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Finding the Drinfeld centre of a category
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I have the following unitary monoidal spherical category C:
Simple objects: $1,x,y$.
Non-trivial Fusion Rules:
$$xotimes y=x=yotimes x$$
$$xotimes x=1 oplus 2x oplus y$$
I would like to find the Drinfeld centre for this category. I am aware that the Drinfeld centre has elements of the form $(x,e_x)$ where $xin C$ and $e_x=e_x(y)in Hom(xy,yx),yin C$ has to satisfy
(i) $fotimes id_x o e_x(y)=e_x(z) o id_x otimes f forall f:yrightarrow z$
(ii) $e_x(yotimes z)=id_yotimes e_z(z) o e_X(y) otimes id_z
forall y,z in C$
(iii) $e_x(1)=id_x$
I am trying to find explicit expressions for the maps $e_x(y)$.
One can consider a basis $(v^yx_x_k)_io(v^x_k_xy)_j$ where $x_k=1,x,y$
and $1leq i,j leq dim(Hom(xy,x_k))$ for the spaces Hom(xy,yx).
For example: $e_y(y)= alpha (v^yy_1 o v^1_yy)$ where $alpha$ is some constant which we have to determine using the conditions above. If we use the first condition and choose z=1 then we get,
$$f otimes id_y o e_y(y)=id_y otimes f$$ where $f:yrightarrow 1$. I am not sure how to proceed from here. Is this information enough to determine $alpha$? I think I am missing something here and any help would be appreciated.
Edit: Since the category is spherical, one could maybe make use of the traces and quantum dimensions of the simple objects. But, I am not sure how to proceed.
category-theory monoidal-categories fusion-categories
asked Jul 27 at 11:55
Rajath Krishna R
862728
add a comment |Â
up vote
1
down vote
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I have the following unitary monoidal spherical category C:
Simple objects: $1,x,y$.
Non-trivial Fusion Rules:
$$xotimes y=x=yotimes x$$
$$xotimes x=1 oplus 2x oplus y$$
I would like to find the Drinfeld centre for this category. I am aware that the Drinfeld centre has elements of the form $(x,e_x)$ where $xin C$ and $e_x=e_x(y)in Hom(xy,yx),yin C$ has to satisfy
(i) $fotimes id_x o e_x(y)=e_x(z) o id_x otimes f forall f:yrightarrow z$
(ii) $e_x(yotimes z)=id_yotimes e_z(z) o e_X(y) otimes id_z
forall y,z in C$
(iii) $e_x(1)=id_x$
I am trying to find explicit expressions for the maps $e_x(y)$.
One can consider a basis $(v^yx_x_k)_io(v^x_k_xy)_j$ where $x_k=1,x,y$
and $1leq i,j leq dim(Hom(xy,x_k))$ for the spaces Hom(xy,yx).
For example: $e_y(y)= alpha (v^yy_1 o v^1_yy)$ where $alpha$ is some constant which we have to determine using the conditions above. If we use the first condition and choose z=1 then we get,
$$f otimes id_y o e_y(y)=id_y otimes f$$ where $f:yrightarrow 1$. I am not sure how to proceed from here. Is this information enough to determine $alpha$? I think I am missing something here and any help would be appreciated.
Edit: Since the category is spherical, one could maybe make use of the traces and quantum dimensions of the simple objects. But, I am not sure how to proceed.
category-theory monoidal-categories fusion-categories
asked Jul 27 at 11:55
Rajath Krishna R
862728
That is not enough information to determine the category or its Drinfeld center; you haven't told us anything about the associator, for example.
– Qiaochu Yuan
Jul 27 at 20:40
@Qiaochu Yuan. The associativity relations can be derived from the fusion rules by solving the Pentagon equation and the explicit form of all the association matrices are known.
– Rajath Krishna R
Jul 28 at 1:02
@QiaochuYuan Can you kindly elaborate on your comment? I am still struggling to find the answer. The appendix of arxiv.org/abs/0710.5761 contains all the half-braidings for the above category. But, I am not sure how to obtain them.
– Rajath Krishna R
Aug 4 at 16:04
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I have the following unitary monoidal spherical category C:
Simple objects: $1,x,y$.
Non-trivial Fusion Rules:
$$xotimes y=x=yotimes x$$
$$xotimes x=1 oplus 2x oplus y$$
I would like to find the Drinfeld centre for this category. I am aware that the Drinfeld centre has elements of the form $(x,e_x)$ where $xin C$ and $e_x=e_x(y)in Hom(xy,yx),yin C$ has to satisfy
(i) $fotimes id_x o e_x(y)=e_x(z) o id_x otimes f forall f:yrightarrow z$
(ii) $e_x(yotimes z)=id_yotimes e_z(z) o e_X(y) otimes id_z
forall y,z in C$
(iii) $e_x(1)=id_x$
I am trying to find explicit expressions for the maps $e_x(y)$.
One can consider a basis $(v^yx_x_k)_io(v^x_k_xy)_j$ where $x_k=1,x,y$
and $1leq i,j leq dim(Hom(xy,x_k))$ for the spaces Hom(xy,yx).
For example: $e_y(y)= alpha (v^yy_1 o v^1_yy)$ where $alpha$ is some constant which we have to determine using the conditions above. If we use the first condition and choose z=1 then we get,
$$f otimes id_y o e_y(y)=id_y otimes f$$ where $f:yrightarrow 1$. I am not sure how to proceed from here. Is this information enough to determine $alpha$? I think I am missing something here and any help would be appreciated.
