On the linear independency of monoidal natural transformations

Author:
Kenichi Shimizu

Journal:
Proc. Amer. Math. Soc. **140** (2012), 1939-1946

MSC (2010):
Primary 18D10

Published electronically:
October 19, 2011

MathSciNet review:
2888181

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Abstract | References | Similar Articles | Additional Information

Abstract: Let , be monoidal functors from a monoidal category to a linear abelian rigid monoidal category over an algebraically closed field . Then the set of natural transformations is naturally a vector space over . Under certain assumptions, we show that the set of monoidal natural transformations is linearly independent as a subset of .

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.

**1.**Bojko Bakalov and Alexander Kirillov Jr.,*Lectures on tensor categories and modular functors*, University Lecture Series, vol. 21, American Mathematical Society, Providence, RI, 2001. MR**1797619****2.**Pavel Etingof and Viktor Ostrik,*Finite tensor categories*, Mosc. Math. J.**4**(2004), no. 3, 627–654, 782–783 (English, with English and Russian summaries). MR**2119143****3.**Saunders Mac Lane,*Categories for the working mathematician*, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR**1712872****4.**Dmitri Nikshych,*On the structure of weak Hopf algebras*, Adv. Math.**170**(2002), no. 2, 257–286. MR**1932332**, 10.1006/aima.2002.2081

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Additional Information

**Kenichi Shimizu**

Affiliation:
Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki, 305-8571, Japan

Email:
shimizu@math.tsukuba.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-2011-11068-2

Keywords:
Monoidal category,
monoidal functor,
finite tensor category.

Received by editor(s):
October 21, 2010

Received by editor(s) in revised form:
February 11, 2011

Published electronically:
October 19, 2011

Communicated by:
Lev Borisov

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.