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''braided monoidal category, quasi-tensor category''
 
''braided monoidal category, quasi-tensor category''
  
A generalization of the notion of tensor product of vector spaces, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b1204201.png" /> is associative only up to isomorphism and the transposition isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b1204202.png" /> need not square to the identity.
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A generalization of the notion of tensor product of vector spaces, in which $\otimes$ is associative only up to isomorphism and the transposition isomorphism $\otimes \rightarrow \otimes ^ { \text{op} }$ need not square to the identity.
  
The full definition is: a [[Category|category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b1204203.png" /> of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b1204204.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b1204205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b1204206.png" />, etc., equipped with a functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b1204207.png" />, and a collection of functorial isomorphisms
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The full definition is: a [[Category|category]] $\mathcal{C}$ of objects $V$, $W$, $Z$, etc., equipped with a functor $\otimes : \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C}$, and a collection of functorial isomorphisms
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b1204208.png" /></td> </tr></table>
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\begin{equation*} \Phi _ { V , W , Z } : ( V \bigotimes W ) \bigotimes Z \rightarrow V \bigotimes ( W \bigotimes Z ) \end{equation*}
  
 
(called the associator) between any three objects and
 
(called the associator) between any three objects and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b1204209.png" /></td> </tr></table>
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\begin{equation*} \Psi _ { V , W } : V \bigotimes W \rightarrow W \bigotimes V \end{equation*}
  
 
(called the braiding) between any two objects.  "Functorial"  means that these isomorphisms commute with any morphisms between objects. Thus,
 
(called the braiding) between any two objects.  "Functorial"  means that these isomorphisms commute with any morphisms between objects. Thus,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042010.png" /></td> </tr></table>
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\begin{equation*} ( \phi \bigotimes \text { id } ) \Psi _ { V , W } = \Psi _ { V , Z } ( \text { id } \bigotimes \phi ) , \forall \phi : W \rightarrow Z, \end{equation*}
  
if the morphism is applied before or after the braiding. Similarly for functoriality in the other arguments of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042011.png" /> and in the arguments of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042012.png" />. The precise definition is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042014.png" /> are natural equivalences, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042015.png" /> is the functor sending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042016.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042017.png" />, etc., and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042018.png" /> is the functor sending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042019.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042020.png" />.
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if the morphism is applied before or after the braiding. Similarly for functoriality in the other arguments of $\Psi$ and in the arguments of $\Phi$. The precise definition is that $\Phi : ( \otimes ) \otimes \rightarrow \otimes ( \otimes )$ and $\Psi : \otimes \rightarrow \otimes ^ { \text{ op} }$ are natural equivalences, where $( \otimes ) \otimes :\mathcal  C \times \mathcal C \times \mathcal C \rightarrow \mathcal C$ is the functor sending $( V , W , Z )$ to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042017.png"/>, etc., and $\otimes ^{\operatorname{op}} : \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C}$ is the functor sending $( V , W )$ to $W \otimes V$.
  
In addition, these functors are required to be coherent. The coherence for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042021.png" /> is the pentagon identity
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In addition, these functors are required to be coherent. The coherence for $\Phi$ is the pentagon identity
  
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b120420a.gif" />
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<img src="https://www.encyclopediaofmath.org/legacyimages/common_img/b120420a.gif" style="border:1px solid;"/>
  
 
Figure: b120420a
 
Figure: b120420a
  
It says that the two ways to reverse the bracketings as shown coincide. MacLane's coherence theorem says that then all other routes between two bracketed tensor products also coincide. In effect, this means that one may generalize constructions in linear algebra exactly as if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042022.png" /> were strictly associative, dropping brackets. Afterwards one may add brackets, for example with brackets accumulating to the left, and then insert applications of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042023.png" /> as needed for the desired compositions to make sense; all different ways to do this will yield the same net result.
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It says that the two ways to reverse the bracketings as shown coincide. MacLane's coherence theorem says that then all other routes between two bracketed tensor products also coincide. In effect, this means that one may generalize constructions in linear algebra exactly as if $\otimes$ were strictly associative, dropping brackets. Afterwards one may add brackets, for example with brackets accumulating to the left, and then insert applications of $\Phi$ as needed for the desired compositions to make sense; all different ways to do this will yield the same net result.
  
One also requires a unit object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042024.png" /> and an associated collection of functorial isomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042026.png" /> obeying a triangle coherence identity
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One also requires a unit object $1$ and an associated collection of functorial isomorphisms $r _{V} : V \rightarrow V \otimes \underline { 1 }$ and $l _ { V } : V \rightarrow \underline { 1 } \otimes V$ obeying a triangle coherence identity
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042027.png" /></td> </tr></table>
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\begin{equation*} \operatorname { id} \bigotimes r _ { W } = \Phi _ { V , 1 , W } \circ ( l _ { V } \bigotimes \text { id } ). \end{equation*}
  
This structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042028.png" /> is called a monoidal category (see also [[Triple|Triple]]).
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This structure $( \mathcal C , \otimes , \Phi , \underline { 1 } , l , r )$ is called a monoidal category (see also [[Triple|Triple]]).
  
