Laurent Demonet

This seminar in organized in collaboration with Tomoki Nakanishi and Osamu Iyama

Remark: for rooms, **多** is the building of Graduate School of Mathematics and **A** is the building Science A.

**2015/03/13 - 14:00 to 15:30 - 多-309 - Gustavo Jasso (Bonn): Higher Auslander-Reiten theory revisited**. I will report on recent joint work with Kvamme and Raedschelders, where we recover basic results in higher Auslander-Reiten theory by exploiting the $d$-abelian structure of $d$-cluster-tilting subcategories. We also investigate the concept of morphisms determined by objects in this context. No familiarity with higher Auslander-Reiten theory will be assumed.

**2013/06/11 - 13:00 - room 多-552 - Tomoki Nakanishi (Nagoya): Wonder of sine-Gordon $Y$-systems**. The sine-Gordon $Y$-systems and the reduced sine-Gordon $Y$-systems were introduced by Tateo in the 90's in the study of the integrable deformation of conformal field theory by the thermodynamic Bethe ansatz method. The periodicity property and the dilogarithm identities concerning these $Y$-systems were conjectured by Tateo, and only a part of them have been proved so far. We formulate these $Y$-systems by the polygon realization of cluster algebras of types $A$ and $D$, and prove the conjectured periodicity and dilogarithm identities in full generality. As it turns out, there is a wonderful interplay among continued fractions, triangulations of polygons, cluster algebras, and $Y$-systems. This is a joint work with Salvatore Stella.

Slides of the talk (from the webpage of Tomoki Nakanishi).

**2013/07/10 - 14:45 - room A-317 - Gustavo Jasso (Nagoya): Skewsymmetric cluster algebras and their categorifications**. The aim of this talk is to give an introduction to cluster algebras and their interaction with representation theory of finite dimensional algebras. For this we will focus on the simpler class of skewsymmetric cluster algebras without coefficients, and briefly describe their categorification by $2$-Calabi-Yau triangulated categories with a cluster tilting objects.

**2013/07/16 - 13:00 - room A-317 - Yann Palu (Amiens - France): Introduction to the Caldero-Chapoton map**. The Caldero-Chapoton map is an important tool in the theory of
categorification of cluster algebras. It gives an "explicit" formula for the cluster variables of a given cluster algebra in terms of the triangulated category which categorifies it. The talk will be an
introduction, illustrated with examples, to the Caldero-Chapoton map for cluster categories and their Hom-finite generalisations.

**2013/07/16 - 14:45 - room A-317 - Pierre-Guy Plamondon (Orsay - France): Cluster categories, $\mathbf{g}$-vectors, and exchange graphs I**. These talks will be concerned with additive categorification of Fomin-Zelevinsky's cluster algebras. We will focus on the interpretation of mutation, $\mathbf{g}$-vectors, coefficients and exchange graphs inside some
triangulated categories, and see how one can deduce informations on the corresponding notions inside cluster algebras.

**2013/07/19 - 13:00 - room A-317 - Xueyu Luo (Nagoya): Ice quivers with potentials associated with triangulations and Cohen-Macaulay modules over orders**. Given a triangulation of a polygon $P$ with $n$ vertices, we associate an ice quiver with potential such that the associated Jacobian algebra has the structure of a Gorenstein tiled $K[x]$-order $\Lambda$. Then we show that the stable category of the category of Cohen-Macaulay $\Lambda$-modules is equivalent to the cluster category $\mathscr{C}$ of Dynkin type $A_{n-3}$.
It gives a natural interpretation of the usual indexation of cluster tilting objects of $\mathscr{C}$ by triangulations of $P$. Moreover, it extends naturally the triangulated categorification by $\mathscr{C}$ of the cluster algebra of type $A_{n-3}$ to an exact categorification by adding coefficients corresponding to the sides of $P$. Finally, we lift the previous equivalence of categories to an equivalence between the stable category of graded Cohen-Macaulay $\Lambda$-modules and the bounded derived category of modules over a quiver of type $A_{n-3}$.

**2013/07/19 - 14:45 - room A-317 - Yann Palu (Amiens - France): Iyama-Yoshino reduction for cluster categories from surfaces**. In order to show that cluster-tilting objects have a mutation theory which resemble that of the clusters in a cluster algebra, Iyama and Yoshino introduced a "reduction" for triangulated categories. This Iyama-Yoshino reduction turned out to be of paramount importance in the categorification of cluster algebras. This talk will serve as an illustration of Iyama-Yoshino reductions in the context of cluster categories associated with marked surfaces. We will also present somme applications to the mutation theory of rigid objects in these cluster categories.

**2013/07/23 - 13:00 - room A-317 - Pierre-Guy Plamondon (Orsay - France): Cluster categories, $\mathbf{g}$-vectors, and exchange graphs II**. These talks will be concerned with additive categorification of Fomin-Zelevinsky's cluster algebras. We will focus on the interpretation of mutation, $\mathbf{g}$-vectors, coefficients and exchange graphs inside some
triangulated categories, and see how one can deduce informations on the corresponding notions inside cluster algebras.

