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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: Let $ F$, $ G: \mathcal {I} \to \mathcal {C}$ be monoidal functors from a monoidal category $ \mathcal {I}$ to a linear abelian rigid monoidal category $ \mathcal {C}$ over an algebraically closed field $ \mathbf {k}$. Then the set $ \mathrm {Nat}(F, G)$ of natural transformations $ F \to G$ is naturally a vector space over $ \mathbf {k}$. Under certain assumptions, we show that the set of monoidal natural transformations $ F \to G$ is linearly independent as a subset of $ \mathrm {Nat}(F, G)$.

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.


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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.