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Tokyo-Nagoya Algebra Seminar
Seminar information archive ～04/19｜Next seminar｜Future seminars 04/20～
| Organizer(s) | Noriyuki Abe, Aaron Chan, Osamu Iyama, Yasuaki Gyoda, Hiroyuki Nakaoka, Ryo Takahashi |
|---|
Seminar information archive
2022/01/18
15:00-16:30 Online
Please see the URL below for details on the online seminar.
Haruhisa Enomoto (Osaka Prefecture University)
Exact-categorical properties of subcategories of abelian categories 1 (Japanese)
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Please see the URL below for details on the online seminar.
Haruhisa Enomoto (Osaka Prefecture University)
Exact-categorical properties of subcategories of abelian categories 1 (Japanese)
[ Abstract ]
Quillen's exact category is a powerful framework for studying extension-closed subcategories of abelian categories, and provides many interesting questions on such subcategories.
In the first talk, I will explain the basics of some properties and invariants of exact categories (e.g. the Jordan-Holder property, simple objects, and Grothendieck monoid).
In the second talk, I will give some results and questions about particular classes of exact categories arising in the representation theory of algebras (e.g. torsion(-free) classes over path algebras and preprojective algebras).
If time permits, I will discuss questions of whether these results can be generalized to extriangulated categories (extension-closed subcategories of triangulated categories).
[ Reference URL ]Quillen's exact category is a powerful framework for studying extension-closed subcategories of abelian categories, and provides many interesting questions on such subcategories.
In the first talk, I will explain the basics of some properties and invariants of exact categories (e.g. the Jordan-Holder property, simple objects, and Grothendieck monoid).
In the second talk, I will give some results and questions about particular classes of exact categories arising in the representation theory of algebras (e.g. torsion(-free) classes over path algebras and preprojective algebras).
If time permits, I will discuss questions of whether these results can be generalized to extriangulated categories (extension-closed subcategories of triangulated categories).
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2021/12/16
16:45-18:15 Online
Please see the URL below for details on the online seminar.
Nicholas Williams (University of Cologne)
Cyclic polytopes and higher Auslander-Reiten theory (English)
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Please see the URL below for details on the online seminar.
Nicholas Williams (University of Cologne)
Cyclic polytopes and higher Auslander-Reiten theory (English)
[ Abstract ]
Oppermann and Thomas show that tilting modules over Iyama’s higher Auslander algebras of type A are in bijection with triangulations of even-dimensional cyclic polytopes. Triangulations of cyclic polytopes are partially ordered in two natural ways known as the higher Stasheff-Tamari orders, which were introduced in the 1990s by Kapranov, Voevodsky, Edelman, and Reiner as higher-dimensional generalisations of the Tamari lattice. These two partial orders were conjectured to be equal in 1996 by Edelman and Reiner, and we prove that this conjecture is true. We further show how the higher Stasheff-Tamari orders correspond in even dimensions to natural orders on tilting modules which were studied by Riedtmann, Schofield, Happel, and Unger. This then allows us to complete the picture of Oppermann and Thomas by showing that triangulations of odd-dimensional cyclic polytopes correspond to equivalence classes of d-maximal green sequences, which we introduce as higher-dimensional analogues of Keller’s maximal green sequences. We show that the higher Stasheff-Tamari orders correspond to natural orders on equivalence classes of d-maximal green sequences, which relate to the no-gap conjecture of Brustle, Dupont, and Perotin. The equality of the higher Stasheff-Tamari orders then implies that these algebraic orders on tilting modules and d-maximal green sequences are equal. If time permits, we will also discuss some results on mutation of cluster-tilting objects and triangulations.
