TY - JOUR
T1 - On the linear independency of monoidal natural transformations
AU - Shimizu, Kenichi
N1 - Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.
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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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 -