arXiv subjects

Kenichi Shimizu

Publications and source records attributed to Kenichi Shimizu.

At least 37 records · Page 2Linked to original sources

arXiv2022

Nakayama functor for monads on finite abelian categories

If $\mathcal{M}$ is a finite abelian category and $\mathbf{T}$ is a linear right exact monad on $\mathcal{M}$, then the category $\mathbf{T}\mbox{-mod}$ of $\mathbf{T}$-modules is a finite abelian category. We give an explicit formula of the Nakayama functor of $\mathbf{T}\mbox{-mod}$ under the assumption that the underlying functor of the monad $\mathbf{T}$ has a double left adjoint and a double right adjoint. As applications, we deduce formulas of the Nakayama functor of the center of a finite bimodule category and the dual of a finite tensor category. Some examples from the Hopf algebra theory are also discussed.

math.QA↗

arXiv2022

Asymptotic properties of Bayesian inference in linear regression with a structural break

This paper studies large sample properties of a Bayesian approach to inference about slope parameters $γ$ in linear regression models with a structural break. In contrast to the conventional approach to inference about $γ$ that does not take into account the uncertainty of the unknown break location $τ$, the Bayesian approach that we consider incorporates such uncertainty. Our main theoretical contribution is a Bernstein-von Mises type theorem (Bayesian asymptotic normality) for $γ$ under a wide class of priors, which essentially indicates an asymptotic equivalence between the conventional frequentist and Bayesian inference. Consequently, a frequentist researcher could look at credible intervals of $γ$ to check robustness with respect to the uncertainty of $τ$. Simulation studies show that the conventional confidence intervals of $γ$ tend to undercover in finite samples whereas the credible intervals offer more reasonable coverages in general. As the sample size increases, the two methods coincide, as predicted from our theoretical conclusion. Using data from Paye and Timmermann (2006) on stock return prediction, we illustrate that the traditional confidence intervals on $γ$ might underrepresent the true sampling uncertainty.

econ.EM↗

arXiv2021

Bayesian Approaches to Shrinkage and Sparse Estimation

In all areas of human knowledge, datasets are increasing in both size and complexity, creating the need for richer statistical models. This trend is also true for economic data, where high-dimensional and nonlinear/nonparametric inference is the norm in several fields of applied econometric work. The purpose of this paper is to introduce the reader to the world of Bayesian model determination, by surveying modern shrinkage and variable selection algorithms and methodologies. Bayesian inference is a natural probabilistic framework for quantifying uncertainty and learning about model parameters, and this feature is particularly important for inference in modern models of high dimensions and increased complexity. We begin with a linear regression setting in order to introduce various classes of priors that lead to shrinkage/sparse estimators of comparable value to popular penalized likelihood estimators (e.g.\ ridge, lasso). We explore various methods of exact and approximate inference, and discuss their pros and cons. Finally, we explore how priors developed for the simple regression setting can be extended in a straightforward way to various classes of interesting econometric models. In particular, the following case-studies are considered, that demonstrate application of Bayesian shrinkage and variable selection strategies to popular econometric contexts: i) vector autoregressive models; ii) factor models; iii) time-varying parameter regressions; iv) confounder selection in treatment effects models; and v) quantile regression models. A MATLAB package and an accompanying technical manual allow the reader to replicate many of the algorithms described in this review.

econ.EM↗

arXiv2021

Modified traces and the Nakayama functor

We organize the modified trace theory with the use of the Nakayama functor of finite abelian categories. For a linear right exact functor $Σ$ on a finite abelian category $\mathcal{M}$, we introduce the notion of a $Σ$-twisted trace on the class $\mathrm{Proj}(\mathcal{M})$ of projective objects of $\mathcal{M}$. In our framework, there is a one-to-one correspondence between the set of $Σ$-twisted traces on $\mathrm{Proj}(\mathcal{M})$ and the set of natural transformations from $Σ$ to the Nakayama functor of $\mathcal{M}$. Non-degeneracy and compatibility with the module structure (when $\mathcal{M}$ is a module category over a finite tensor category) of a $Σ$-twisted trace can be written down in terms of the corresponding natural transformation. As an application of this principal, we give existence and uniqueness criteria for modified traces. In particular, a unimodular pivotal finite tensor category admits a non-zero two-sided modified trace if and only if it is spherical. Also, a ribbon finite tensor category admits such a trace if and only if it is unimodular.

