Search

Recently, Morier-Genoud and Ovsienko introduced a $q$-deformation of rational numbers. More precisely, for an irreducible fraction $\frac{r}s>0$, they constructed coprime polynomials $\mathcal{R}_{\frac{r}s}(q),~ \mathcal{S}_{\frac{r}s}(q) \in {\mathbb Z}[q]$ with $\mathcal{R}_{\frac{r}s}(1)=r,~\mathcal{S}_{\frac{r}s}(1)=s$. Their theory has a rich background and many applications. By definition, if $r \equiv r' \pmod{s}$, then $\mathcal{S}_{\frac{r}s}(q)=\mathcal{S}_{\frac{r'}s}(q)$. We show that $rr'{\equiv} -1 \pmod{s}$ implies $\mathcal{S}_{\frac{r}s}(q)=\mathcal{S}_{\frac{r'}s}(q)$, and it is conjectured that the converse holds if $s$ is prime (and $r \not \equiv r' \pmod{s}$). We also show that $s$ is a multiple of 3 (resp. 4) if and only if $\mathcal{S}_{\frac{r}s}(\zeta)=0$ for $\zeta=(-1+\sqrt{-3})/2$ (resp. $\zeta=i$). We give applications to the representation theory of quivers of type $A$ and the Jones polynomials of rational links., Comment: 34 pages, typos fixed; exposition improved. To appear in Arnold Math. J

In connection with cluster algebras, snake graphs and q-integers, Kyungyong Lee and Ralf Schiffler recently found a formula for computing the (normalized) Jones polynomials of rational links in terms of continued fraction expansion of rational numbers. Sophie Morier-Genoud and Valentin Ovsienko introduced q-deformed continued fractions, and showed that by using them each coefficient of the normalized Jones polynomial counted quiver representations of type A_n. In this paper we introduce q-deformed integers defined by pairs of coprime integers, which are motivated by the denominators and the numerators of their q-deformed continued fractions, and give an efficient algorithm for computing the (normalized) Jones polynomials of rational links. Various properties of q-integers defined by pairs of coprime integers are investigated and shown its applications.

The 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., Comment: 26 pages

In this note we give a precise statement and a detailed proof for reconstruction problem of weak bialgebra maps. As an application we characterize indecomposability of weak algebras in categorical setting., Comment: 20 pages

In the present paper, we build a bridge between Conway-Coxeter friezes and rational tangles through the Kauffman bracket polynomials. One can compute a Kauffman bracket polynomials attached to rational links by using Conway-Coxeter friezes. As an application one can give a complete invariant on Conway-Coxeter friezes of zigzag-type., Comment: This article is a revised version of the previous one (arXiv: 1806.04840v1)

Braided 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 my previous paper "The coribbon structures of some finite dimensional braided Hopf algebras generated by $2\times 2$-matrix coalgebras" which is published in Banach Center Publication., Comment: to appear in Proceedings of the meeting for study of number theory, Hopf algebras and related topics, which was held in Toyama, Japan in 2017

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 the Drinfel'd double, which he called the Schr\"odinger representation. We study this representation from the viewpoint of the theory of monoidal 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 F from the module category over A to the module category over B of k-linear monoidal categories, then the equivalence between the module categories over D(A) and D(B) induced by F preserves the Schr\"odinger representation. 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 Schr\"odinger 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., Comment: 32 pages, 3 figures

We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field by using the braiding structures of A. We investigate basic properties of the polynomial invariants including stability under extension of the base field. 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., Comment: 64 pages

In this article, we discuss a (2+1)-dimensional topological quantum field theory, for short TQFT, with a Verlinde basis. As a conclusion of this general theory, we have a Dehn surgery formula. We show that Turaev-Viro-Ocneanu TQFT has a Verlinde basis. Several applications of this theorem are exposed. Based on Izumi's data of subfactors, we list several computations of Turaev-Viro-Ocneanu invariants for some 3-manifolds., Comment: Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon4/paper18.abs.html Version 2: Corrections made to calculations on page 292

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., Comment: 32 pages, 20 figures

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 TQFT, 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 the forthcoming paper, we will exbit computations of Turaev-Viro-Ocneanu invariants for several ``basic 3-manifolds ''. In Appendix, we discuss the relationship between the system of M_{infinity}-M_{infinity} bimodules arising from the asymptotic inclusion M V M^{op} subset M_{infinity} constructed from N subset M and the tube system obtained from a subfactor N subset M., Comment: 37 pages, 36 figures