The coherence conditions for the additional structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042029.png" /> of a braided category are the so-called hexagon identities
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The coherence conditions for the additional structure $\Psi$ of a braided category are the so-called hexagon identities
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042030.png" /></td> </tr></table>
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\begin{equation*} \Psi _ { V \bigotimes W , Z } = \Psi _ { V , Z } \circ \Psi _ { W , Z }, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042031.png" /></td> </tr></table>
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\begin{equation*} \Psi _ { V , W \bigotimes  Z } = \Psi _ { V ,\, Z } \circ \Psi _ { V , W } . \end{equation*}
  
Here and below, the bracketings and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042032.png" /> needed to make sense of these identities are omitted. (When they are inserted, each identity corresponds to a diagram with six arrows.) One can show that compatibility of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042033.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042034.png" /> is then automatic.
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Here and below, the bracketings and $\Phi$ needed to make sense of these identities are omitted. (When they are inserted, each identity corresponds to a diagram with six arrows.) One can show that compatibility of $\Psi$ with $1$ is then automatic.
  
Notice that although <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042035.png" /> generalizes the concept of transposition of vector spaces, one does not demand that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042036.png" />. If this does hold for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042038.png" />, then a symmetric monoidal category or tensor category is obtained. In this case the two hexagon identities are equivalent and ensure that all ways to go
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Notice that although $\Psi$ generalizes the concept of transposition of vector spaces, one does not demand that $\Psi _ { V , W } = \Psi _ { W , V } ^ { - 1 }$. If this does hold for all $V$, $W$, then a symmetric monoidal category or tensor category is obtained. In this case the two hexagon identities are equivalent and ensure that all ways to go
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042039.png" /></td> </tr></table>
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\begin{equation*} V _ { 1 } \bigotimes \ldots \bigotimes V _ { n } \rightarrow V _ { \sigma ( 1 ) } \bigotimes \ldots \bigotimes V _ { \sigma ( n ) } \end{equation*}
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042040.png" /> a permutation) by composing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042041.png" /> (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042042.png" />) yield the same result. In particular, there is an action of the [[Symmetric group|symmetric group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042043.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042044.png" /> for any object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042045.png" />.
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($\sigma$ a permutation) by composing $\Psi$ (and $\Phi$) yield the same result. In particular, there is an action of the [[Symmetric group|symmetric group]] $S _ { n }$ in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042044.png"/> for any object $V$.
  
For a general braided category there is a similar result in terms of braids. To explain this, the following notational device is used: instead of writing the morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042047.png" /> in the usual way as arrows, one writes them as braids and assumes that they point downwards:
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For a general braided category there is a similar result in terms of braids. To explain this, the following notational device is used: instead of writing the morphisms $\Psi _ { V , W }$ and $\Psi _ { W , V } ^ { - 1 }$ in the usual way as arrows, one writes them as braids and assumes that they point downwards:
  
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b120420b.gif" />
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<img src="https://www.encyclopediaofmath.org/legacyimages/common_img/b120420b.gif" style="border:1px solid;"/>
  
 
Figure: b120420b
 
Figure: b120420b
  
One writes another morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042048.png" />, say, as a node on a strand connecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042049.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042050.png" /> and assumed to be pointing downwards. Functoriality says that morphisms can be pulled through braid crossings as
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One writes another morphism $\phi : W \rightarrow Z$, say, as a node on a strand connecting $W$ to $Z$ and assumed to be pointing downwards. Functoriality says that morphisms can be pulled through braid crossings as
  
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b120420c.gif" />
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<img src="https://www.encyclopediaofmath.org/legacyimages/common_img/b120420c.gif" style="border:1px solid;"/>
  
 
Figure: b120420c
 
Figure: b120420c
Line 59: Line 67:
 
Similarly for morphisms on the other strand. When the tensor product of several morphisms is applied, one writes their strands separately side by side, connecting the relevant objects in the tensor product. On the other hand, one is free to group some of the objects in the tensor product together as a single object and represent morphisms to and from it by a single strand. This notation is consistent precisely because of the hexagon conditions, which become
 
Similarly for morphisms on the other strand. When the tensor product of several morphisms is applied, one writes their strands separately side by side, connecting the relevant objects in the tensor product. On the other hand, one is free to group some of the objects in the tensor product together as a single object and represent morphisms to and from it by a single strand. This notation is consistent precisely because of the hexagon conditions, which become
  
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b120420d.gif" />
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<img src="https://www.encyclopediaofmath.org/legacyimages/common_img/b120420d.gif" style="border:1px solid;"/>
  
 
Figure: b120420d
 
Figure: b120420d
Line 65: Line 73:
 
The doubled strands on the left-hand side could be replaced by single strands for the composite morphisms.
 
The doubled strands on the left-hand side could be replaced by single strands for the composite morphisms.
  