Edit: Since the category is spherical, one could maybe make use of the traces and quantum dimensions of the simple objects. But, I am not sure how to proceed.
category-theory monoidal-categories fusion-categories
asked Jul 27 at 11:55
Rajath Krishna R
862728
I have the following unitary monoidal spherical category C:
Simple objects: $1,x,y$.
Non-trivial Fusion Rules:
$$xotimes y=x=yotimes x$$
$$xotimes x=1 oplus 2x oplus y$$
I would like to find the Drinfeld centre for this category. I am aware that the Drinfeld centre has elements of the form $(x,e_x)$ where $xin C$ and $e_x=e_x(y)in Hom(xy,yx),yin C$ has to satisfy
(i) $fotimes id_x o e_x(y)=e_x(z) o id_x otimes f forall f:yrightarrow z$
(ii) $e_x(yotimes z)=id_yotimes e_z(z) o e_X(y) otimes id_z
forall y,z in C$
(iii) $e_x(1)=id_x$
I am trying to find explicit expressions for the maps $e_x(y)$.
One can consider a basis $(v^yx_x_k)_io(v^x_k_xy)_j$ where $x_k=1,x,y$
and $1leq i,j leq dim(Hom(xy,x_k))$ for the spaces Hom(xy,yx).
For example: $e_y(y)= alpha (v^yy_1 o v^1_yy)$ where $alpha$ is some constant which we have to determine using the conditions above. If we use the first condition and choose z=1 then we get,
$$f otimes id_y o e_y(y)=id_y otimes f$$ where $f:yrightarrow 1$. I am not sure how to proceed from here. Is this information enough to determine $alpha$? I think I am missing something here and any help would be appreciated.
Edit: Since the category is spherical, one could maybe make use of the traces and quantum dimensions of the simple objects. But, I am not sure how to proceed.
category-theory monoidal-categories fusion-categories
asked Jul 27 at 11:55
Rajath Krishna R
862728
edited Jul 27 at 12:14
asked Jul 27 at 11:55
Rajath Krishna R
862728
asked Jul 27 at 11:55
Rajath Krishna R
862728
asked Jul 27 at 11:55
Rajath Krishna R
862728
862728
That is not enough information to determine the category or its Drinfeld center; you haven't told us anything about the associator, for example.
– Qiaochu Yuan
Jul 27 at 20:40
@Qiaochu Yuan. The associativity relations can be derived from the fusion rules by solving the Pentagon equation and the explicit form of all the association matrices are known.
– Rajath Krishna R
Jul 28 at 1:02
@QiaochuYuan Can you kindly elaborate on your comment? I am still struggling to find the answer. The appendix of arxiv.org/abs/0710.5761 contains all the half-braidings for the above category. But, I am not sure how to obtain them.
– Rajath Krishna R
Aug 4 at 16:04
add a comment |Â
That is not enough information to determine the category or its Drinfeld center; you haven't told us anything about the associator, for example.
– Qiaochu Yuan
Jul 27 at 20:40
@Qiaochu Yuan. The associativity relations can be derived from the fusion rules by solving the Pentagon equation and the explicit form of all the association matrices are known.
– Rajath Krishna R
Jul 28 at 1:02
@QiaochuYuan Can you kindly elaborate on your comment? I am still struggling to find the answer. The appendix of arxiv.org/abs/0710.5761 contains all the half-braidings for the above category. But, I am not sure how to obtain them.
– Rajath Krishna R
Aug 4 at 16:04
That is not enough information to determine the category or its Drinfeld center; you haven't told us anything about the associator, for example.
– Qiaochu Yuan
Jul 27 at 20:40
That is not enough information to determine the category or its Drinfeld center; you haven't told us anything about the associator, for example.
– Qiaochu Yuan
Jul 27 at 20:40
@Qiaochu Yuan. The associativity relations can be derived from the fusion rules by solving the Pentagon equation and the explicit form of all the association matrices are known.
– Rajath Krishna R
Jul 28 at 1:02
@Qiaochu Yuan. The associativity relations can be derived from the fusion rules by solving the Pentagon equation and the explicit form of all the association matrices are known.
– Rajath Krishna R
Jul 28 at 1:02
@QiaochuYuan Can you kindly elaborate on your comment? I am still struggling to find the answer. The appendix of arxiv.org/abs/0710.5761 contains all the half-braidings for the above category. But, I am not sure how to obtain them.
– Rajath Krishna R
Aug 4 at 16:04
@QiaochuYuan Can you kindly elaborate on your comment? I am still struggling to find the answer. The appendix of arxiv.org/abs/0710.5761 contains all the half-braidings for the above category. But, I am not sure how to obtain them.
– Rajath Krishna R
Aug 4 at 16:04
add a comment |Â
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That is not enough information to determine the category or its Drinfeld center; you haven't told us anything about the associator, for example.
– Qiaochu Yuan
Jul 27 at 20:40
@Qiaochu Yuan. The associativity relations can be derived from the fusion rules by solving the Pentagon equation and the explicit form of all the association matrices are known.
– Rajath Krishna R
Jul 28 at 1:02
@QiaochuYuan Can you kindly elaborate on your comment? I am still struggling to find the answer. The appendix of arxiv.org/abs/0710.5761 contains all the half-braidings for the above category. But, I am not sure how to obtain them.
– Rajath Krishna R
Aug 4 at 16:04