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Kenichi Shimizu

Publications and source records attributed to Kenichi Shimizu.

43 records · Page 3Linked to original sources
arXiv2013

Schrödinger representations from the viewpoint of monoidal categories

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 Schrödinger 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 Schrödinger 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 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.

math.RA↗
arXiv2013

Frobenius-Schur theorem for $C^*$-categories

We generalize the Frobenius-Schur theorem to $C^*$-categories. From this category-theoretical point of view, we introduce the notions of real, complex and quaternionic representations of Hopf $C^*$-algebras. Based on these definitions, we give another type of the Hopf-algebraic analogue of the Frobenius-Schur theorem, originally due to Linchenko and Montgomery. Also given are similar results for weak Hopf $C^*$-algebras, table algebras and compact quantum groups.

math.RT↗
arXiv2012

Frobenius-Schur indicator for categories with duality

We introduce the Frobenius-Schur indicator for categories with duality to give a category-theoretical understanding of various generalizations of the Frobenius-Schur theorem, including that for semisimple quasi-Hopf algebras, weak Hopf C*-algebras and association schemes. Our framework also clarifies a mechanism how the `twisted' theory arises from the ordinary case. As a demonstration, we give a twisted Frobenius-Schur theorem for semisimple quasi-Hopf algebras. We also give several applications to the quantum SL_2.

math.RT↗
arXiv2010

On the linear independency of monoidal natural transformations

Let $F, G: \mathcal{I} \to \mathcal{C}$ be strong monoidal functors from a skeletally small monoidal category $\mathcal{I}$ to a tensor category $\mathcal{C}$ over an algebraically closed field $k$. The set $Nat(F, G)$ of natural transformations $F \to G$ is naturally a vector space over $k$. We show that the set $Nat_\otimes(F, G)$ of monoidal natural transformations $F \to G$ is linearly independent as a subset of $Nat(F, G)$. As a corollary, we can show that the group of monoidal natural automorphisms on the identity functor on a finite tensor category is finite. We can also show that the set of pivotal structures on a finite tensor category is finite.

math.CT↗
arXiv2009

Monoidal Morita invariants for finite group algebras

Two Hopf algebras are called monoidally Morita equivalent if module categories over them are equivalent as linear monoidal categories. We introduce monoidal Morita invariants for finite-dimensional Hopf algebras based on certain braid group representations arising from the Drinfeld double construction. As an application, we show, for any integer $n$, the number of elements of order $n$ is a monoidal Morita invariant for finite group algebras. We also describe relations between our construction and invariants of closed 3-manifolds due to Reshetikhin and Turaev.

math.QA↗