The coherence theorem for braided categories then asserts that different routes between tensor product expressions by repeated applications of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042052.png" /> and their inverses compose to the same morphism if the corresponding braids are the same. In particular, for any object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042053.png" /> there is an action of the pure braid group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042054.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042055.png" />. The former can be presented as
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The coherence theorem for braided categories then asserts that different routes between tensor product expressions by repeated applications of $\Phi$, $\Psi$ and their inverses compose to the same morphism if the corresponding braids are the same. In particular, for any object $V$ there is an action of the pure braid group $B _ { n }$ on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042055.png"/>. The former can be presented as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042056.png" /></td> </tr></table>
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\begin{equation*} b _ { i } b _ { i  + 1} b _ { i } = b _ { i  + 1} b _ { i } b _ { i } + 1 , b _ { i } b _ { j } = b _ { j } b _ { i } , \quad | i - j | \geq 2, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042057.png" /> is represented by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042058.png" /> acting in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042059.png" />th copies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042060.png" />. The representation usually has a kernel and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042061.png" /> modulo this kernel is the Hecke algebra associated to an object in a braided category.
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where $b _ { i }$ is represented by $\Psi$ acting in the $( i , i + 1 )$th copies of $V$. The representation usually has a kernel and $B _ { n }$ modulo this kernel is the Hecke algebra associated to an object in a braided category.
  
Finally, an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042062.png" /> in a braided category is rigid if there are another object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042063.png" /> and morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042065.png" /> such that
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Finally, an object $V$ in a braided category is rigid if there are another object $V ^ { * }$ and morphisms $\operatorname{ev} _ { V } : V ^ { * } \otimes V \rightarrow \underline { 1 }$ and $\operatorname{coev}_V  : \underline { 1 } \rightarrow V \otimes V ^ { * }$ such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042066.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042066.png"/></td> </tr></table>
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042067.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042067.png"/></td> </tr></table>
  
(suppressing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042070.png" />). Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042071.png" /> is called a left dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042072.png" /> and is unique up to isomorphism (there is a similar notion of right dual). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042073.png" /> is called rigid if every object has such duals. Using diagrammatic notation one writes
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(suppressing $\Phi$, $l$, $r$). Here, $V ^ { * }$ is called a left dual of $V$ and is unique up to isomorphism (there is a similar notion of right dual). $\mathcal{C}$ is called rigid if every object has such duals. Using diagrammatic notation one writes
  
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b120420e.gif" />
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<img src="https://www.encyclopediaofmath.org/legacyimages/common_img/b120420e.gif" style="border:1px solid;"/>
  
 
Figure: b120420e
 
Figure: b120420e
Line 85: Line 93:
 
and the above axioms become
 
and the above axioms become
  
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b120420f.gif" />
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<img src="https://www.encyclopediaofmath.org/legacyimages/common_img/b120420f.gif" style="border:1px solid;"/>
  
 
Figure: b120420f
 
Figure: b120420f
  
Note that in a rigid braided category, morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042074.png" /> look like knots (cf. also [[Knot theory|Knot theory]]; [[Braid theory|Braid theory]]). Every knot presented on paper is in fact the closure of some braid, so with a little more structure (notably, both left and right duals) one can arrange that every oriented knot can be read from top to bottom as a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042075.png" />. Fixing an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042076.png" />,
+
Note that in a rigid braided category, morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042074.png"/> look like knots (cf. also [[Knot theory|Knot theory]]; [[Braid theory|Braid theory]]). Every knot presented on paper is in fact the closure of some braid, so with a little more structure (notably, both left and right duals) one can arrange that every oriented knot can be read from top to bottom as a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042075.png"/>. Fixing an object $V$,
  
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
+
<img src="https://www.encyclopediaofmath.org/legacyimages/common_img/" style="border:1px solid;"/>
  
can be read as a left or right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042077.png" /> and
+
can be read as a left or right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042077.png"/> and
  
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/" />
+
<img src="https://www.encyclopediaofmath.org/legacyimages/common_img/" style="border:1px solid;"/>
  
can be read as a left or right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042078.png" /> (in accordance with orientation). One can read braid crossings as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042079.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042080.png" />. The result is not quite a knot invariant but can usually be adjusted to become one.
+
can be read as a left or right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042078.png"/> (in accordance with orientation). One can read braid crossings as $\Psi$ or $\Psi ^ { - 1 }$. The result is not quite a knot invariant but can usually be adjusted to become one.
  
 
==Examples.==
 
==Examples.==
Some standard examples of braided categories are provided by the following constructions over a ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042081.png" />. Its invertible elements are denoted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042082.png" />.
+
Some standard examples of braided categories are provided by the following constructions over a ground field $k$. Its invertible elements are denoted $k ^ { * }$.
  
1) Fix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042083.png" />, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042084.png" />th root of unity. The category of anyspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042085.png" /> consists of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042086.png" />-graded vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042087.png" /> (where the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042088.png" /> have degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042089.png" />). Morphisms are degree-preserving linear mappings. One takes the associator trivial and the braiding
+
1) Fix $q \in k$, an $n$th root of unity. The category of anyspaces $\operatorname{Vec}_n$ consists of $\mathbf{Z} / n \mathbf{Z}$-graded vector spaces $V = \oplus _ { i = 0 } ^ { n - 1 } V _ { i }$ (where the elements of $V _ { i }$ have degree $i$). Morphisms are degree-preserving linear mappings. One takes the associator trivial and the braiding
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042090.png" /></td> </tr></table>
+
\begin{equation*} \Psi _ { V , W } ( v \bigotimes w ) = q ^ { |v| | w | } w \bigotimes v \end{equation*}
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042091.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042092.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042093.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042094.png" />, respectively.
+
for $v \in V$ and $w \in W$ of degree $v$, $w $, respectively.
  