[ Reference URL ]Oppermann and Thomas show that tilting modules over Iyama’s higher Auslander algebras of type A are in bijection with triangulations of even-dimensional cyclic polytopes. Triangulations of cyclic polytopes are partially ordered in two natural ways known as the higher Stasheff-Tamari orders, which were introduced in the 1990s by Kapranov, Voevodsky, Edelman, and Reiner as higher-dimensional generalisations of the Tamari lattice. These two partial orders were conjectured to be equal in 1996 by Edelman and Reiner, and we prove that this conjecture is true. We further show how the higher Stasheff-Tamari orders correspond in even dimensions to natural orders on tilting modules which were studied by Riedtmann, Schofield, Happel, and Unger. This then allows us to complete the picture of Oppermann and Thomas by showing that triangulations of odd-dimensional cyclic polytopes correspond to equivalence classes of d-maximal green sequences, which we introduce as higher-dimensional analogues of Keller’s maximal green sequences. We show that the higher Stasheff-Tamari orders correspond to natural orders on equivalence classes of d-maximal green sequences, which relate to the no-gap conjecture of Brustle, Dupont, and Perotin. The equality of the higher Stasheff-Tamari orders then implies that these algebraic orders on tilting modules and d-maximal green sequences are equal. If time permits, we will also discuss some results on mutation of cluster-tilting objects and triangulations.
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2021/11/19
17:00-18:30 Online
Please see the URL below for details on the online seminar.
Yuta Kozakai (Tokyo University of Science)
有限群のブロック上の$\tau$-傾理論 (Japanese) (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Please see the URL below for details on the online seminar.
Yuta Kozakai (Tokyo University of Science)
有限群のブロック上の$\tau$-傾理論 (Japanese) (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2021/07/08
16:00-17:30 Online
Please see the URL below for details on the online seminar.
Tsukasa Ishibashi (RIMS, Kyoto University)
Sign-stable mutation loops and pseudo-Anosov mapping classes (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Please see the URL below for details on the online seminar.
Tsukasa Ishibashi (RIMS, Kyoto University)
Sign-stable mutation loops and pseudo-Anosov mapping classes (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2021/06/24
16:00-17:30 Online
Please see the URL below for details on the online seminar.
Kohei Kikuta (Chuo University)
Rank 2 free subgroups in autoequivalence groups of Calabi-Yau categories
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Please see the URL below for details on the online seminar.
Kohei Kikuta (Chuo University)
Rank 2 free subgroups in autoequivalence groups of Calabi-Yau categories
[ Abstract ]
Via homological mirror symmetry, there is a relation between autoequivalence groups of derived categories of coherent sheaves on Calabi-Yau varieties, and the symplectic mapping class groups of symplectic manifolds.
In this talk, as an analogue of mapping class groups of closed oriented surfaces, we study autoequivalence groups of Calabi-Yau triangulated categories. In particular, we consider embeddings of rank 2 (non-commutative) free groups generated by spherical twists. It is interesting that the proof of main results is almost similar to that of corresponding results in the theory of mapping class groups.
[ Reference URL ]Via homological mirror symmetry, there is a relation between autoequivalence groups of derived categories of coherent sheaves on Calabi-Yau varieties, and the symplectic mapping class groups of symplectic manifolds.
In this talk, as an analogue of mapping class groups of closed oriented surfaces, we study autoequivalence groups of Calabi-Yau triangulated categories. In particular, we consider embeddings of rank 2 (non-commutative) free groups generated by spherical twists. It is interesting that the proof of main results is almost similar to that of corresponding results in the theory of mapping class groups.
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2021/06/02
16:00-17:30 Online
Please see the URL below for details on the online seminar.
Satoshi Murai (Waseda University)
An equivariant Hochster's formula for $S_n$-invariant monomial ideals (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Please see the URL below for details on the online seminar.
Satoshi Murai (Waseda University)
An equivariant Hochster's formula for $S_n$-invariant monomial ideals (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2021/05/20
16:00-17:30 Online
Please see the URL below for details on the online seminar.
