WAKUI,Michihisa - Faculty of Engineering Science Department of Mathematics

WAKUI, Michihisa

Faculty, Department/Institute

Faculty of Engineering Science Department of Mathematics

Academic status (qualification)

Professor Apr. 1,2021

Academic Degrees

Master's degree Mar. 1992 Kyushu University
Doctorate of Science Dec. 2002 Osaka University

Homepage Address, E-mail Address

Homepage Address：http://www2.itc.kansai-u.ac.jp/~wakui/

Research fields	keyword
Algebra	;
Topology	;;;

research topic	Stùdy on topologićal invariants oƒ low dimensional maniƒolds
Study theme state	Individual Research
research duration
Research Programs
keyword	Knots,Topologial ield theory,Hop algebra,
Research field
Research Topics Overview

Research Publications

No.	Type of publication	Date of publication (Date of presentation)	Title	Type of research result	Jointly authored or single authored	Publisher and journal name	Volume number
1	Papers1	2022/4～2022/42022,04,00,2022,04,00	A characterization of Conway-Coxeter friezes of zigzag type by rational links	Academic Journal	Co-author	Osaka Journal of Mathematics	59, 341--362
2	Textbook23	2021/3/19～2021,03,19,,,	Introduction to algebraic topology Fundamental groups and homology groups	Monograph	Single-Author
3	International academic conference8	2019/9/9～2019/9/92019,09,09,2019,09,09	Indecomposability of weak Hopf algebras	Other	Single-Author
4	Academic presentation7	2019/3/26～2019/3/262019,03,26,2019,03,26	On structures of low-dimensional weak Hopf algebras	Other	Single-Author
5	Papers1	2019～20192019,00,00,2019,00,00	Kauffman bracket polynomials for Conway Coxeter Friezes	Monograph	Co-author	"Proceedings of the Meeting for Study of Number Theory, Hopf algebras and Related Topoics" edited by H. Yamane, T. Kogiso, Y. Koga and I. Kimura, Yokohama Publ.	51--79
6	Papers1	2019～20192019,00,00,2019,00,00	Braided Morita equivalence for finite-dimensional semisimple and cosemisimple Hopf algebras	Monograph	Single-Author	"Proceedings of the Meeting for Study of Number Theory, Hopf algebras and Related Topoics" edited by H. Yamane, T. Kogiso, Y. Koga and I. Kimura, Yokohama Publ.	157--183
7	Papers1	2015～20152015,00,00,2015,00,00	Schrödinger representations from the viewpoint of monoidal categories	Academic Journal	Co-author	Algebras and Representation Theory	18 (2015), 1623--1647
8	Academic presentation7	2010/10/18～2010,10,18,,,	Tensor Morita invariants of finite-dimensinal Hopf algebras assocated with the Hopf link		Single-Author
9	Papers1	2010～2010,,,,,	Polynomial invariants for a semisimple and cosemisimple Hopf algebra of finite dimension	Academic Journal	Single-Author	J. Pure Appl. Algebra	214, 701—728
10	Papers1	2010～2010,,,,,	Triangular structures of Hopf algebras and tensor Morita equivalences	Academic Journal	Single-Author	Revista de la Union Matematica Argentina	51, 193—210
11	Papers1	2009/2～2009,02,,,,	Polynomial invariants of finite-dimensional Hopf algebras derived from braiding structures			Proceedings of the 41st Symposium on Ring Theory and Representation Theory	96-105
12	Academic presentation7	2008/9～2008,09,,,,	Polynomial invariants of representation categories of semisimple and cosemisimple Hopf algebras
13	Academic presentation7	2007/8～2007,08,,,,	On the Turaev-Viro-Ocneanu invariant of 3-manifolds derived from generalized E_6-subfactors
14	Presentations99	2007～2007,,,,,	On the Turaev-Viro-Ocneanu invariant of 3-manifolds derived from generalized E_6-subfactors
15	Papers1	2007～2007,,,,,	On the Turaev-Viro-Ocneanu invariants of 3-manifolds derived from generalized E_6-subfactors			The proceedings of the conference “Intelligence of low dimensional topology”	21-30
16	Papers1	2005～2005,,,,,	(2+1)-dimensional topological quantum field theory from subfactors and Dehn surgery formula for 3-manifold invariant	Academic Journal	Co-author	Advances in Math.	195, 165-204
17	Papers1	2004～2004,,,,,	Various structures associated to the representation categories of eight-dimensional nonsemisimple Hopf algebras	Academic Journal	Single-Author	Algebras and Representation Theory	7, 491-515
18	Papers1	2003～2003,,,,,	Computatioins of Turaev-Viro-Ocneanu invariants of 3-manifolds fom subfactors	Academic Journal	Co-author	J. Knot Theory and its Ramif.	12, 543--574,
19	Papers1	2003～2003,,,,,	The coribbon structures of some finite dimensional braided Hopf algebras generated by $2\times2$-matrix coalgebras	In-house publication	Single-Author	Banach Center Publications,	61, 333-344,
20	Papers1	2003～2003,,,,,	On representation rings of non-semisimple Hopf algebras of low dimension	Other	Single-Author	Proceedings of the 35th Symposium on Ring Theory and Representation Theory	9月14日
21	Papers1	2002～2002,,,,,	On Turaev-Viro-Ocneanu invariant of 3-manifolsds derived from the E_6-subfactor	Academic Journal	Co-author	kyushu J.Math.	56, 59-81
22	Papers1	2002～2002,,,,,	(2+1)-dimensional topological quantum field theory with Verlinde basis and Turaev-Viro-Ocneanu invariants of 3-manifolds	Monograph	Co-author	Geom. Topol. Monogr.	4, 281-294
23	Papers1	1998～1998,,,,,	On the Universal R-matrices of the dihedral groups	In-house publication	Single-Author	Surikaiseki Kenkyusho kokyuroku	1057, 41-53
24	Papers1	1994～1994,,,,,	Fusion algebras of o orbifold models (A survey)	Monograph	Single-Author	World Scientific, ”Topology, Geometry and Field theory” edited by K. Fukaya, M. Furuta, T. Kohno and D. Kotschick	225-235
25	Papers1	1992～1992,,,,,	On Dijkgraaf-Witten invariant of 3-manifolds	Academic Journal	Single-Author	Osaka Journal of Mathematics	29, 675-696