math.QA↗

arXiv2021

Tannaka theory and the FRT construction over non-commutative algebras

Let $A$ be an algebra over a commutative ring $k$. We introduce the notion of a coquasitriangular left bialgebroid over $A$ and show that the category of left comodules over such a bialgebroid has a braiding. We also investigate a Tannaka type construction of bimonads and bialgebroids. As an application, the Faddeev-Reshetikhin-Takhtajan (FRT) construction over the algebra $A$ is established. Our construction associates a coquasitriangular bialgebroid to a braided object $(M, c)$ in the category of $A$-bimodules such that $M$ is finitely generated and projective as a left $A$-module. A Hopf algebroid version of this construction is also provided.

math.QA↗

arXiv2021

Ribbon structures of the Drinfeld center of a finite tensor category

We classify the ribbon structures of the Drinfeld center $\mathcal{Z}(\mathcal{C})$ of a finite tensor category $\mathcal{C}$. Our result generalizes Kauffman and Radford's classification result of the ribbon elements of the Drinfeld double of a finite-dimensional Hopf algebra. As a consequence, we see that $\mathcal{Z}(\mathcal{C})$ is a modular tensor category in the sense of Lyubashenko if $\mathcal{C}$ is a spherical finite tensor category in the sense of Douglas, Schommer-Pries and Snyder.

math.QA↗

arXiv2020

Categorical aspects of cointegrals on quasi-Hopf algebras

We discuss relations between some category-theoretical notions for a finite tensor category and cointegrals on a quasi-Hopf algebra. Specifically, for a finite-dimensional quasi-Hopf algebra $H$, we give an explicit description of categorical cointegrals of the category ${}_H \mathscr{M}$ of left $H$-modules in terms of cointegrals on $H$. Provided that $H$ is unimodular, we also express the Frobenius structure of the `adjoint algebra' in the Yetter-Drinfeld category ${}^H_H \mathscr{YD}$ by using an integral in $H$ and a cointegral on $H$. Finally, we give a description of the twisted module trace for projective $H$-modules in terms of cointegrals on $H$.

math.QA↗

arXiv2018

Pivotal structures of the Drinfeld center of a finite tensor category

We classify the pivotal structures of the Drinfeld center $\mathcal{Z}(\mathcal{C})$ of a finite tensor category $\mathcal{C}$. As a consequence, every pivotal structure of $\mathcal{Z}(\mathcal{C})$ can be obtained from a pair $(β, j)$ consisting of an invertible object $β$ of $\mathcal{C}$ and an isomorphism $j: β\otimes (-) \otimes β^* \to (-)^{**}$ of monoidal functors.

math.CT↗

arXiv2018

Further results on the structure of (co)ends in finite tensor categories

Let $\mathcal{C}$ be a finite tensor category, and let $\mathcal{M}$ be an exact left $\mathcal{C}$-module category. The action of $\mathcal{C}$ on $\mathcal{M}$ induces a functor $ρ: \mathcal{C} \to \mathrm{Rex}(\mathcal{M})$, where $\mathrm{Rex}(\mathcal{M})$ is the category of $k$-linear right exact endofunctors on $\mathcal{M}$. Our key observation is that $ρ$ has a right adjoint $ρ^{\mathrm{ra}}$ given by the end $ρ^{\mathrm{ra}}(F) = \int_{M \in \mathcal{M}} \underline{\mathrm{Hom}}(M, M)$. As an application, we establish the following results: (1) We give a description of the composition of the induction functor $\mathcal{C}_{\mathcal{M}}^* \to \mathcal{Z}(\mathcal{C}_{\mathcal{M}}^*)$ and Schauenburg's equivalence $\mathcal{Z}(\mathcal{C}_{\mathcal{M}}^*) \approx \mathcal{Z}(\mathcal{C})$. (2) We introduce the space $\mathrm{CF}(\mathcal{M})$ of `class functions' of $\mathcal{M}$ and initiate the character theory for pivotal module categories. (3) We introduce a filtration for $\mathrm{CF}(\mathcal{M})$ and discuss its relation with some ring-theoretic notions, such as the Reynolds ideal and its generalizations. (4) We show that $\mathrm{Ext}_{\mathcal{C}}^{\bullet}(1, ρ^{\mathrm{ra}}(\mathrm{id}_{\mathcal{M}}))$ is isomorphic to the Hochschild cohomology of $\mathcal{M}$. As an application, we show that the modular group acts projectively on the Hochschild cohomology of a modular tensor category.