Ryo Kanda (Osaka city University)
This talk is based on joint work with Tsutomu Nakamura. For a module-finite algebra over a commutative noetherian ring, we give a complete description of flat cotorsion modules in terms of prime ideals of the algebra, as a generalization of Enochs' result for a commutative noetherian ring. As a consequence, we show that pointwise Matlis duality gives a bijective correspondence between the isoclasses of indecomposable flat cotorsion right modules and the isoclasses of indecomposable injective left modules. This correspondence is an explicit realization of Herzog's homeomorphism induced from elementary duality between Ziegler spectra.
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Please see the URL below for details on the online seminar.
Ryo Kanda (Osaka city University)
This talk is based on joint work with Tsutomu Nakamura. For a module-finite algebra over a commutative noetherian ring, we give a complete description of flat cotorsion modules in terms of prime ideals of the algebra, as a generalization of Enochs' result for a commutative noetherian ring. As a consequence, we show that pointwise Matlis duality gives a bijective correspondence between the isoclasses of indecomposable flat cotorsion right modules and the isoclasses of indecomposable injective left modules. This correspondence is an explicit realization of Herzog's homeomorphism induced from elementary duality between Ziegler spectra.
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2021/05/06
16:00-17:30 Online
Please see the URL below for details on the online seminar.
Liran Shaul (Charles University)
Derived quotients of Cohen-Macaulay rings (English)
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Please see the URL below for details on the online seminar.
Liran Shaul (Charles University)
Derived quotients of Cohen-Macaulay rings (English)
[ Abstract ]
It is well known that if A is a Cohen-Macaulay ring and $a_1,\dots,a_n$ is an $A$-regular sequence, then the quotient ring $A/(a_1,\dots,a_n)$ is also a Cohen-Macaulay ring. In this talk we explain that by deriving the quotient operation, if A is a Cohen-Macaulay ring and $a_1,\dots,a_n$ is any sequence of elements in $A$, the derived quotient of $A$ with respect to $(a_1,\dots,a_n)$ is Cohen-Macaulay. Several applications of this result are given, including a generalization of Hironaka's miracle flatness theorem to derived algebraic geometry.
[ Reference URL ]It is well known that if A is a Cohen-Macaulay ring and $a_1,\dots,a_n$ is an $A$-regular sequence, then the quotient ring $A/(a_1,\dots,a_n)$ is also a Cohen-Macaulay ring. In this talk we explain that by deriving the quotient operation, if A is a Cohen-Macaulay ring and $a_1,\dots,a_n$ is any sequence of elements in $A$, the derived quotient of $A$ with respect to $(a_1,\dots,a_n)$ is Cohen-Macaulay. Several applications of this result are given, including a generalization of Hironaka's miracle flatness theorem to derived algebraic geometry.
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2021/04/22
16:00-17:30 Online
Please see the URL below for details on the online seminar.
Julian Külshammer (Uppsala University)
Exact categories via A-infinity algebras (English)
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Please see the URL below for details on the online seminar.
Julian Külshammer (Uppsala University)
Exact categories via A-infinity algebras (English)
[ Abstract ]
Many instances of extension closed subcategories appear throughout representation theory, e.g. filtered modules, Gorenstein projectives, as well as modules of finite projective dimension. In the first part of the talk, I will outline a general strategy to realise such subcategories as categories of induced modules from a subalgebra using A-infinity algebras. In the second part, I will describe how this strategy has been successfully applied for the exact category of filtered modules over a quasihereditary algebra. In particular I will present compatibility results of this approach with heredity ideals in a quasihereditary algebra from joint work with Teresa Conde.
[ Reference URL ]Many instances of extension closed subcategories appear throughout representation theory, e.g. filtered modules, Gorenstein projectives, as well as modules of finite projective dimension. In the first part of the talk, I will outline a general strategy to realise such subcategories as categories of induced modules from a subalgebra using A-infinity algebras. In the second part, I will describe how this strategy has been successfully applied for the exact category of filtered modules over a quasihereditary algebra. In particular I will present compatibility results of this approach with heredity ideals in a quasihereditary algebra from joint work with Teresa Conde.
