700 research outputs found
Relative Serre functor for comodule algebras
Let be a finite tensor category, and let be an
exact left -module category. A relative Serre functor of
, introduced by Fuchs, Schaumann and Schweigert, is an endofunctor
on together with a natural isomorphism
for , where is
the internal Hom functor of . In this paper, we discuss the case
where and
for a finite-dimensional Hopf algebra and a finite-dimensional exact left
-comodule algebra . We give an explicit description of a relative Serre
functor of and its twisted module structure in terms of
integrals of and the Frobenius structure of . We also study pivotal
structures on and give some explicit examples.Comment: 48 page
Frobenius-Schur indicators in Tambara-Yamagami categories
We introduce formulae of Frobenius-Schur indicators of simple objects of
Tambara-Yamagami categories. By using techniques of the Fourier transform on
finite abelian groups, we study some arithmetic properties of indicators.Comment: 21 page
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 , the number of
elements of order 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.Comment: 25 pages; To appear in J. of Algebra. Main modifications are the
following: (i) Verbose parts of the paper were summarized. (ii) Theorem 6.3
is added. (iii) The relation between Theorem 1.1 and works of Ng and
Schauenburg is adde
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