PapersA characterization of Conway-Coxeter friezes of zigzag type by rational linksIn refereedAcademic JournalCo-authorWAKUI,Michihisa;KOGISO, TakeyoshiJones polynomialOsaka Journal of Mathematics59, 341--3622022/4～2022/4The present paper show that Conway-Coxeter friezes of zigzag type are characterized by (unoriented) rational links. As an application of this characterization Jones polynomial can be defined for Conway-Coxeter friezes of zigzag type. This gives a new method for computing the Jones polynomial for oriented rational links.

PapersIn refereedMonographSingle-AuthorWAKUI,Michihisa2021/7～

TextbookIntroduction to algebraic topology Fundamental groups and homology groupsMonographSingle-AuthorWAKUI,Michihisa2021/3/19～

International academic conferenceIndecomposability of weak Hopf algebrasUnrefereedOtherSingle-AuthorWAKUI,Michihisa2019/9/9～2019/9/9International Workshop on Hopf Algebras and Tensor CategoriesNanjing, ChainaIn this talk we study on the direct sum construction of weak Hopf algebras, which is a generalization of Kaplansky type construction studied by Chebel and Makhlouf. Any finite-dimensional weak Hopf algebra can be uniquely decomposed into finitely many indecomposable weak Hopf algebras up to isomorphism. So, indecomposable weak Hopf algebras are fundamental and important. We determine the indecomposable low-dimensional weak Hopf algebras listed by Chebel and Makhlouf, and show that a finite-dimensional Hopf algebra is always indecomposable as a weak Hopf algebra. A categorical viewpoint for indecomposability is also discussed.

Academic presentationOn structures of low-dimensional weak Hopf algebrasUnrefereedOtherSingle-AuthorWAKUI,Michihisa2019/3/26～2019/3/26Hopf Algbera Conference in Tsukuba 2019筑波大学Chebel and Makhlouf are classified 2- and 3-dimensional weak bialgebras up to isomorphism. These weak bialgebras can be constructed by a generalization of the Kaplansky's type construction for bialgebras. In this talk, we compare a Hopf algera H to the weak Hopf algebra by the Kaplansky's type construction from H, and show that there is one-to-one correspondences between the sets of their quasitriangular structures and their integrals, and so on, respectively.

