philos. Mag. 6 (1961), pp. 1061–1063.
J. Vitória. Mutations vs. Seiberg duality. J. Algebra 321 (2009), pp. 816–828. (arXiv:0709.3939[math.RA]).

Matthew Pressland, School of Mathematics & Statistics, University of Glasgow, Glasgow G20 8LR,
United Kingdom
Email address: Matthew.Pressland@glasgow.ac.uk

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Crystals and Preprojective Algebra Modules
Anne Dranowski
Abstract. 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 placeof 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 tableauxand 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.
USC Dana and David Dornsife College of Letters, Arts and Sciences 3551 Trousdale Pkwy.
ADM 304 Los Angeles
Email address: dranowsk@usc.edu

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Generalized preprojective algebras
Bernard Leclerc
Abstract. Classical preprojective algebras were introduced by GelfandPonomarev 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 ringsof certain finite-dimensional unipotent subgroups of symmetric Kac-Moodygroups 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 tojoint work with Hernandez on representations of quantum aﬃne algebras. Finally if time allows, I will discuss some recent developments by Murakami and
Fujita-Murakami.
LMNO, Univ. de Caen, CNRS, UMR 6139, F-14032 Caen Cedex, FRANCE
Email address: bernard.leclerc@unicaen.fr

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Calabi-Yau algebras and canonical bundles
Akishi Ikeda
Abstract. 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 therelationship between double gradings of Calabi-Yau completions and weights
of torus actions on canonical bundles. This is a joint work with Yu Qiu.
Department of Mathematics, Josai University, 1-1 Keyakidai,Sakado-shi,Saitama 350-0295
Japan
Email address: akishi@josai.ac.jp

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Deformed Cartan matrices and generalized
preprojective algebras
Kota Murakami
Abstract. 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 theoryof a bigraded version of the generalized preprojective algebra introduced byGeiss-Leclerc-Schröer. In particular, we will interpret several numerical properties of the (q, t)-deformed Cartan matrix as homological properties from therepresentation theory of the generalized preprojective algebra. This talk isbased on a joint work with Ryo Fujita.
Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
Email address: k-murakami@math.kyoto-u.ac.jp

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