Clearly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042095.png" /> is the usual category of vector spaces, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042096.png" /> is the category of linear superspaces or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042097.png" />-graded vector spaces. The category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042098.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042099.png" />-graded spaces is braided similarly for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420100.png" />.
+
Clearly, $\operatorname{Vec}_{1}$ is the usual category of vector spaces, while $\text{Vec}_2$ is the category of linear superspaces or ${\bf Z} / 2 {\bf Z}$-graded vector spaces. The category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b12042098.png"/> of $\mathbf{Z}$-graded spaces is braided similarly for any $q \in k ^ { * }$.
  
A similar construction works for grading by any Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420101.png" /> equipped with a bicharacter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420102.png" /> (a function multiplicative in each argument) and braiding
+
A similar construction works for grading by any Abelian group $G$ equipped with a bicharacter $\beta : G \times G \rightarrow k ^ { * }$ (a function multiplicative in each argument) and braiding
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420103.png" /></td> </tr></table>
+
\begin{equation*} \Psi _ { V , W } ( v \bigotimes w ) = \beta ( | v | , | w | ) w \bigotimes v \end{equation*}
  
on elements of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420104.png" />.
+
on elements of degree $| v | , | w | \in G$.
  
2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420105.png" /> is a group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420106.png" /> a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420107.png" />-cocycle, then the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420108.png" />-graded spaces is monoidal with associator
+
2) If $G$ is a group and $\phi : G \times G \times G \rightarrow k ^ { * }$ a group $3$-cocycle, then the category of $G$-graded spaces is monoidal with associator
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420109.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420109.png"/></td> </tr></table>
  
on elements of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420110.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420111.png" /> is Abelian and equipped with a quasi-bicharacter with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420112.png" />, then the category is braided.
+
on elements of degree $| v | , | w | , | z | \in G$. If $G$ is Abelian and equipped with a quasi-bicharacter with respect to $\phi$, then the category is braided.
  
For example, the octonion algebra lives naturally in such a category with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420113.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420114.png" /> a certain coboundary.
+
For example, the octonion algebra lives naturally in such a category with $G = \mathbf{Z} _ { 2 } \times \mathbf{Z} _ { 2 } \times \mathbf{Z} _ { 2 }$ and $\phi$ a certain coboundary.
  
3) The category of finite-dimensional representations of the quantum enveloping algebras (cf. also [[Quantum groups|Quantum groups]]; [[Universal enveloping algebra|Universal enveloping algebra]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420115.png" /> associated to a semi-simple [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420116.png" />. Also, certain classes of infinite-dimensional representations. Associated to the standard representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420117.png" /> is the standard Hecke algebra defined by the additional relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420118.png" />. The knot invariant associated to this same representation is the Jones knot polynomial.
+
3) The category of finite-dimensional representations of the quantum enveloping algebras (cf. also [[Quantum groups|Quantum groups]]; [[Universal enveloping algebra|Universal enveloping algebra]]) $U _ { q } ( \mathfrak { g } )$ associated to a semi-simple [[Lie algebra|Lie algebra]] $\frak g$. Also, certain classes of infinite-dimensional representations. Associated to the standard representation of $U _ { q } ( \operatorname{sl} _ { 2 } )$ is the standard Hecke algebra defined by the additional relation $( b _ { i } - q ) ( b _ { i } + q ^ { - 1 } ) = 0$. The knot invariant associated to this same representation is the Jones knot polynomial.
  
In an algebraic formulation, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420119.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420120.png" />, is a [[Quasi-triangular Hopf algebra|quasi-triangular Hopf algebra]], then its category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420121.png" /> of representations is braided with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420122.png" />, where one first acts with the quasi-triangular structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420123.png" /> and then applies the usual transposition mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420124.png" />.
+
In an algebraic formulation, if $H$, $\mathcal{R} \in H \otimes H$, is a [[Quasi-triangular Hopf algebra|quasi-triangular Hopf algebra]], then its category $\square _ { H } \cal M$ of representations is braided with $\Psi = \tau \circ \mathcal{R}$, where one first acts with the quasi-triangular structure $\mathcal{R} \in H \otimes H$ and then applies the usual transposition mapping $\tau$.
  
4) The category of co-modules under the matrix quantum groups (cf. [[Quantum groups|Quantum groups]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420125.png" />. For example, the quantum plane generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420126.png" /> modulo the relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420127.png" /> is covariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420128.png" /> and hence lives as an algebra in its braided category of co-modules. Likewise, the category of co-modules under a quantum matrix bi-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420129.png" /> associated to an invertible solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420130.png" /> of the [[Yang–Baxter equation|Yang–Baxter equation]] is braided.
+
4) The category of co-modules under the matrix quantum groups (cf. [[Quantum groups|Quantum groups]]) $G_q$. For example, the quantum plane generated by $x , y$ modulo the relations $y x = q x y$ is covariant under $\operatorname{SL} _ { q } ( 2 )$ and hence lives as an algebra in its braided category of co-modules. Likewise, the category of co-modules under a quantum matrix bi-algebra $A ( R )$ associated to an invertible solution $R$ of the [[Yang–Baxter equation|Yang–Baxter equation]] is braided.
  
