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 |
|---|---|
| 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.
In: Proceedings of the American Mathematical Society, Vol. 140, No. 6, 2012, p. 1939-1946.Research output: Contribution to journal › Article
}
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.
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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 -