PapersKauffman bracket polynomials for Conway Coxeter FriezesUnrefereedMonographCo-authorWAKUI,Michihisa;Kogiso, Takeyoshi"Proceedings of the Meeting for Study of Number Theory, Hopf algebras and Related Topoics" edited by H. Yamane, T. Kogiso, Y. Koga and I. Kimura, Yokohama Publ.51--792019～2019In this paper, we construct Kauffman bracket polynomials associated with Conway Coxeter Friezes based on Yamada's ancestor triangles and we denote the relation between Conway Coxeter Friezes and Yamada's ancestor triangles. Furthermore we also explain relations between Conway Coxeter Friezes and Markov triples.

PapersBraided Morita equivalence for finite-dimensional semisimple and cosemisimple Hopf algebrasIn refereedMonographSingle-AuthorWAKUI,Michihisa"Proceedings of the Meeting for Study of Number Theory, Hopf algebras and Related Topoics" edited by H. Yamane, T. Kogiso, Y. Koga and I. Kimura, Yokohama Publ.157--1832019～2019Braided Morita invariants of finite-dimensional semisimple and cosemisimple Hopf algebras with braidings are constructed by refining the polynomial invariants introduced by the author. The invariants are computed for the duals of Suzuki's braided Hopf algebras, and as an application of that, the braided Morita equivalence classes over the $8$-dimensional Kac-Paljutkin algebra are determined. This paper also includes the modified results and proofs on determination of the coribbon elements of Suzuki's braided Hopf algebras, that are discussed and given in the published paper from Banach Center Publication.

PapersIn refereedAcademic JournalCo-authoredWAKUI,Michihisa;KOGISO, Takeyoshi2019～

Academic presentationSingle-AuthorWAKUI,Michihisa2017/2/14～2017/2/14

PapersSchrödinger representations from the viewpoint of monoidal categoriesIn refereedAcademic JournalCo-authorWAKUI,Michihisa;SHIMIZU, KENICHIAlgebras and Representation Theory18 (2015), 1623--16472015～2015The Drinfel'd double D(A) of a finite-dimensional Hopf algebra A is a Hopf algebraic counterpart of the monoidal center construction. Majid introduced an important representation of D(A), which he called the Schrödinger representation. We study this representation from the viewpoint of the theory of tensor categories. One of our main results is as follows: If two finite-dimensional Hopf algebras A and B over a field k are monoidally Morita equivalent, i.e., there exists an equivalence between the module categories over A and B as k-linear monoidal categories, then the equivalence between the module categories over D(A) and D(B) induced by F preserves the Schrödinger representation. As an application, we construct a family of invariants of finite-dimensional Hopf algebras under the monoidal Morita equivalence. This family is parametrized by braids. The invariant associated to a braid b is, roughly speaking, defined by "coloring'' the closure of b by the Schrödinger representation. We investigate what algebraic properties this family have and, in particular, show that the invariant associated to a certain braid closely relates to the number of irreducible representations.

UnrefereedIn-house publicationSingle-AuthorMichihisa Wakui;2013/6～

TextbookMonographSingle-AuthorMichihisa Wakui;2013/3/20～

Academic presentationUnrefereedSingle-AuthorMichihisa Wakui;2012/12/10～

Academic presentationMichihisa Wakui;2011/8/25～

Academic presentationTensor Morita invariants of finite-dimensinal Hopf algebras assocated with the Hopf linkSingle-AuthorMichihisa Wakui;2010/10/18～

PapersPolynomial invariants for a semisimple and cosemisimple Hopf algebra of finite dimensionIn refereedAcademic JournalSingle-AuthorMichihisa Wakui;J. Pure Appl. Algebra214, 701—7282010～We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra $A$ over a field $¥boldsymbol{k}$ by using the braiding structures of $A$. The coefficients of polynomial invariants are integers if $¥boldsymbol{k}$ is a finite Galois extension of $¥mathbb{Q}$, and $A$ is a scalar extension of some finite-dimensional semisimple Hopf algebra over $¥mathbb{Q}$. Furthermore, we show that our polynomial invariants are indeed tensor invariants of the representation category of $A$, and recognize the difference of the representation category and the representation ring of $A$. Actually, by computing and comparing polynomial invariants, we find new examples of pairs of Hopf algebras whose representation rings are isomorphic, but representation categories are distinct.

PapersTriangular structures of Hopf algebras and tensor Morita equivalencesIn refereedAcademic JournalSingle-AuthorMichihisa Wakui;Revista de la Union Matematica Argentina51, 193—2102010～In this paper, the triangular structures of a Hopf algebra $A$ are discussed as a tensor Morita invariant. It is shown by many examples that triangular structures are useful for detecting whether module categories are monoidally equivalent or not. By counting and comparing the numbers of triangular structures, we give simple proofs of some results obtained in ¥cite{W} without polynomial invariants.