In a general algebraic formulation, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420131.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420132.png" />, is a dual (or co-) quasi-triangular Hopf algebra, then its category of co-modules is braided by
+
In a general algebraic formulation, if $H$, $\mathcal{R} : H \otimes H \rightarrow k $, is a dual (or co-) quasi-triangular Hopf algebra, then its category of co-modules is braided by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420133.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420133.png"/></td> </tr></table>
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420134.png" /> denotes the result of the co-action <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420135.png" />.
+
where $\sum _ { V } v ^ { \overline{( 1 ) }} \otimes v ^ { \overline{( 2 ) } }$ denotes the result of the co-action $V \rightarrow H \otimes V$.
  
5) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420136.png" /> is any [[Hopf algebra|Hopf algebra]] with invertible antipode, then its category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420137.png" /> of crossed modules (also called Drinfel'd–Radford–Yetter modules) is braided. Here, objects are vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420138.png" /> which are both modules and co-modules under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420139.png" />, the two being compatible in the sense
+
5) If $H$ is any [[Hopf algebra|Hopf algebra]] with invertible antipode, then its category $\square _ { H } ^ { H } \mathcal{M}$ of crossed modules (also called Drinfel'd–Radford–Yetter modules) is braided. Here, objects are vector spaces $V$ which are both modules and co-modules under $H$, the two being compatible in the sense
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420140.png" /></td> </tr></table>
+
\begin{equation*} \sum h_{ ( 1 )} v ^ { \overline{( 1 )} } \bigotimes h_{ ( 2 )} \rhd v ^ { \overline{( 2 )} } = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420141.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420141.png"/></td> </tr></table>
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420142.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420143.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420144.png" /> denotes the action and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420145.png" /> denotes the co-product. The braiding is
+
for $h \in H$ and $v \in V$, where $\triangleright$ denotes the action and $\sum h _ { ( 1 ) } \otimes h _ { ( 2 ) }$ denotes the co-product. The braiding is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420146.png" /></td> </tr></table>
+
\begin{equation*} \Psi _ { V , W } ( v \bigotimes w ) = \sum v ^ { \overline{( 1 )} } \rhd w \bigotimes v ^ { \overline{( 2 )} }. \end{equation*}
  
For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420147.png" /> is any Lie algebra and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420148.png" /> is its enveloping algebra (cf. also [[Universal enveloping algebra|Universal enveloping algebra]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420149.png" /> is a crossed module by the adjoint action and the co-product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420150.png" />. The braiding induced on it is
+
For example, if $\frak g$ is any Lie algebra and $U ( \mathfrak { g } )$ is its enveloping algebra (cf. also [[Universal enveloping algebra|Universal enveloping algebra]]), then $V = k 1 \oplus \mathfrak g \subset U ( \mathfrak g )$ is a crossed module by the adjoint action and the co-product of $U ( \mathfrak { g } )$. The braiding induced on it is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420151.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420151.png"/></td> </tr></table>
  
6) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420152.png" /> be any monoidal category. Then its representation-theoretic dual category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420153.png" /> (also called the double <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420154.png" /> or centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420155.png" />) is a braided category. Objects are pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420156.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420157.png" /> is an object of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420158.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420159.png" /> is a collection of functorial isomorphisms representing the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420160.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420161.png" /> in the sense
+
6) Let $\mathcal{C}$ be any monoidal category. Then its representation-theoretic dual category $\mathcal{C} ^ { \circ }$ (also called the double $D ( \mathcal{C} )$ or centre $Z (\cal C )$) is a braided category. Objects are pairs $( V , \lambda )$ where $V$ is an object of $\mathcal{C}$ and $\lambda _ { W } : V \otimes W \rightarrow W \otimes V$ is a collection of functorial isomorphisms representing the $\otimes$ of $\mathcal{C}$ in the sense
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420162.png" /></td> </tr></table>
+
\begin{equation*} \lambda _ { \underline{1} } = \operatorname {id} , \lambda _ { W  \bigotimes Z} = \lambda_{Z} \circ \lambda _ { W } \end{equation*}
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420163.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420164.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420165.png" /> suppressed in this notation). The braiding is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420166.png" />.
+
($\Phi$, $l$, $r$ suppressed in this notation). The braiding is $\Psi _ { ( V , \lambda ) , ( W , \mu ) } = \lambda _ { W }$.
  
