TY - JOUR

T1 - Schrödinger Representations from the Viewpoint of Tensor Categories

AU - Shimizu, Kenichi

AU - Wakui, Michihisa

N1 - Funding Information:
For this research the first author (K.S.) is supported by Grant-in-Aid for JSPS Fellows (24⋅3606), and the second author (M.W.) is partially supported by Grant-in-Aid for Scientific Research (No. 22540058), JSPS.
Publisher Copyright:
© 2015, Springer Science+Business Media Dordrecht.

PY - 2015/12/1

Y1 - 2015/12/1

N2 - 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:(Formula presented.) of k-linear monoidal categories, then the equivalence (Formula presented.) induced by F preserves the Schrödinger representation. Here, (Formula presented.) 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.

AB - 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:(Formula presented.) of k-linear monoidal categories, then the equivalence (Formula presented.) induced by F preserves the Schrödinger representation. Here, (Formula presented.) 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.

KW - Braiding

KW - Drinfel’d double

KW - Hopf algebra

KW - Monoidal Morita invariant

KW - Monoidal category

KW - Schrödinger representation

UR - http://www.scopus.com/inward/record.url?scp=84937053629&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84937053629&partnerID=8YFLogxK

U2 - 10.1007/s10468-015-9554-7

DO - 10.1007/s10468-015-9554-7

M3 - Article

AN - SCOPUS:84937053629

SN - 1386-923X

VL - 18

SP - 1623

EP - 1647

JO - Algebras and Representation Theory

JF - Algebras and Representation Theory

IS - 6

ER -