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2021/04/08
16:00-17:30 Online
Please see the URL below for details on the online seminar.
Kevin Coulembier (Univeristy of Sydney)
Abelian envelopes of monoidal categories (English)
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Please see the URL below for details on the online seminar.
Kevin Coulembier (Univeristy of Sydney)
Abelian envelopes of monoidal categories (English)
[ Abstract ]
For the purposes of this talk, a ‘tensor category’ is an abelian rigid monoidal category, linear over some field. I will try to argue that there are good reasons (inspired by classification attempts of tensor categories, by motives, by Frobenius twists on tensor categories and by the idea of universal tensor categories), to try to associate tensor categories to non-abelian rigid monoidal categories. Then I will comment on some of the recent progress made on such constructions (in work of Benson, Comes, Entova, Etingof, Heidersdof, Hinich, Ostrik, Serganova and myself).
[ Reference URL ]For the purposes of this talk, a ‘tensor category’ is an abelian rigid monoidal category, linear over some field. I will try to argue that there are good reasons (inspired by classification attempts of tensor categories, by motives, by Frobenius twists on tensor categories and by the idea of universal tensor categories), to try to associate tensor categories to non-abelian rigid monoidal categories. Then I will comment on some of the recent progress made on such constructions (in work of Benson, Comes, Entova, Etingof, Heidersdof, Hinich, Ostrik, Serganova and myself).
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2021/03/11
16:00-17:30 Online
Please see the URL below for details on the online seminar.
Akihito Wachi (Hokkaido University of Education)
相対不変式で生成されるゴレンスタイン環のレフシェッツ性 (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Please see the URL below for details on the online seminar.
Akihito Wachi (Hokkaido University of Education)
相対不変式で生成されるゴレンスタイン環のレフシェッツ性 (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2021/02/24
16:00-17:30 Online
Please see the URL below for details on the online seminar.
Shunya Saito (Nagoya University)
周期三角圏上の傾理論 (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Please see the URL below for details on the online seminar.
Shunya Saito (Nagoya University)
周期三角圏上の傾理論 (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2021/02/10
16:00-17:30 Online
Please see the URL below for details on the online seminar.
Akishi Ikeda (Josai University)
Gentle代数の2重次数付きCalabi-Yau完備化と曲面の幾何学 (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Please see the URL below for details on the online seminar.
Akishi Ikeda (Josai University)
Gentle代数の2重次数付きCalabi-Yau完備化と曲面の幾何学 (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2021/01/21
17:00-18:30 Online
Please see the URL below for details on the online seminar.
Hideya Watanabe (Kyoto University)
Based modules over the i-quantum group of type AI (Japanese)
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Please see the URL below for details on the online seminar.
Hideya Watanabe (Kyoto University)
Based modules over the i-quantum group of type AI (Japanese)
[ Abstract ]
In recent years, i-quantum groups are intensively studied because of their importance in various branches of mathematics and physics. Although i-quantum groups are thought of as generalizations of Drinfeld-Jimbo quantum groups, their representation theory is much more difficult than that of quantum groups. In this talk, I will focus on the i-quantum group of type AI. It is a non-standard quantization of the special orthogonal Lie algebra so_n. I will report my recent research on based modules, which are modules equipped with distinguished bases, called the i-canonical bases. The first main result is a new combinatorial formula describing the branching rule from sl_n to so_n. The second one is the irreducibility of cell modules associated with the i-canonical bases.
[ Reference URL ]In recent years, i-quantum groups are intensively studied because of their importance in various branches of mathematics and physics. Although i-quantum groups are thought of as generalizations of Drinfeld-Jimbo quantum groups, their representation theory is much more difficult than that of quantum groups. In this talk, I will focus on the i-quantum group of type AI. It is a non-standard quantization of the special orthogonal Lie algebra so_n. I will report my recent research on based modules, which are modules equipped with distinguished bases, called the i-canonical bases. The first main result is a new combinatorial formula describing the branching rule from sl_n to so_n. The second one is the irreducibility of cell modules associated with the i-canonical bases.
