# The 2nd Meeting for Study of Number theory, Hopf algebras and related topics

## Date/Venue

Date
2020年2月15日(土)～2020年2月18日(火)
Venue

Organizers

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## Program

15th Feb (Sat)富山大学理学部多目的ホール
Fac. Sci., Multiple Purpose Hall (B243)
TimeSpeakerタイトル
13:25Opening Remark
13:30 ~ 14:15Iwao Kimura 木村 巌
（Univ. Toyama, Fac. Sci., 富山大学 理学部）
Algebraic and Modular Construction of unramified Abelian $l$-extensions over cyclotomic extensions
14:25 ~ 15:10Yusuke Ohkubo 大久保 勇輔
（Univ. Tokyo, Grad. School of Math. Sci., 東京大学, 数理科学研究科）
Matrix element formula for the intertwiners of the Ding-Iohara-Miki algebra and its application
15:25-16:10Yudai Otsuto 乙戸 勇大
（Hokkaido Univ., Fac. Sci., 北海道大学, 理学研究院）
Two FRT bialgebroids and their relations
（Univ. Tsukuba, Doctral Prgram in Math., 筑波大学, 数理物質系）
Forms of differential Lie algebras over $\mathbb{C}(t)$
17:15-18:00Nur Hamid (Nurul Jadid Univ. and Kanazawa Univ.)Jacobi polynomials of binary codes
16th Feb (Sun)富山大学理学部多目的ホール
Fac. Sci., Multiple Purpose Hall (B243)
TimeSpeakerTitle
9:25Opening Remark
9:30-10:15Yoshiyuki Koga 古閑 義之
（Univ. Fukui, School of Engineering, 福井大学, 工学研究科）
BGG resolutions for the affine Lie superalgebra $\widehat{sl}(n,1)$
10:25-11:10Yuki Kanakubo 金久保 有輝
（Sophia Univ., Fac. Sci. and Technology, 上智大学, 理工学部）
Adapted Sequence for Polyhedral Realization of Crystal Bases
11:20-12:05Yuichi Sakai 境 優一
（Kyushu Univ., Multiple Zeta Research Center, 九州大学, 多重ゼータ研究センター）
Modular linear differential equations : application to vertex operator algebras
13:40-14:25Taiki Shibata 柴田 大樹
（Okayama Univ. Science, Fac. Sci., 岡山理科大学, 理学部）
A Hopf-algebraic study of Quasireductive supergroups
14:35-15:20Takuya Matsumoto 松本 拓也
（Nagoya Univ., Graduate School of Math., 名古屋大学, 多元数理科学研究科）
The screening operators for the extended W-algebras of type $A_1$ at positive rational levels
15:35-16:20Shun'ichi Yokoyama 横山 俊一
（Tokyo Metropolitan Univ., Graduate School of Science, 首都大学東京, 理学研究科）
Number theory with CAS: for performant and flexible computation
16:30-17:15Michihisa Wakui 和久井 道久
（Kansai Univ., Fac. of Engineering Science, 関西大学, システム理工学部）
The Jones polynomial of frieze patterns by Conway and Coxeter
17:25-18:10Takeyoshi Kogiso 小木曽 岳義
（Josai Univ., Fac. of Sci., 城西大学, 理学部）
$q$-Deformation of continued fractions and its application to the Markov equation
17th Feb (Mon)共通教育棟D11教室
General Education Bldg. D11-room
TimeSpeakerTitle
9:25Opening Remark
9:30-10:15Kazuyuki Oshima 大島 和幸
（Aichi Institute of Technology, Center for General Education, 愛知工業大学、基礎教育センター）
Representations of elliptic quantum toroidal algebras
10:25-11:10Hajime Nagoya 名古屋 創
(Kanazawa Univ., School of Math. and Physics, 金沢大学, 数物科学系)
Irregular conformal blocks and Painlevé tau functions
11:20-12:05Ryo Sato 佐藤 僚
Logarithmic extension of the Kazama-Suzuki coset model
14:00-14:45Ken Ito 伊藤 健
（Aichi Institute of Technology, Center for General Education, 愛知工業大学、基礎教育センター）
Summary of realization of the twisted affine Lie algebras and its root systems
14:55-15:40Masaya Tomie 冨江 雅也
（Morioka Univ., Fac. of Humanities,盛岡大学, 文学部）
Reflectable bases for Weyl groups of classical finite types, locally finite types and an application
16:00-17:00Jun Morita 森田 純
（University of Tsukuba, Doctral Prgram in Mathematics, 筑波大学, 数理物質系）
Chevalley groups over Dedekind domains and $K_2$ groups
17:10-17:55Yoji Yoshii 吉井 洋二（Iwate University, Fac. Edu., 岩手大学, 教育学部）Minimal locally affine Lie algebras
18th Feb (Tue)共通教育棟D11教室
General Education Bldg. D11-room
TimeSpeakerTitle
9:10Opening Remark
9:15-10:00Tatsuya Kawabe 川部達哉
(Univ. Toyama, Fac. Sci., 富山大学, 理学部)
On the virtually solvability of discontinuous subgroups of affine motions
10:10-10:55Kenichi Shimizu 清水 健一
(Shibaura Institute of Technology, Systems Engineering and Science, 芝浦工業大学, システム理工学部)
Categorical aspects of cointegrals on quasi-Hopf algebras
11:05-11:50Hiroyuki Yamane 山根 宏之
（Univ. Toyama, Fac. Sci., 富山大学, 理学部）
Typical character for the generalized quantum groups
（closing: 11:50 ~ 12:00）
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## Titles and Abstracts

