Ribbon Hopf algebra
A ribbon Hopf algebra ( A , ∇ , η , Δ , ε , S , R , ν ) {\displaystyle (A,\nabla ,\eta ,\Delta ,\varepsilon ,S,{\mathcal {R}},\nu )} is a quasitriangular Hopf algebra which possess an invertible central element ν {\displaystyle \nu } more commonly known as the ribbon element, such that the following conditions hold: ν 2 = u S ( u ) , S ( ν ) = ν , ε ( ν ) = 1 {\displaystyle \nu ^{2}=uS(u),\;S(\nu )=\nu ,\;\varepsilon (\nu )=1} Δ ( ν ) = ( R 21 R 12 ) − 1 ( ν ⊗ ν ) {\displaystyle \Delta (\nu )=({\mathcal {R}}_{21}{\mathcal {R}}_{12})^{-1}(\nu \otimes \nu )} where u = ∇ ( S ⊗ id ) ( R 21 ) {\displaystyle u=\nabla (S\otimes {\text{id}})({\mathcal {R}}_{21})} . Note that the element u exists for any quasitriangular Hopf algebra, and u S ( u ) {\displaystyle uS(u)} must always be central and satisfies S ( u S ( u ) ) = u S ( u ) , ε ( u S ( u ) ) = 1 , Δ ( u S ( u ) ) = ( R 21 R 12 ) − 2 ( u S ( u ) ⊗ u S ( u ) ) {\displaystyle S(uS(u))=uS(u),\varepsilon (uS(u))=1,\Delta (uS(u))=({\mathcal {R}}_{21}{\mathcal {R}}_{12})^{-2}(uS(u)\otimes uS(u))} , so that all that is required is that it have a central square root with the above properties.