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2021/01/14
16:00-17:30 Online
Please see the URL below for details on the online seminar.
Ryo Ohkawa (Kobe University)
$(-2)$ blow-up formula (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Please see the URL below for details on the online seminar.
Ryo Ohkawa (Kobe University)
$(-2)$ blow-up formula (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2020/12/17
16:00-17:30 Online
Please see the URL below for details on the online seminar.
Xiao-Wu Chen (University of Science and Technology of China)
The finite EI categories of Cartan type (English)
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Please see the URL below for details on the online seminar.
Xiao-Wu Chen (University of Science and Technology of China)
The finite EI categories of Cartan type (English)
[ Abstract ]
We will recall the notion of a finite free EI category introduced by Li. To each Cartan triple, we associate a finite free EI category, called the finite EI category of Cartan type. The corresponding category algebra is isomorphic to the 1-Gorenstein algebra, introduced by Geiss-Leclerc-Schroer, that is associated to possibly another Cartan triple. The construction of the second Cartan triple is related to the well-known unfolding of valued graphs. We will apply the obtained algebra isomorphism to re-interpret some tau-locally free modules as induced modules over a certain skew group algebra. This project is joint with Ren Wang.
[ Reference URL ]We will recall the notion of a finite free EI category introduced by Li. To each Cartan triple, we associate a finite free EI category, called the finite EI category of Cartan type. The corresponding category algebra is isomorphic to the 1-Gorenstein algebra, introduced by Geiss-Leclerc-Schroer, that is associated to possibly another Cartan triple. The construction of the second Cartan triple is related to the well-known unfolding of valued graphs. We will apply the obtained algebra isomorphism to re-interpret some tau-locally free modules as induced modules over a certain skew group algebra. This project is joint with Ren Wang.
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2020/12/10
16:30-18:00 Online
Please see the URL below for details on the online seminar.
Hiroki Matsui (University of Tokyo)
Subcategories of module/derived categories and subsets of Zariski spectra (Japanese)
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Please see the URL below for details on the online seminar.
Hiroki Matsui (University of Tokyo)
Subcategories of module/derived categories and subsets of Zariski spectra (Japanese)
[ Abstract ]
The classification problem of subcategories has been well considered in many areas. This problem is initiated by Gabriel in 1962 by giving a classification of localizing subcategories of the module category Mod R via specialization-closed subsets of the Zariski spectrum Spec R for a commutative noetherian ring. After that several authors tried to generalize this result in many ways. For example, four decades later, Krause introduced the notion of coherent subsets of Spec R and used it to classify wide subcategories of Mod R. In this talk, I will introduce the notions of n-wide subcategories of Mod R and n-coherent subsets of Spec R for a (possibly infinite) non-negative integer n. I will also introduce the notion of n-uniform subcategories of the derived category D(Mod R) and prove the correspondences among these classes. This result unifies/generalizes many known results such as the classification given by Gabriel, Krause, Neeman, Takahashi, Angeleri Hugel-Marks-Stovicek-Takahashi-Vitoria. This talk is based on joint work with Ryo Takahashi.
[ Reference URL ]The classification problem of subcategories has been well considered in many areas. This problem is initiated by Gabriel in 1962 by giving a classification of localizing subcategories of the module category Mod R via specialization-closed subsets of the Zariski spectrum Spec R for a commutative noetherian ring. After that several authors tried to generalize this result in many ways. For example, four decades later, Krause introduced the notion of coherent subsets of Spec R and used it to classify wide subcategories of Mod R. In this talk, I will introduce the notions of n-wide subcategories of Mod R and n-coherent subsets of Spec R for a (possibly infinite) non-negative integer n. I will also introduce the notion of n-uniform subcategories of the derived category D(Mod R) and prove the correspondences among these classes. This result unifies/generalizes many known results such as the classification given by Gabriel, Krause, Neeman, Takahashi, Angeleri Hugel-Marks-Stovicek-Takahashi-Vitoria. This talk is based on joint work with Ryo Takahashi.