PapersPolynomial invariants of finite-dimensional Hopf algebras derived from braiding structuresWAKUI,Michihisa;;Proceedings of the 41st Symposium on Ring Theory and Representation Theory96-1052009/2～

Academic presentationPolynomial invariants of representation categories of semisimple and cosemisimple Hopf algebrasWAKUI,Michihisa;;2008/9～

Academic presentationOn the Turaev-Viro-Ocneanu invariant of 3-manifolds derived from generalized E_6-subfactorsWAKUI,Michihisa;;2007/8～

PresentationsOn the Turaev-Viro-Ocneanu invariant of 3-manifolds derived from generalized E_6-subfactorsWAKUI Michihisa;;2007～Grant-in-Aid for Scientific Research

PresentationsWAKUI,Michihisa;;2007～

PapersOn the Turaev-Viro-Ocneanu invariants of 3-manifolds derived from generalized E_6-subfactorsMchihisa Wakui;Nobuya Sato;;The proceedings of the conference “Intelligence of low dimensional topology”21-302007～The proceedings of the conference “Intelligence of low dimensional topology”Osaka City University

Papers(2+1)-dimensional topological quantum field theory from subfactors and Dehn surgery formula for 3-manifold invariantIn refereedAcademic JournalCo-authorWAKUI Michihisa;Yasuyuki Kawahigashi;Nobuya Sato;;Advances in Math.195, 165-2042005～

PapersVarious structures associated to the representation categories of eight-dimensional nonsemisimple Hopf algebrasIn refereedAcademic JournalSingle-AuthorWAKUI Michihisa;;Algebras and Representation Theory7, 491-5152004～Grant-in-Aid for Encouragement of Young Scientists

PapersComputatioins of Turaev-Viro-Ocneanu invariants of 3-manifolds fom subfactorsIn refereedAcademic JournalCo-authorWAKUI Michihisa;Nobuya Sato;;J. Knot Theory and its Ramif.12, 543--574,2003～

PapersThe coribbon structures of some finite dimensional braided Hopf algebras generated by $2\times2$-matrix coalgebrasIn refereedIn-house publicationSingle-AuthorWAKUI Michihisa;;Banach Center Publications,61, 333-344,2003～

PapersOn representation rings of non-semisimple Hopf algebras of low dimensionUnrefereedOtherSingle-AuthorWAKUI Michihisa;;Proceedings of the 35th Symposium on Ring Theory and Representation Theory9月14日2003～

PapersOn Turaev-Viro-Ocneanu invariant of 3-manifolsds derived from the E_6-subfactorIn refereedAcademic JournalCo-authorWAKUI Michihisa;Kotarou Suzuki;;kyushu J.Math.56, 59-812002～

Papers(2+1)-dimensional topological quantum field theory with Verlinde basis and Turaev-Viro-Ocneanu invariants of 3-manifoldsIn refereedMonographCo-authorWAKUI Michihisa;Nobuya Sato;;Geom. Topol. Monogr.4, 281-2942002～

PapersOn the Universal R-matrices of the dihedral groupsUnrefereedIn-house publicationSingle-AuthorWAKUI Michihisa;;Surikaiseki Kenkyusho kokyuroku1057, 41-531998～

PapersFusion algebras of o orbifold models (A survey)UnrefereedMonographSingle-AuthorWAKUI Michihisa;;World Scientific, ”Topology, Geometry and Field theory” edited by K. Fukaya, M. Furuta, T. Kohno and D. Kotschick225-2351994～

PapersOn Dijkgraaf-Witten invariant of 3-manifoldsIn refereedAcademic JournalSingle-AuthorWAKUI Michihisa;;Osaka Journal of Mathematics29, 675-6961992～In 1990, Dijkgraaf and Witten introduced an invariant for closed 3-manifolds using a finite group and its 3-cocycle, and suggested a method of construction of a topological quantum field theory based on triangulations. In this paper, we formulate the topological invariance of their invariant for compact 3-manifolds possibly with boundary in a rigorous way, and prove it by checking the invariance of their invariant under the relative Alexander moves. Furthermore, in the same manner developed by Turaev and Viro, we prove that the method of construction of Dijkgraaf-Witten invariant for compact 3-manifolds possibly with boundary gives rise to an example of topological quantum field theory due to Atiyah.

WAKUI, Michihisa

Faculty, Department/Institute

Academic status (qualification)

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Homepage Address, E-mail Address

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