These constructions are in roughly increasing order of generality. Thus, the categorical dual or double construction applied to the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420167.png" />-modules yields the category of crossed modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420168.png" />. These, in turn, are an elementary reformulation (and thereby a slight generalization to infinite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420169.png" />) of the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420170.png" />-modules, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420171.png" /> is the quantum double quasi-triangular Hopf algebra associated to any finite-dimensional Hopf algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120420/b120420172.png" />. Meanwhile, a bicharacter on an Abelian group extends by linearity to a dual-quasi-triangular structure on the corresponding group algebra.
+
These constructions are in roughly increasing order of generality. Thus, the categorical dual or double construction applied to the category of $H$-modules yields the category of crossed modules $\square _ { H } ^ { H } \mathcal{M}$. These, in turn, are an elementary reformulation (and thereby a slight generalization to infinite-dimensional $H$) of the category of $D ( H )$-modules, where $D ( H )$ is the quantum double quasi-triangular Hopf algebra associated to any finite-dimensional Hopf algebra $H$. Meanwhile, a bicharacter on an Abelian group extends by linearity to a dual-quasi-triangular structure on the corresponding group algebra.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. MacLane,  "Categories for the working mathematician" , ''GTM'' , '''5''' , Springer  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Joyal,  R. Street,  "Braided monoidal categories"  ''Math. Reports Macquarie Univ.'' , '''86008'''  (1986)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Majid,  "Foundations of quantum group theory" , Cambridge Univ. Press  (1995)</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  S. MacLane,  "Categories for the working mathematician" , ''GTM'' , '''5''' , Springer  (1974)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  A. Joyal,  R. Street,  "Braided monoidal categories"  ''Math. Reports Macquarie Univ.'' , '''86008'''  (1986)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  S. Majid,  "Foundations of quantum group theory" , Cambridge Univ. Press  (1995)</td></tr></table>

Revision as of 17:02, 1 July 2020

braided monoidal category, quasi-tensor category

A generalization of the notion of tensor product of vector spaces, in which $\otimes$ is associative only up to isomorphism and the transposition isomorphism $\otimes \rightarrow \otimes ^ { \text{op} }$ need not square to the identity.

The full definition is: a category $\mathcal{C}$ of objects $V$, $W$, $Z$, etc., equipped with a functor $\otimes : \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C}$, and a collection of functorial isomorphisms

\begin{equation*} \Phi _ { V , W , Z } : ( V \bigotimes W ) \bigotimes Z \rightarrow V \bigotimes ( W \bigotimes Z ) \end{equation*}

(called the associator) between any three objects and

\begin{equation*} \Psi _ { V , W } : V \bigotimes W \rightarrow W \bigotimes V \end{equation*}

(called the braiding) between any two objects. "Functorial" means that these isomorphisms commute with any morphisms between objects. Thus,

\begin{equation*} ( \phi \bigotimes \text { id } ) \Psi _ { V , W } = \Psi _ { V , Z } ( \text { id } \bigotimes \phi ) , \forall \phi : W \rightarrow Z, \end{equation*}

if the morphism is applied before or after the braiding. Similarly for functoriality in the other arguments of $\Psi$ and in the arguments of $\Phi$. The precise definition is that $\Phi : ( \otimes ) \otimes \rightarrow \otimes ( \otimes )$ and $\Psi : \otimes \rightarrow \otimes ^ { \text{ op} }$ are natural equivalences, where $( \otimes ) \otimes :\mathcal C \times \mathcal C \times \mathcal C \rightarrow \mathcal C$ is the functor sending $( V , W , Z )$ to , etc., and $\otimes ^{\operatorname{op}} : \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C}$ is the functor sending $( V , W )$ to $W \otimes V$.

In addition, these functors are required to be coherent. The coherence for $\Phi$ is the pentagon identity

Figure: b120420a

It says that the two ways to reverse the bracketings as shown coincide. MacLane's coherence theorem says that then all other routes between two bracketed tensor products also coincide. In effect, this means that one may generalize constructions in linear algebra exactly as if $\otimes$ were strictly associative, dropping brackets. Afterwards one may add brackets, for example with brackets accumulating to the left, and then insert applications of $\Phi$ as needed for the desired compositions to make sense; all different ways to do this will yield the same net result.

One also requires a unit object $1$ and an associated collection of functorial isomorphisms $r _{V} : V \rightarrow V \otimes \underline { 1 }$ and $l _ { V } : V \rightarrow \underline { 1 } \otimes V$ obeying a triangle coherence identity

\begin{equation*} \operatorname { id} \bigotimes r _ { W } = \Phi _ { V , 1 , W } \circ ( l _ { V } \bigotimes \text { id } ). \end{equation*}

This structure $( \mathcal C , \otimes , \Phi , \underline { 1 } , l , r )$ is called a monoidal category (see also Triple).

The coherence conditions for the additional structure $\Psi$ of a braided category are the so-called hexagon identities

\begin{equation*} \Psi _ { V \bigotimes W , Z } = \Psi _ { V , Z } \circ \Psi _ { W , Z }, \end{equation*}

\begin{equation*} \Psi _ { V , W \bigotimes Z } = \Psi _ { V ,\, Z } \circ \Psi _ { V , W } . \end{equation*}

Here and below, the bracketings and $\Phi$ needed to make sense of these identities are omitted. (When they are inserted, each identity corresponds to a diagram with six arrows.) One can show that compatibility of $\Psi$ with $1$ is then automatic.

Notice that although $\Psi$ generalizes the concept of transposition of vector spaces, one does not demand that $\Psi _ { V , W } = \Psi _ { W , V } ^ { - 1 }$. If this does hold for all $V$, $W$, then a symmetric monoidal category or tensor category is obtained. In this case the two hexagon identities are equivalent and ensure that all ways to go

\begin{equation*} V _ { 1 } \bigotimes \ldots \bigotimes V _ { n } \rightarrow V _ { \sigma ( 1 ) } \bigotimes \ldots \bigotimes V _ { \sigma ( n ) } \end{equation*}

($\sigma$ a permutation) by composing $\Psi$ (and $\Phi$) yield the same result. In particular, there is an action of the symmetric group $S _ { n }$ in for any object $V$.

