### Abstract

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

Original language | English |
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Pages (from-to) | 1939-1946 |

Number of pages | 8 |

Journal | Proceedings of the American Mathematical Society |

Volume | 140 |

Issue number | 6 |

DOIs | |

Publication status | Published - 2012 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**On the linear independency of monoidal natural transformations.** / Shimizu, Kenichi.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 140, no. 6, pp. 1939-1946. https://doi.org/10.1090/S0002-9939-2011-11068-2

}

TY - JOUR

T1 - On the linear independency of monoidal natural transformations

AU - Shimizu, Kenichi

PY - 2012

Y1 - 2012

N2 - Let F, G:I → C be monoidal functors from a monoidal category I to a linear abelian rigid monoidal category C over an algebraically closed field k. Then the set Nat(F,G) of natural transformations F → G is naturally a vector space over k. Under certain assumptions, we show that the set of monoidal natural transformations F → G is linearly independent as a subset of 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.

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

UR - http://www.scopus.com/inward/record.url?scp=84857295058&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84857295058&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-2011-11068-2

DO - 10.1090/S0002-9939-2011-11068-2

M3 - Article

VL - 140

SP - 1939

EP - 1946

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 6

ER -