### 15/Feb/2020 (Sat)

Speaker:
Iwao Kimura 木村 巌（University of Toyama, Facultiy of Science, 富山大学 理学部）
Title:
Algebraic and Modular Construction of unramified Abelian $l$-extensions over cyclotomic extensions
Abstract:
We discuss some technics to construct an unramified Abelian $l$-extension over the $l$-th cyclotomic field or the $l$-th cyclotomic extension over a real quadratic field ($l$ is an odd prime). One is an "algebraic" method, which is basing on Kummer theory, due mainly to N. Nakagoshi. Another is a "modular" method, which is basing on the existence of mod-$l$ Galois representation associated to Hecke eigenforms, due to K. Ribet and others.
Speaker:
Yusuke Ohkubo 大久保 勇輔（University of Tokyo, Graduate School of Mathematical Sciences, 東京大学, 数理科学研究科）
Title:
Matrix element formula for the intertwiners of the Ding-Iohara-Miki algebra and its application
Abstract:
As a generalization of Affine quantum groups, there is a Hopf algebra called the Ding-Iohara-Miki algebra (DIM algebra). This algebra is closely related to the $q$-W algebras and plays an important role in the AGT correspondence. In this talk, I will explain a certain matrix element formula for the intertwiners of the DIM algebra. This formula indicates a proof of the $q$-deformed AGT correspondence and the S-duality of the string theory. Moreover, as its application, I will give several combinatorial formulas for non-stationary Ruijsenaars functions which were introduced by Shiraishi last year. The non-stationary Ruijsenaars functions were given as an affine analogue of the Macdonald functions, and they have two more parameters p and κ in addition to the ones in Macdonald functions. In the case that κ is specialized, the non-stationary Ruijsenaars functions can be reproduced by the intertwiners of the DIM algebra, and it can be shown that these are the essentially same as a sort of elliptic analogue of the Macdonald functions.
Speaker:
Yudai Otsuto 乙戸 勇大（Hokkaido University, Facultiy of Science, 北海道大学, 理学研究院）
Title:
Two FRT bialgebroids and their relations
Abstract:
Faddeev-Reshetikhin-Takhtajan introduced construction of bialgebras using solutions to the Yang-Baxter equation (YBE), called FRT construction. This construction was developed by Hayashi to theface type YBE and by Shibukawa-Takeuchi to the dynamical Yang-Baxter map. Our aim is to discuss relations between Hayashi's construction and Shibukawa-Takeuchi' s construction. Matsumoto-Shimizu gave a homomorphism between the two weak bialgebras obtained by the two constructions. We generalize these two constructions and this homomorphism. This talk is based in part on a joint work with Youichi Shibukawa.
Speaker:
Yuta Shimada 島田 祐汰（University of Tsukuba, Doctral Prgram in Mathematics,筑波大学, 数理物質系）
Title:
Forms of differential Lie algebras over $\mathbb{C}(t)$
Abstract:
As an answer to a problem personally posed by Prof. Pianzola, we determine all forms of the differential $\mathbb{C}(t)$-Lie algebras $\mathfrak{g}_0 \otimes_{\mathbb{C}} \mathbb{C}(t)$, where $\mathfrak{g}_0$ is any simple Lie algebra over $\mathbb{C}$. The rational function field $\mathbb{C}(t)$ in one variable is regarded as a differential field with respect to the standard differential operator $\delta t=1$, $\delta c=0$, $c \in \mathbb{C}$. A differential $\mathbb{C}(t)$-Lie algebra is a Lie algebra $\mathfrak{g}$ over $\mathbb{C}(t)$ which is equipped with a differential operator such that $\delta(aX)=\delta(a)X+a \delta(X)$, $\delta[X,Y]=[\delta X,Y]+[X,\delta Y]$, where $a \in \mathbb{C}(t)$, $X, Y \in \mathfrak{g}$. In our differential context, forms are defined as in the non-differential situation, where base extensions are supposed to be faithfully flat morphisms of differential rings. The main tools to prove our result are descent theory and Hopf-Galois Theory modified into the differential context. This talk is based on a joint work with Akira Masuoka (preprint arXiv:1912.10705).
Speaker:
Nur Hamid (Nurul Jadid University and Kanazawa University)
Title:
Jacobi polynomials of binary codes
Abstract:
The Jacobi polynomial belongs to the invariant ring of a fixed finite group. In this talk, under the connection between coding theory and invariant theory of a finite group, we show that the invariant ring of a group given can be generated by the Jacobi polynomials of the binary codes in genus 1.