**2013/07/30 - 13:00 - room A-317 - Yuya Mizuno (Nagoya): Introduction to $\mathbf{g}$-vectors of support $\tau$-tilting modules.**.
In this talk, we give an introduction to $\mathbf{g}$-vectors (which is also called index) of support $\tau$-tilting modules. We explain basic properties of them and a deep connection of $\mathbf{g}$-vectors of cluster-tilting objects in a 2-CY triangulated category.
Moreover, we study $\mathbf{g}$-vectors of support $\tau$-tilting modules over preprojective algebras of Dynkin type.
We give an explicit description of them and show a relationship of chambers of the associated root system.

**2013/08/01 - 13:00 - room A-332 - Yu Liu (Nagoya): Hearts of twin cotorsion pairs on exact categories**. In the papers of Nakaoka, he introduced the notion of hearts of (twin) cotorsion pair on triangulated categories and showed that they have structures of (semi-)abelian categories. We study in this talk a twin cotorsion pair $(\mathscr{S},\mathscr{T}),(\mathscr{U},\mathscr{V})$ on an exact category $\mathscr{B}$ with enough projectives and injectives and introduce a notion of the heart. First we show that its heart is preabelian. Moreover we show the heart of a single cotorsion pair is abelian. These results are analog of Nakaoka's results in triangulated categories. We will also construct cohomological functors from the exact category to the hearts of cotorsion pairs on it.

**2013/08/01 - 14:15 - room A-332 - Laurent Demonet (Nagoya): Positivity for cluster algebras**. The aim of this talk is to report the recent work of Lee and Schiffler proving the positivity for skew-symmetric cluster algebras. Precisely speaking, any cluster variable can be expanded as a Laurent polynomial
$$\sum_{\mathbf{i} \in \mathbb{Z}^n} a_\mathbf{i} \mathbf{x}^\mathbf{i}$$
where $\mathbf{x} = \{x_1, x_2, \dots, x_n\}$ is a given cluster and the $a_\mathbf{i}$ non-negative integer coefficients (finitely many of which are non-zero).

See the article on arXiv.

**2014/04/08 - 13:00 - room A-317 - Tomoki Nakanishi (Nagoya): Signed mutations and singed pops in cluster algebras**. We propose two kinds of extensions of mutations in cluster algebras. The first one is the signed mutation. As the name suggests, it is the mutation with sign, and if the sign coincides with the tropical sign, it is the ordinary mutation, but, if it does not, it is twisted by coefficients. The second one is the signed pop. This operation is defined only for a cluster algebra with surface realization, and it is defined for a puncture inside self-folded triangle. These operations naturally appeared in the study of the mutation of Stokes graphs in the exact WKB analysis, but there seems to be a possibility of application for representation theory as well. In this talk we focus on the cluster algebraic aspect of these operations.

**2014/04/20 - 16:00 - room A-428 - Akishi Ikeda (Tokyo): $A$型箙に付随した$N$-Calabi-Yau圏の安定性条件の空間について**.

**2014/11/10 - 11:00 and 14:00 - room 多-552 - Takuma Aihara: Brauer graph algebras I**. Reading seminar (mainly internal).

**2014/11/17 - 10:30 - room 多-409 - Tomoki Nakanishi (Nagoya): Generalized cluster algebras (1): generalized mutations**. Generalized cluster algebras were introduced by Chekhov and Shapiro recently. They are generalizations of ordinary cluster algebras by replacing the familiar binomial term in the exchange relation of cluster variables with the polynomial one. It was shown by Chekhov and Shapiro that this generalization preserves the Laurent property and also the finite-type classification. In this series of talks I will demonstrate that it preserves essentially every aspect of ordinary cluster algebras.

See the article on arXiv.

**2014/12/01 - 13:00 - room 多-409 - Tomoki Nakanishi (Nagoya): Generalized cluster algebras (2): seed structure**. See previous abstract.

**2014/12/08 - 11:00 and 14:00 - room 多-552 - Takahide Adachi (Nagoya): Brauer graph algebras II**. Reading seminar (mainly internal).

**2015/01/16 - 13:00 - room 多-309 - Tomoki Nakanishi (Nagoya): Generalized cluster algebras (3): quantization**. Generalized cluster algebras were introduced by Chekhov and Shapiro recently. They are generalizations of ordinary cluster algebras by replacing the familiar binomial term in the exchange relation of cluster variables with the polynomial one. It was shown by Chekhov and Shapiro that this generalization preserves the Laurent property and also the finite-type classification. In this series of talks I will demonstrate that it preserves essentially every aspect of ordinary cluster algebras.

See the article on arXiv.

**2015/01/16 - 14:45 - room 多-309 - Bernard Leclerc (Caen - France): Quivers with relations associated with symmetrizable Cartan matrices**. Let $C$ be a symetrizable generalized Cartan matrix. In a joint work with Geiss and Schröer we have associated with $C$ a class of finite-dimensional algebras $H=H(C,D,\Omega)$ defined by quivers with relations, and we have started to study their representation theory. In this talk I will explain how we can now generalize a classical theorem of Schofield, and give in terms of varieties of $H$-modules a geometric constrution of the enveloping algebra $U(\mathfrak{n})$ of the positive part of the symmetrizable Kac-Moody algebra associated with $C$.

See the article on arXiv.