A simple property of the $S$-matrix of a pre-modular category
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I am following Pavel Etingof et al's book on tensor categories. In order to get used to the $S$-matrix of a pre-modular category and related concepts, I am trying to prove the following simple fact:
The elements $s_{XY}$ of the $S$-matrix satisfy:
$forall A,Bin mathcal O(mathcal C), s_{A^*B^*}=s_{AB}$
I tried to play around with the definitions:
$
s_{AB}=text{Tr}(b_{BA}b_{AB}) := text{Tr}^L(psi_{Aotimes B}b_{BA}b_{AB}):=ev_{(Aotimes
B)^*}circ (psi_{Aotimes B}b_{BA}b_{AB})otimes id_{(Aotimes B)^*} circ coev_{Aotimes B}
$
$
s_{A^*B^*}=text{Tr}(b_{B^*A^*}b_{A^*B^*}) := text{Tr}^R(b_{B^*A^*}b_{A^*B^*}psi^{-1}_{A^*otimes B^*}):=ev_{(A^*otimes B^*)^{veevee}}circ id_{(A^*otimes B^*)^vee}otimes (b_{B^*A^*}b_{A^*B^*}psi^{-1}_{A^*otimes B^*}) circ coev_{(A^*otimes B^*)^vee}
$
But got nowhere.
Notation: $X^vee$ is the right dual of $X$.
category-theory monoidal-categories
asked Dec 1 at 13:38
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1,006515
add a comment |
up vote
1
down vote
favorite
I am following Pavel Etingof et al's book on tensor categories. In order to get used to the $S$-matrix of a pre-modular category and related concepts, I am trying to prove the following simple fact:
The elements $s_{XY}$ of the $S$-matrix satisfy:
$forall A,Bin mathcal O(mathcal C), s_{A^*B^*}=s_{AB}$
I tried to play around with the definitions:
$
s_{AB}=text{Tr}(b_{BA}b_{AB}) := text{Tr}^L(psi_{Aotimes B}b_{BA}b_{AB}):=ev_{(Aotimes
B)^*}circ (psi_{Aotimes B}b_{BA}b_{AB})otimes id_{(Aotimes B)^*} circ coev_{Aotimes B}
$
$
s_{A^*B^*}=text{Tr}(b_{B^*A^*}b_{A^*B^*}) := text{Tr}^R(b_{B^*A^*}b_{A^*B^*}psi^{-1}_{A^*otimes B^*}):=ev_{(A^*otimes B^*)^{veevee}}circ id_{(A^*otimes B^*)^vee}otimes (b_{B^*A^*}b_{A^*B^*}psi^{-1}_{A^*otimes B^*}) circ coev_{(A^*otimes B^*)^vee}
$
But got nowhere.
Notation: $X^vee$ is the right dual of $X$.
category-theory monoidal-categories
asked Dec 1 at 13:38
Soap
1,006515
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I am following Pavel Etingof et al's book on tensor categories. In order to get used to the $S$-matrix of a pre-modular category and related concepts, I am trying to prove the following simple fact:
The elements $s_{XY}$ of the $S$-matrix satisfy:
$forall A,Bin mathcal O(mathcal C), s_{A^*B^*}=s_{AB}$
I tried to play around with the definitions:
$
s_{AB}=text{Tr}(b_{BA}b_{AB}) := text{Tr}^L(psi_{Aotimes B}b_{BA}b_{AB}):=ev_{(Aotimes
B)^*}circ (psi_{Aotimes B}b_{BA}b_{AB})otimes id_{(Aotimes B)^*} circ coev_{Aotimes B}
$
$
s_{A^*B^*}=text{Tr}(b_{B^*A^*}b_{A^*B^*}) := text{Tr}^R(b_{B^*A^*}b_{A^*B^*}psi^{-1}_{A^*otimes B^*}):=ev_{(A^*otimes B^*)^{veevee}}circ id_{(A^*otimes B^*)^vee}otimes (b_{B^*A^*}b_{A^*B^*}psi^{-1}_{A^*otimes B^*}) circ coev_{(A^*otimes B^*)^vee}
$
But got nowhere.
Notation: $X^vee$ is the right dual of $X$.
category-theory monoidal-categories
asked Dec 1 at 13:38
Soap
1,006515
I am following Pavel Etingof et al's book on tensor categories. In order to get used to the $S$-matrix of a pre-modular category and related concepts, I am trying to prove the following simple fact:
The elements $s_{XY}$ of the $S$-matrix satisfy:
$forall A,Bin mathcal O(mathcal C), s_{A^*B^*}=s_{AB}$
I tried to play around with the definitions:
$
s_{AB}=text{Tr}(b_{BA}b_{AB}) := text{Tr}^L(psi_{Aotimes B}b_{BA}b_{AB}):=ev_{(Aotimes
B)^*}circ (psi_{Aotimes B}b_{BA}b_{AB})otimes id_{(Aotimes B)^*} circ coev_{Aotimes B}
$
$
s_{A^*B^*}=text{Tr}(b_{B^*A^*}b_{A^*B^*}) := text{Tr}^R(b_{B^*A^*}b_{A^*B^*}psi^{-1}_{A^*otimes B^*}):=ev_{(A^*otimes B^*)^{veevee}}circ id_{(A^*otimes B^*)^vee}otimes (b_{B^*A^*}b_{A^*B^*}psi^{-1}_{A^*otimes B^*}) circ coev_{(A^*otimes B^*)^vee}
$
But got nowhere.
Notation: $X^vee$ is the right dual of $X$.
category-theory monoidal-categories
category-theory monoidal-categories
asked Dec 1 at 13:38
Soap
1,006515
asked Dec 1 at 13:38
Soap
1,006515
asked Dec 1 at 13:38
Soap
1,006515
asked Dec 1 at 13:38
Soap
1,006515
asked Dec 1 at 13:38
Soap
1,006515
1,006515
add a comment |
add a comment |
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