For a general braided category there is a similar result in terms of braids. To explain this, the following notational device is used: instead of writing the morphisms $\Psi _ { V , W }$ and $\Psi _ { W , V } ^ { - 1 }$ in the usual way as arrows, one writes them as braids and assumes that they point downwards:

Figure: b120420b

One writes another morphism $\phi : W \rightarrow Z$, say, as a node on a strand connecting $W$ to $Z$ and assumed to be pointing downwards. Functoriality says that morphisms can be pulled through braid crossings as

Figure: b120420c

Similarly for morphisms on the other strand. When the tensor product of several morphisms is applied, one writes their strands separately side by side, connecting the relevant objects in the tensor product. On the other hand, one is free to group some of the objects in the tensor product together as a single object and represent morphisms to and from it by a single strand. This notation is consistent precisely because of the hexagon conditions, which become

Figure: b120420d

The doubled strands on the left-hand side could be replaced by single strands for the composite morphisms.

The coherence theorem for braided categories then asserts that different routes between tensor product expressions by repeated applications of $\Phi$, $\Psi$ and their inverses compose to the same morphism if the corresponding braids are the same. In particular, for any object $V$ there is an action of the pure braid group $B _ { n }$ on . The former can be presented as

\begin{equation*} b _ { i } b _ { i + 1} b _ { i } = b _ { i + 1} b _ { i } b _ { i } + 1 , b _ { i } b _ { j } = b _ { j } b _ { i } , \quad | i - j | \geq 2, \end{equation*}

where $b _ { i }$ is represented by $\Psi$ acting in the $( i , i + 1 )$th copies of $V$. The representation usually has a kernel and $B _ { n }$ modulo this kernel is the Hecke algebra associated to an object in a braided category.

Finally, an object $V$ in a braided category is rigid if there are another object $V ^ { * }$ and morphisms $\operatorname{ev} _ { V } : V ^ { * } \otimes V \rightarrow \underline { 1 }$ and $\operatorname{coev}_V : \underline { 1 } \rightarrow V \otimes V ^ { * }$ such that

(suppressing $\Phi$, $l$, $r$). Here, $V ^ { * }$ is called a left dual of $V$ and is unique up to isomorphism (there is a similar notion of right dual). $\mathcal{C}$ is called rigid if every object has such duals. Using diagrammatic notation one writes

Figure: b120420e

and the above axioms become

Figure: b120420f

Note that in a rigid braided category, morphisms look like knots (cf. also Knot theory; Braid theory). Every knot presented on paper is in fact the closure of some braid, so with a little more structure (notably, both left and right duals) one can arrange that every oriented knot can be read from top to bottom as a morphism . Fixing an object $V$,

can be read as a left or right and

can be read as a left or right (in accordance with orientation). One can read braid crossings as $\Psi$ or $\Psi ^ { - 1 }$. The result is not quite a knot invariant but can usually be adjusted to become one.

Examples.

Some standard examples of braided categories are provided by the following constructions over a ground field $k$. Its invertible elements are denoted $k ^ { * }$.

1) Fix $q \in k$, an $n$th root of unity. The category of anyspaces $\operatorname{Vec}_n$ consists of $\mathbf{Z} / n \mathbf{Z}$-graded vector spaces $V = \oplus _ { i = 0 } ^ { n - 1 } V _ { i }$ (where the elements of $V _ { i }$ have degree $i$). Morphisms are degree-preserving linear mappings. One takes the associator trivial and the braiding

\begin{equation*} \Psi _ { V , W } ( v \bigotimes w ) = q ^ { |v| | w | } w \bigotimes v \end{equation*}

for $v \in V$ and $w \in W$ of degree $v$, $w $, respectively.

Clearly, $\operatorname{Vec}_{1}$ is the usual category of vector spaces, while $\text{Vec}_2$ is the category of linear superspaces or ${\bf Z} / 2 {\bf Z}$-graded vector spaces. The category of $\mathbf{Z}$-graded spaces is braided similarly for any $q \in k ^ { * }$.

A similar construction works for grading by any Abelian group $G$ equipped with a bicharacter $\beta : G \times G \rightarrow k ^ { * }$ (a function multiplicative in each argument) and braiding

\begin{equation*} \Psi _ { V , W } ( v \bigotimes w ) = \beta ( | v | , | w | ) w \bigotimes v \end{equation*}

on elements of degree $| v | , | w | \in G$.

2) If $G$ is a group and $\phi : G \times G \times G \rightarrow k ^ { * }$ a group $3$-cocycle, then the category of $G$-graded spaces is monoidal with associator

on elements of degree $| v | , | w | , | z | \in G$. If $G$ is Abelian and equipped with a quasi-bicharacter with respect to $\phi$, then the category is braided.