### 16/Feb/2020 (Sun)

Speaker:
Yoshiyuki Koga 古閑 義之（University of Fukui, School of Engineering, 福井大学, 工学研究科）
Title:
BGG resolutions for the affine Lie superalgebra $\widehat{sl}(n,1)$
Abstract:
A.M.Semikhatov and A.Taormina introduced certain generalized Verma modules and constructed BGG resolutions of admissible representations over the affine Lie superalgebra $\widehat{sl}(2,1)$. In this talk, I will explain similar BGG resolutions of integrable highest weight representations over $\widehat{sl}(n,1)$.
Speaker:
Yuki Kanakubo 金久保 有輝（Sophia University, Faculty of Science and Technology, 上智大学, 理工学部）
Title:
Adapted Sequence for Polyhedral Realization of Crystal Bases
Abstract:
Polyhedral realization is a kind of description of crystal bases as integer points in some polyhedral convex cone invented by Nakashima and Zelevinsky. To construct the polyhedral realization, we need an infinite sequence $\iota$ of indices. In the case $\iota$ satisfies a positivity condition' (resp. ample condition'), there is a method to obtain an explicit form of the polyhedral realization for $B(\infty)$ (resp. $B(\lambda)$) associated with $\iota$. However, it seems to be difficult to confirm whether $\iota$ satisfies the positivity (resp. ample) condition or not. In this talk, I will give a sufficient condition of $\iota$ for the positivity (resp. ample) condition in the case the associated Lie algebra is classical type. I will also give explicit forms of the polyhedral realizations in terms of column tableaux for sequences which satisfy the sufficient condition. This is a joint work with Toshiki Nakashima in Sophia University.
Speaker:
Yuichi Sakai 境 優一（Kyushu University, Multiple Zeta Research Center, 九州大学, 多重ゼータ研究センター）
Title:
Modular linear differential equations : application to vertex operator algebras
Abstract:
For simple, rational, $C_2$-cofinite vertex operator algebras(VOAs), characters of its modules are holomorphic functions on the complex upper -half plane, and they are solutions of a modular linear differential equation(MLDE). In this talk, we give a classification of such VOAs by using MLDEs. This talk is based on a joint work with Kiyokazu Nagatomo.
Speaker:
Taiki Shibata 柴田 大樹（Okayama University of Science, Faculty of Science, 岡山理科大学, 理学部）
Title:
A Hopf-algebraic study of Quasireductive supergroups
Abstract:
An algebraic supergroup is a group-valued functor on the category of commutative superalgebras represented by a finitely generated commutative Hopf superalgebra. It has been known that representations of algebraic supergroups can be applied in non-super (modular) representation theory. In 2011, V. Serganova introduced the notion of quasireductive supergroups as a super version of the notion of split reductive groups. This forms an important class of algebraic supergroups including Chevalley supergroups (introduced by R. Fioresi and F. Gavarini, 2012) and queer supergroups $Q(n)$.
In this talk, I explain a Hopf-algebraic approach to the study of structure and representation of quasireductive supergroups defined over a field.
Speaker:
Takuya Matsumoto 松本 拓也 （Nagoya University, Graduate School of Mathematics, 名古屋大学, 多元数理科学研究科）
Title:
The screening operators for the extended W-algebras of type $A_1$ at positive rational levels
Abstract:
The screening operators are the operators commuting with the Virasoro generators and the W-algebra generators, in more generic case. The extended W-algebras of type $A_1$ at positive rational levels is a vertex operator algebra defined by the intersection of their kernels with the particular central charges. It is known that the screening operators are relating to the Jack polynomials though the free field realization. In this talk, we will report the recent progress on the explicit constructions on the screening operators, the Virasoro singular vectors, and the corresponding field generators.
Speaker:
Shun'ichi Yokoyama 横山 俊一（Tokyo Metropoplitan University, Graduate School of Science, 首都大学東京, 理学研究科）
Title:
Number theory with CAS: for performant and flexible computation
Abstract:
Nemo (distributed at the TU Kaiserslautern) is a software designed for computations in polynomial algebra and number theory. In this talk, we will introduce NemoCAS project and extensive packages (especially a prototype Julia package Hecke for algebraic number theory) with some benchmarks vs. Magma.
Speaker:
Michihisa Wakui 和久井 道久（Kansai University, Faculty of Engineering Science, 関西大学, システム理工学部）
Title:
The Jones polynomial of frieze patterns by Conway and Coxeter
Abstract:
In 1973 Conway and Coxeter introduced some frieze pattern displayed as an array of (positive) integers, which are generated by the unimodular rule or the determinant condition ad-bc=1. Conway-Coxeter friezes are closely related to triangulated triangles, rational numbers, cluster algebras and so on. In this talk we give a direct connection between Conway-Coxeter friezes of zigzag type and rational links in the 3-sphere. As an application of it we show that "the Jones polynomial" can be defined for Conway-Coxeter friezes of zigzag type. We also supply a useful method of computation of the Jones polynomial by Farey sums and negative continued-fraction expressions of rational numbers. This talk is based on a joint work with T. Kogiso (Josai University).
Speaker:
Takeyoshi Kogiso 小木曽 岳義（Josai University, Faculty of Science, 城西大学, 理学部）
Title:
$q$-Deformation of continued fractions and its application to the Markov equation.
Abstract:
$q$-Deformation of continued fractions was introduced by Morier-Genoud and Ovsienko. The most important application of this $q$-deformation of regular (or negative) continued fraction expansion of rational number r/s is to calculate the Jones polynomial of the rational link of r/s. In this talk, I introduce a certain application of this to solutions of Markov equation and quadratic irrational numbers. If time permit, I introduce further generalization of them to cluster variable-version.

