caa a22 4500 606160671 CHVBK 20210128100631.0 cr unu---uuuuu 210128e20151201xx s 000 0 eng 10.1007/s10468-015-9554-7 doi (NATIONALLICENCE)springer-10.1007/s10468-015-9554-7 Schrödinger Representations from the Viewpoint of Tensor Categories [Elektronische Daten] [Kenichi Shimizu, Michihisa Wakui] 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 Schrödinger 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 F : A M → B M $F: {~}_{A}{\mathbf{M}} \to {~}_{B}{\mathbf{M}}$ of k-linear monoidal categories, then the equivalence D ( A ) M ≈ D ( B ) M ${~}_{D(A)}{\mathbf{M}} \approx {~}_{D(B)}{\mathbf{M}}$ induced by F preserves the Schrödinger representation. Here, A M ${~}_{A}\mathbf{M}$ 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 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. Springer Science+Business Media Dordrecht, 2015 Hopf algebra nationallicence Monoidal category nationallicence Drinfel'd double nationallicence Schrödinger representation nationallicence Monoidal Morita invariant nationallicence Braiding nationallicence Shimizu Kenichi Graduate School of Mathematics, Nagoya University, Furo-cho, 464-8602, Nagoya, Chikusa-ku, Japan aut Wakui Michihisa Department of Mathematics, Faculty of Engineering Science, Kansai University, 564-8680, Osaka, Suita-shi, Japan aut Algebras and Representation Theory Springer Netherlands 18/6(2015-12-01), 1623-1647 1386-923X 18:6<1623 2015 18 10468 https://doi.org/10.1007/s10468-015-9554-7 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10468-015-9554-7 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Shimizu Kenichi Graduate School of Mathematics, Nagoya University, Furo-cho, 464-8602, Nagoya, Chikusa-ku, Japan aut NATIONALLICENCE 700 1- Wakui Michihisa Department of Mathematics, Faculty of Engineering Science, Kansai University, 564-8680, Osaka, Suita-shi, Japan aut NATIONALLICENCE 773 0- Algebras and Representation Theory Springer Netherlands 18/6(2015-12-01), 1623-1647 1386-923X 18:6<1623 2015 18 10468