For example, the octonion algebra lives naturally in such a category with $G = \mathbf{Z} _ { 2 } \times \mathbf{Z} _ { 2 } \times \mathbf{Z} _ { 2 }$ and $\phi$ a certain coboundary.

3) The category of finite-dimensional representations of the quantum enveloping algebras (cf. also Quantum groups; Universal enveloping algebra) $U _ { q } ( \mathfrak { g } )$ associated to a semi-simple Lie algebra $\frak g$. Also, certain classes of infinite-dimensional representations. Associated to the standard representation of $U _ { q } ( \operatorname{sl} _ { 2 } )$ is the standard Hecke algebra defined by the additional relation $( b _ { i } - q ) ( b _ { i } + q ^ { - 1 } ) = 0$. The knot invariant associated to this same representation is the Jones knot polynomial.

In an algebraic formulation, if $H$, $\mathcal{R} \in H \otimes H$, is a quasi-triangular Hopf algebra, then its category $\square _ { H } \cal M$ of representations is braided with $\Psi = \tau \circ \mathcal{R}$, where one first acts with the quasi-triangular structure $\mathcal{R} \in H \otimes H$ and then applies the usual transposition mapping $\tau$.

4) The category of co-modules under the matrix quantum groups (cf. Quantum groups) $G_q$. For example, the quantum plane generated by $x , y$ modulo the relations $y x = q x y$ is covariant under $\operatorname{SL} _ { q } ( 2 )$ and hence lives as an algebra in its braided category of co-modules. Likewise, the category of co-modules under a quantum matrix bi-algebra $A ( R )$ associated to an invertible solution $R$ of the Yang–Baxter equation is braided.

In a general algebraic formulation, if $H$, $\mathcal{R} : H \otimes H \rightarrow k $, is a dual (or co-) quasi-triangular Hopf algebra, then its category of co-modules is braided by

where $\sum _ { V } v ^ { \overline{( 1 ) }} \otimes v ^ { \overline{( 2 ) } }$ denotes the result of the co-action $V \rightarrow H \otimes V$.

5) If $H$ is any Hopf algebra with invertible antipode, then its category $\square _ { H } ^ { H } \mathcal{M}$ of crossed modules (also called Drinfel'd–Radford–Yetter modules) is braided. Here, objects are vector spaces $V$ which are both modules and co-modules under $H$, the two being compatible in the sense

\begin{equation*} \sum h_{ ( 1 )} v ^ { \overline{( 1 )} } \bigotimes h_{ ( 2 )} \rhd v ^ { \overline{( 2 )} } = \end{equation*}

for $h \in H$ and $v \in V$, where $\triangleright$ denotes the action and $\sum h _ { ( 1 ) } \otimes h _ { ( 2 ) }$ denotes the co-product. The braiding is

\begin{equation*} \Psi _ { V , W } ( v \bigotimes w ) = \sum v ^ { \overline{( 1 )} } \rhd w \bigotimes v ^ { \overline{( 2 )} }. \end{equation*}

For example, if $\frak g$ is any Lie algebra and $U ( \mathfrak { g } )$ is its enveloping algebra (cf. also Universal enveloping algebra), then $V = k 1 \oplus \mathfrak g \subset U ( \mathfrak g )$ is a crossed module by the adjoint action and the co-product of $U ( \mathfrak { g } )$. The braiding induced on it is

6) Let $\mathcal{C}$ be any monoidal category. Then its representation-theoretic dual category $\mathcal{C} ^ { \circ }$ (also called the double $D ( \mathcal{C} )$ or centre $Z (\cal C )$) is a braided category. Objects are pairs $( V , \lambda )$ where $V$ is an object of $\mathcal{C}$ and $\lambda _ { W } : V \otimes W \rightarrow W \otimes V$ is a collection of functorial isomorphisms representing the $\otimes$ of $\mathcal{C}$ in the sense

\begin{equation*} \lambda _ { \underline{1} } = \operatorname {id} , \lambda _ { W \bigotimes Z} = \lambda_{Z} \circ \lambda _ { W } \end{equation*}

($\Phi$, $l$, $r$ suppressed in this notation). The braiding is $\Psi _ { ( V , \lambda ) , ( W , \mu ) } = \lambda _ { W }$.

These constructions are in roughly increasing order of generality. Thus, the categorical dual or double construction applied to the category of $H$-modules yields the category of crossed modules $\square _ { H } ^ { H } \mathcal{M}$. These, in turn, are an elementary reformulation (and thereby a slight generalization to infinite-dimensional $H$) of the category of $D ( H )$-modules, where $D ( H )$ is the quantum double quasi-triangular Hopf algebra associated to any finite-dimensional Hopf algebra $H$. Meanwhile, a bicharacter on an Abelian group extends by linearity to a dual-quasi-triangular structure on the corresponding group algebra.

References

[a1] S. MacLane, "Categories for the working mathematician" , GTM , 5 , Springer (1974)
[a2] A. Joyal, R. Street, "Braided monoidal categories" Math. Reports Macquarie Univ. , 86008 (1986)
[a3] S. Majid, "Foundations of quantum group theory" , Cambridge Univ. Press (1995)
How to Cite This Entry:
Braided category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Braided_category&oldid=50446
This article was adapted from an original article by S. Majid (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article