### 17/Feb/2020 (Mon)

Speaker:
Kazuyuki Oshima 大島 和幸（Aichi Institute of Technology, Center for General Education, 愛知工業大学、基礎教育センター）
Title:
Representations of elliptic quantum toroidal algebras
Abstract:
Quantum toroidal algebras are generalizations of quantum affine algebras. They have been studied in connection with Macdonald functions, $W$-algebras, double affine Hecke algebras etc. We introduce an elliptic analogue of the quantum toroidal algebra for $\mathrm{gl}_N$. We also construct some representations for elliptic quantum toroidal algebras. This is a joint work with Hitoshi Konno (Tokyo Univ. of Marine Science and Technology).
Speaker:
Hajime Nagoya 名古屋 創 (Kanazawa University, School of Mathematics and Physics, 金沢大学, 数物科学系)
Title:
Irregular conformal blocks and Painlevé tau functions
Abstract:
Gamayun, Iorgov, Lisovyy conjectured that the tau function of the sixth Painlevé equation is a Fourier transformation of a (regular) Virasoro conformal block in 2012 and Iorgov, Lisovyy, Teschner, and Bershtein, Shchechkin proved it by different methods in 2014. We introduce irregular vertex operator of the super Virasoro algebra and prove that Fourier transformations of irregular conformal blocks solve bilinear equations satisfied by the tau functions of the fourth and fifth Painlevé equations.
Speaker:
Ryo Sato 佐藤 僚 (Academia Sinica, 中央研究院)
Title:
Logarithmic extension of the Kazama-Suzuki coset model
Abstract:
Representation theory of supersymmetric (SUSY) vertex algebras, which are $\mathbb{Z}/2\mathbb{Z}$-graded generalizations of vertex operator algebras, plays a fundamental role in the study of 2-dimensional superconformal field theory. In particular, semisimple finite abelian categories of some specific representations closed under the fusion tensor product have been of particular interest in connection with rational superconformal field theories. In this talk, we provide a new example of non-semisimple ("irrational") finite braided tensor category based on the Kazama-Suzuki coset construction associated with the orthosymplectic Lie superalgebra $\mathfrak{osp}(1|2)$.
Speaker:
Ken Ito 伊藤 健 （Aichi Institute of Technology, Center for General Education, 愛知工業大学、基礎教育センター）
Title:
Summary of realization of the twisted affine Lie algebras and its root systems.
Abstract:
In this talk, we will summarize the well-known realizations of the twisted affine Lie algebras and review their relationship to the structure of the root systems.
Speaker:
Masaya Tomie 冨江 雅也（Morioka Universty, Faculty of Humanities, 盛岡大学, 文学部）
Title:
Reflectable bases for Weyl groups of classical finite types, locally finite types and an application
Abstract:
In this talk, we discuss the reflectable bases for classical finite or locally finite Weyl groups. This part is joint work with Yoji Yoshii. After that, we give some applications for Hamiltonicity for the Cayley graphs from classical finite Weyl groups with several generators.