math.QA↗

arXiv2017

Recent developments of the categorical Verlinde formula

I review recent developments of "non-semisimple" modular tensor categories in the sense of Lyubashenko and the categorical Verlinde formula for such categories (this is a proceedings article for Meeting for Study of Number theory, Hopf algebras and related topics held at University of Toyama, Japan, 12-15 February 2017).

math.QA↗

arXiv2017

Quiver-theoretical approach to dynamical Yang-Baxter maps

A dynamical Yang-Baxter map, introduced by Shibukawa, is a solution of the set-theoretical analogue of the dynamical Yang-Baxter equation. In this paper, we initiate a quiver-theoretical approach for the study of dynamical Yang-Baxter maps. Our key observation is that the category of dynamical sets over a set $Λ$, introduced by Shibukawa to establish a categorical framework to deal with dynamical Yang-Baxter maps, can be embedded into the category of quivers with vertices $Λ$. By using this embedding, we shed light on Shibukawa's classification result of a certain class of dynamical Yang-Baxter maps and extend his construction to obtain a new class of dynamical Yang-Baxter maps. We also discuss a relation between Shibukawa's bialgebroid associated to a dynamical Yang-Baxter map and Hayashi's weak bialgebra associated to a star-triangular face model.

math.QA↗

arXiv2017

Integrals for finite tensor categories

We introduce the notions of categorical integrals and categorical cointegrals of a finite tensor category $\mathcal{C}$ by using a certain adjunction between $\mathcal{C}$ and its Drinfeld center $\mathcal{Z}(\mathcal{C})$. These notions can be identified with integrals and cointegrals of a finite-dimensional Hopf algebra $H$ if $\mathcal{C}$ is the representation category of $H$. We generalize basic results on integrals and cointegrals of a finite-dimensional Hopf algebra (such as the existence, the uniqueness, and the Maschke theorem) to finite tensor categories. Motivated by results of Lorenz, we also investigate relations between categorical integrals and morphisms factoring through projective objects. Finally, we extend the $n$-th indicator of a finite-dimensional Hopf algebra introduced by Kashina, Montgomery and Ng to finite tensor categories.

math.CT↗

arXiv2016

The monoidal center and the character algebra

For a pivotal finite tensor category $\mathcal{C}$ over an algebraically closed field $k$, we define the algebra $\mathsf{CF}(\mathcal{C})$ of class functions and the internal character $\mathsf{ch}(X) \in \mathsf{CF}(\mathcal{C})$ for an object $X \in \mathcal{C}$ by using an adjunction between $\mathcal{C}$ and its monoidal center $\mathcal{Z}(\mathcal{C})$. We also develop the integral theory in a unimodular finite tensor category by using the same adjunction. By utilizing these tools, we extend some results in the character theory of finite-dimensional Hopf algebras to this category-theoretical setting. Our main result is that the map $\mathsf{ch}: \mathsf{Gr}_k(\mathcal{C}) \to \mathsf{CF}(\mathcal{C})$ given by taking the internal character is a well-defined injective algebra map, where $\mathsf{Gr}_k(\mathcal{C})$ is the scalar extension of the Grothendieck ring of $\mathcal{C}$ to $k$. Moreover, under the assumption that $\mathcal{C}$ is unimodular, the map $\mathsf{ch}$ is an isomorphism if and only if $\mathcal{C}$ is semisimple. As an application, we show that the algebra $\mathsf{Gr}_{k}(\mathcal{C})$ is semisimple if $\mathcal{C}$ is a non-degenerate pivotal fusion category. If, moreover, $\mathsf{Gr}_k(\mathcal{C})$ is commutative, then the character table of $\mathcal{C}$ is defined based on the integral theory. It turns out that the character table is obtained from the $S$-matrix if $\mathcal{C}$ is a modular tensor category. Generalizing corresponding results in the finite group theory, we prove the orthogonality relations and the integrality.