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2020/12/03
16:00-17:30 Online
Please see the URL below for details on the online seminar.
Yuki Hirano (Kyoto University)
Full strong exceptional collections for invertible polynomials of chain type
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Please see the URL below for details on the online seminar.
Yuki Hirano (Kyoto University)
Full strong exceptional collections for invertible polynomials of chain type
[ Abstract ]
Constructing a tilting object in the stable category of graded maximal Cohen-Macaulay modules over a given graded Gorenstein ring is an important problem in the representation theory of graded Gorenstein rings. For a hypersurface S/f in a graded regular ring S, this problem is equivalent to constructing a tilting object in the homotopy category of graded matrix factorizations of f. In this talk, we discuss this problem in the case when S is a polynomial ring, f is an invertible polynomial of chain type and S has a rank one abelian group grading (called the maximal grading of f), and in this case we show the existence of a tilting object arising from a full strong exceptional collection. This is a joint work with Genki Ouchi.
[ Reference URL ]Constructing a tilting object in the stable category of graded maximal Cohen-Macaulay modules over a given graded Gorenstein ring is an important problem in the representation theory of graded Gorenstein rings. For a hypersurface S/f in a graded regular ring S, this problem is equivalent to constructing a tilting object in the homotopy category of graded matrix factorizations of f. In this talk, we discuss this problem in the case when S is a polynomial ring, f is an invertible polynomial of chain type and S has a rank one abelian group grading (called the maximal grading of f), and in this case we show the existence of a tilting object arising from a full strong exceptional collection. This is a joint work with Genki Ouchi.
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2020/11/12
16:00-17:30 Online
Please see the URL below for details on the online seminar.
Arashi Sakai (Nagoya University)
ICE-closed subcategories and wide tau-tilting modules (Japanese)
Please see the URL below for details on the online seminar.
Arashi Sakai (Nagoya University)
ICE-closed subcategories and wide tau-tilting modules (Japanese)
2020/10/27
16:30-18:00 Online
Please see the URL below for details on the online seminar.
Yasuaki Gyoda (Nagoya University)
Positive cluster complex and $\tau$-tilting complex (Japanese)
Please see the URL below for details on the online seminar.
Yasuaki Gyoda (Nagoya University)
Positive cluster complex and $\tau$-tilting complex (Japanese)
[ Abstract ]
In cluster algebra theory, cluster complexes are actively studied as simplicial complexes, which represent the structure of a seed and its mutations. In this talk, I will discuss a certain subcomplex, called positive cluster complex, of a cluster complex. This is a subcomplex whose vertex set consists of all cluster variables except for those in the initial seed. I will also introduce another simplicial complex in this talk - the tau-tilting complex, which has vertices given by all indecomposable tau-rigid modules, and simplices given by basic tau-rigid modules. In the case of a cluster-tilted algebra, it turns out that a tau-tilting complex corresponds to some positive cluster complex. Due to this fact, we can investigate the structure of a tau-tilting complex of tau-tilting finite type by using the tools of cluster algebra theory. This is joint work with Haruhisa Enomoto.
In cluster algebra theory, cluster complexes are actively studied as simplicial complexes, which represent the structure of a seed and its mutations. In this talk, I will discuss a certain subcomplex, called positive cluster complex, of a cluster complex. This is a subcomplex whose vertex set consists of all cluster variables except for those in the initial seed. I will also introduce another simplicial complex in this talk - the tau-tilting complex, which has vertices given by all indecomposable tau-rigid modules, and simplices given by basic tau-rigid modules. In the case of a cluster-tilted algebra, it turns out that a tau-tilting complex corresponds to some positive cluster complex. Due to this fact, we can investigate the structure of a tau-tilting complex of tau-tilting finite type by using the tools of cluster algebra theory. This is joint work with Haruhisa Enomoto.
