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Ribbon Hopf algebra

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Algebraic structure

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.

Here

A {\displaystyle A} is a vector space
∇ {\displaystyle \nabla } is the multiplication map ∇ : A ⊗ A → A {\displaystyle \nabla :A\otimes A\rightarrow A}
Δ {\displaystyle \Delta } is the co-product map Δ : A → A ⊗ A {\displaystyle \Delta :A\rightarrow A\otimes A}
η {\displaystyle \eta } is the unit operator η : C → A {\displaystyle \eta :\mathbb {C} \rightarrow A}
ε {\displaystyle \varepsilon } is the co-unit operator ε : A → C {\displaystyle \varepsilon :A\rightarrow \mathbb {C} }
S {\displaystyle S} is the antipode S : A → A {\displaystyle S:A\rightarrow A}
R {\displaystyle {\mathcal {R}}} is a universal R matrix

We assume that the underlying field K {\displaystyle K} is C {\displaystyle \mathbb {C} }

If A {\displaystyle A} is finite-dimensional, one could equivalently call it ribbon Hopf if and only if its category of (say, left) modules is ribbon; if A {\displaystyle A} is finite-dimensional and quasi-triangular, then it is ribbon if and only if its category of (say, left) modules is pivotal.

See also

References

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