Kenichi Shimizu
On the linear independency of monoidal natural transformations.
Wed, 08/11/2010 - 15:01 — SomNambulAuthors: Kenichi Shimizu
Subjects: Category Theory
AbstractLet $F, G: \mathcal{I} \to \mathcal{C}$ 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 $Nat(F, G)$.Some computations of Frobenius-Schur indicators of the regular representations of Hopf algebras.
Tue, 02/23/2010 - 16:01 — SomNambulAuthors: Kenichi Shimizu
Subjects: Quantum Algebra
AbstractWe study Frobenius-Schur indicators of the regular representations of
finite-dimensional semisimple Hopf algebras, especially group-theoretical ones.
Those of various Hopf algebras are computed explicitly. In view of our
computational results, we formulate the theorem of Frobenius for semisimple
Hopf algebras and give some partial results on this problem.Monoidal Morita invariants for finite group algebras.
Wed, 10/21/2009 - 14:16 — SomNambulAuthors: Kenichi Shimizu
Subjects: Quantum Algebra
AbstractTwo 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.
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