math.QA↗

arXiv2016

Non-degeneracy conditions for braided finite tensor categories

For a braided finite tensor category $\mathcal{C}$ with unit object $1 \in \mathcal{C}$, Lyubashenko considered a certain Hopf algebra $\mathbb{F} \in \mathcal{C}$ endowed with a Hopf pairing $ω: \mathbb{F} \otimes \mathbb{F} \to 1$ to define the notion of a `non-semisimple' modular tensor category. We say that $\mathcal{C}$ is non-degenerate if the Hopf pairing $ω$ is non-degenerate. In this paper, we show that $\mathcal{C}$ is non-degenerate if and only if it is factorizable in the sense of Etingof, Nikshych and Ostrik, if and only if its Müger center is trivial, if and only if the linear map $Ω: \mathrm{Hom}_{\mathcal{C}}(1, \mathbb{F}) \to \mathrm{Hom}_{\mathcal{C}}(\mathbb{F}, 1)$ induced by the pairing $ω$ is invertible. As an application, we prove that the category of Yetter-Drinfeld modules over a Hopf algebra in $\mathcal{C}$ is non-degenerate if and only if $\mathcal{C}$ is.

math.QA↗

arXiv2016

The relative modular object and Frobenius extensions of finite Hopf algebras

For a certain kind of tensor functor $F: \mathcal{C} \to \mathcal{D}$, we define the relative modular object $χ_F \in \mathcal{D}$ as the "difference" between a left adjoint and a right adjoint of $F$. Our main result claims that, if $\mathcal{C}$ and $\mathcal{D}$ are finite tensor categories, then $χ_F$ can be written in terms of a categorical analogue of the modular function on a Hopf algebra. Applying this result to the restriction functor associated to an extension $A/B$ of finite-dimensional Hopf algebras, we recover the result of Fischman, Montgomery and Schneider on the Frobenius type property of $A/B$. We also apply our results to obtain a "braided" version and a "bosonization" version of the result of Fischman et al.

math.CT↗

arXiv2015

On unimodular finite tensor categories

Let $\mathcal{C}$ be a finite tensor category with simple unit object, let $\mathcal{Z}(\mathcal{C})$ denote its monoidal center, and let $L$ and $R$ be a left adjoint and a right adjoint of the forgetful functor $U: \mathcal{Z}(\mathcal{C}) \to \mathcal{C}$. We show that the following conditions are equivalent: (1) $\mathcal{C}$ is unimodular, (2) $U$ is a Frobenius functor, (3) $L$ preserves the duality, (4) $R$ preserves the duality, (5) $L(1)$ is self-dual, and (6) $R(1)$ is self-dual, where $1 \in \mathcal{C}$ is the unit object. We also give some other equivalent conditions. As an application, we give a categorical understanding of some topological invariants arising from finite-dimensional unimodular Hopf algebras.

math.QA↗

arXiv2015

The pivotal cover and Frobenius-Schur indicators

In this paper, we introduce the notion of the pivotal cover $\mathcal{C}^{\mathsf{piv}}$ of a left rigid monoidal category $\mathcal{C}$ to develop a theoretical foundation for the theory of Frobenius-Schur (FS) indicators in "non-pivotal" settings. For an object $\mathbf{V} \in \mathcal{C}^{\mathsf{piv}}$, the $(n, r)$-th FS indicator $ν_{n, r}(\mathbf{V})$ is defined by generalizing that of an object of a pivotal monoidal category. This notion gives a categorical viewpoint to some recent results on generalizations of FS indicators. Based on our framework, we also study the FS indicators of the "adjoint object" in a finite tensor category, which can be considered as a generalization of the adjoint representation of a Hopf algebra. The indicators of this object closely relate to the space of endomorphisms of the iterated tensor product functor.

math.QA↗

arXiv2014

On indicators of Hopf algebras

Kashina, Montgomery and Ng introduced the $n$-th indicator $ν_n(H)$ of a finite-dimensional Hopf algebra $H$ and showed that the indicators have some interesting properties such as the gauge invariance. The aim of this paper is to investigate the properties of $ν_n$'s. In particular, we obtain the cyclotomic integrality of $ν_n$ and a formula for $ν_n$ of the Drinfeld double. Our results are applied to the finite-dimensional pointed Hopf algebra $u(\mathcal{D}, λ, μ)$ introduced by Andruskiewitsch and Schneider. As an application, we obtain the second indicator of $u_q(\mathfrak{sl}_2)$ and show that if $p$ and $q$ are roots of unity of the same order, then $u_p(\mathfrak{sl}_2)$ and $u_q(\mathfrak{sl}_2)$ are gauge equivalent if and only if $q = p$, where $p$ and $q$ are roots of unity of the same odd order.

math.RT↗