TY - JOUR

T1 - On the linear independency of monoidal natural transformations

AU - Shimizu, Kenichi

PY - 2012/2/27

Y1 - 2012/2/27

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

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

M3 - Article

AN - SCOPUS:84857295058

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 -