Quasi-triangular quasi-Hopf algebra

A quasi-triangular quasi-Hopf algebra is a specialized form of a <a href="/facts/Quasi-Hopf_algebra/yQetQh7d">quasi-Hopf algebra</a> defined by the <a href="/facts/Ukraine/Qof0uqAj">Ukrainian</a> mathematician <a href="/facts/Vladimir_Drinfeld/xGhJ8NAv">Vladimir Drinfeld</a> in 1989. It is also a generalized form of a <a href="/facts/Quasi-triangular_Hopf_algebra/Y6acUkBu">quasi-triangular Hopf algebra</a>.
A quasi-triangular quasi-Hopf algebra is a set 
 
 
 
 
 
 
 H
 
 A
 
 
 
 
 =
 (
 
 
 A
 
 
 ,
 R
 ,
 Δ
 ,
 ε
 ,
 Φ
 )
 
 
 {\displaystyle {\mathcal {H_{A}}}=({\mathcal {A}},R,\Delta ,\varepsilon ,\Phi )}
 
 where 
 
 
 
 
 
 
 B
 
 A
 
 
 
 
 =
 (
 
 
 A
 
 
 ,
 Δ
 ,
 ε
 ,
 Φ
 )
 
 
 {\displaystyle {\mathcal {B_{A}}}=({\mathcal {A}},\Delta ,\varepsilon ,\Phi )}
 
 is a <a href="/facts/Quasi-Hopf_algebra/yQetQh7d">quasi-Hopf algebra</a> and 
 
 
 
 R
 ∈
 
 
 A
 ⊗
 A
 
 
 
 
 {\displaystyle R\in {\mathcal {A\otimes A}}}
 
 known as the R-matrix, is an invertible element such that

R
        Δ
        (
        a
        )
        =
        σ
        ∘
        Δ
        (
        a
        )
        R
      
    
    {\displaystyle R\Delta (a)=\sigma \circ \Delta (a)R}

for all 
 
 
 
 a
 ∈
 
 
 A
 
 
 
 
 {\displaystyle a\in {\mathcal {A}}}
 
, where 
 
 
 
 σ
 :
 
 
 A
 ⊗
 A
 
 
 →
 
 
 A
 ⊗
 A
 
 
 
 
 {\displaystyle \sigma \colon {\mathcal {A\otimes A}}\rightarrow {\mathcal {A\otimes A}}}
 
 is the switch map given by 
 
 
 
 x
 ⊗
 y
 →
 y
 ⊗
 x
 
 
 {\displaystyle x\otimes y\rightarrow y\otimes x}
 
, and

(
        Δ
        ⊗
        id
        )
        R
        =
        
          Φ
          
            231
          
        
        
          R
          
            13
          
        
        
          Φ
          
            132
          
          
            −
            1
          
        
        
          R
          
            23
          
        
        
          Φ
          
            123
          
        
      
    
    {\displaystyle (\Delta \otimes \operatorname {id} )R=\Phi _{231}R_{13}\Phi _{132}^{-1}R_{23}\Phi _{123}}

(
        id
        ⊗
        Δ
        )
        R
        =
        
          Φ
          
            312
          
          
            −
            1
          
        
        
          R
          
            13
          
        
        
          Φ
          
            213
          
        
        
          R
          
            12
          
        
        
          Φ
          
            123
          
          
            −
            1
          
        
      
    
    {\displaystyle (\operatorname {id} \otimes \Delta )R=\Phi _{312}^{-1}R_{13}\Phi _{213}R_{12}\Phi _{123}^{-1}}

where 
 
 
 
 
 Φ
 
 a
 b
 c
 
 
 =
 
 x
 
 a
 
 
 ⊗
 
 x
 
 b
 
 
 ⊗
 
 x
 
 c
 
 
 
 
 {\displaystyle \Phi _{abc}=x_{a}\otimes x_{b}\otimes x_{c}}
 
 and 
 
 
 
 
 Φ
 
 123
 
 
 =
 Φ
 =
 
 x
 
 1
 
 
 ⊗
 
 x
 
 2
 
 
 ⊗
 
 x
 
 3
 
 
 ∈
 
 
 A
 ⊗
 A
 ⊗
 A
 
 
 
 
 {\displaystyle \Phi _{123}=\Phi =x_{1}\otimes x_{2}\otimes x_{3}\in {\mathcal {A\otimes A\otimes A}}}
 
.
The quasi-Hopf algebra becomes triangular if in addition, 
 
 
 
 
 R
 
 21
 
 
 
 R
 
 12
 
 
 =
 1
 
 
 {\displaystyle R_{21}R_{12}=1}
 
.
The twisting of 
 
 
 
 
 
 
 H
 
 A
 
 
 
 
 
 
 {\displaystyle {\mathcal {H_{A}}}}
 
 by 
 
 
 
 F
 ∈
 
 
 A
 ⊗
 A
 
 
 
 
 {\displaystyle F\in {\mathcal {A\otimes A}}}
 
 is the same as for a quasi-Hopf algebra, with the additional definition of the twisted R-matrix
A quasi-triangular (resp. triangular) quasi-Hopf algebra with 
 
 
 
 Φ
 =
 1
 
 
 {\displaystyle \Phi =1}
 
 is a <a href="/facts/Quasi-triangular_Hopf_algebra/Y6acUkBu">quasi-triangular (resp. triangular) Hopf algebra</a> as the latter two conditions in the definition reduce the conditions of quasi-triangularity of a Hopf algebra.
Similarly to the <a href="/facts/Quasi-bialgebra/5n7nU0EN">twisting</a> properties of the <a href="/facts/Quasi-Hopf_algebra/yQetQh7d">quasi-Hopf algebra</a>, the property of being quasi-triangular or triangular quasi-Hopf algebra is preserved by twisting.

Quasi-triangular quasi-Hopf algebra open-in-new

Quasi-triangular quasi-Hopf algebra