Speaker:
Jun Morita 森田 純（University of Tsukuba, Doctral Prgram in Mathematics, 筑波大学, 数理物質系）
Title:
Chevalley groups over Dedekind domains and $K_2$ groups
Abstract:
We make a review of Chevalley groups over Dedekind domains, namely we discuss Tits systems, presentations, $K_2$, Schur multipliers and ''Heredity Theorem'' etc. Especially we remember these structures in case of $SL(2,Z_S)$. Then, we show several new examples.
Speaker:
Yoji Yoshii 吉井 洋二（Iwate University, Faculty of Education, 岩手大学, 教育学部）
Title:
Minimal locally affine Lie algebras
Abstract:
As a natural generalization of affine Lie algebras, we introduce a minimal locally affine Lie algebra. Such a Lie algebra consists of a universal central extension of a locally loop algebra with 1-dimensional derivations, as the structure of affine Lie algebras. A new phenomenon of this Lie algebra is that the 1-dimensional derivations are not necessarily degree derivations. We explain the classification of the seven types of minimal locally affine Lie algebras and especially about the 1-dimensional derivations of each type. This talk is based on a joint work with Jun Morita.

### 18/Feb/2020 (Tue)

Speaker:
Tatsuya Kawabe 川部達哉 (University of Toyama, Faculty of Science, 富山大学, 理学部)
Title:
On the virtually solvability of discontinuous subgroups of affine motions
Abstract:
Let $\Gamma$ be a discrete subgroup of the group of all affine transformations of $\mathbf{R}^n$. The group $\Gamma$ is called an affine crystallographic group if $\Gamma$ acts properly discontinuously and uniformly on $\mathbf{R}^n$. There is a long standing conjecture: Every affine crystallographic group is virtually solvable. In this talk, we introduce the various aspects about the properly discontinuity of the action of $\Gamma$, and discuss the virtually solvability of an affine crystallographic group in more general situation.
Speaker:
Kenichi Shimizu 清水 健一 (Shibaura Institute of Technology, Systems Engineering and Science, 芝浦工業大学, システム理工学部)
Title:
Categorical aspects of cointegrals on quasi-Hopf algebras
Abstract:
I explain relations between some category-theoretical notions for a finite tensor category and cointegrals on a quasi-Hopf algebra. This talk is based on a joint work with Taiki Shibata.
Speaker:
Hiroyuki Yamane 山根 宏之（University of Toyama, Facultiy of Science, 富山大学, 理学部）
Title:
Typical character for the generalized quantum groups
Abstract:
In the talk, we introduced the typical character formula for the generalized quantum groups. This is a $q$-deformation of the Kac's typical character formula for the basic classical Lie superalgebras. I emphasis that there are many irreducible modules not allowed to take $q$ to 1 for them but allowed to compute their character formula by our typical character formula. (arXiv:1909.08881)
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