In this paper, we build a bridge between Conway–Coxeter friezes (CCFs) and rational tangles through the Kauffman bracket polynomials. One can compute a Kauffman bracket polynomial attached to rational links by using CCFs. As an application, one can give a complete invariant on CCFs of zigzag-type.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

The 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 Schrodinger 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 of k-linear monoidal categories, then the equivalence induced by F preserves the Schrodinger representation. Here, for an algebra A means the category of left A-modules. As an application, we construct a family of invariants of finite-dimensional Hopf algebras under the monoidal Morita equivalence. This family is parameterized by braids. The invariant associated to a braid b is, roughly speaking, defined by "coloring" the closure of b by the Schrodinger 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.

DOI： 10.1007/s10468-015-9554-7

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field k by using the braiding structures of A. The coefficients of polynomial invariants are integers if k is a finite Galois extension of Q, and A is a scalar extension of some finite-dimensional semisimple Hopf algebra over Q. Furthermore, we show that our polynomial invariants are indeed tensor invariants of the representation category of A, and recognize the difference between 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 whose representation categories are distinct. (C) 2009 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.jpaa.2009.07.016

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：UNION MATEMATICA ARGENTINA  

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 [25] without polynomial invariants.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

In this paper, we establish the general theory of (2 + 1)-dimensional topological quantum field theory (in short, TQFT) with a Verlinde basis. It is a consequence that we have a Dehn surgery formula for 3-manifold invariants for this kind of TQFT's. We will show that Turaev-Viro-Ocneanu unitary TQFT's obtained from subfactors satisfy the axioms of TQFT's with Verlinde bases. Hence, in a Turaev-Viro-Ocneanu TQFr, we have a Dehn surgery formula for 3-manifolds. It turns out that this Dehn surgery formula is nothing but the formula of the Reshetikhin-Turaev invariant constructed from a tube system, which is a modular category corresponding to the quantum double construction of a C*-tensor category. In Sato and Wakui (J. Knot Theory Ramif. 12 (2003) 543), we will exhibit computations of Turaev-Viro-Ocneanu invariants for several "basic 3-manifolds". In the appendix, we discuss the relationship between the system of M-infinity-M-infinity bimodules arising from the asymptotic inclusion M v M-op not superset of M-infinity constructed from N subset of M and the tube system obtained from a subfactor N subset of M. (c) 2004 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.aim.2004.07.008

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：KLUWER ACADEMIC PUBL  

We determine various additional structures on all nonsemisimple Hopf algebras of dimension 8 over an algebraically closed field k of characteristic 0, including their representation rings and quasitriangular structures. As a consequence, it is shown that for two such Hopf algebras, the tensor categories of their representations are monoidally equivalent if and only if the representation rings of them are isomorphic as rings.

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：WORLD SCIENTIFIC PUBL CO PTE LTD  

In this paper, we establish a rigorous correspondence between the two tube algebras, that one comes from the Turaev-Viro-Ocneanu TQFT introduced by Ocneanu and another comes from the sector theory introduced by Izumi, and construct a canonical isomorphism between the centers of the two tube algebras, which is a conjugate linear isomorphism preserving the products of the two algebras and commuting with the actions of SL(2, Z). Via this correspondence and the Dehn surgery formula, we compute Turaev-Viro-Ocneanu invariants from several subfactors for basic 3-manifolds including lens spaces and Brieskorn 3-manifolds by using Izumi's data written in terms of sectors.

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Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.2206/kyushujm.56.59

Scopus

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Language：Japanese   Publisher：Kyoto University  

CiNii Books

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：OSAKA JOURNAL OF MATHEMATICS  

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Event date： 2010.10

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Event date： 2008.9

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Event date： 2007.8

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