Ribbon Hopf algebra

Template:Short description A ribbon Hopf algebra $(A,\mathrm{\nabla },\eta ,\mathrm{\Delta },\epsilon ,S,ℛ,\nu )$ is a quasitriangular Hopf algebra which possess an invertible central element $\nu$ more commonly known as the ribbon element, such that the following conditions hold:

${\nu }^2=uS(u),S(\nu )=\nu ,\epsilon (\nu )=1$

$\mathrm{\Delta }(\nu )=({ℛ}_{21}{ℛ}_{12}{)}^{-1}(\nu \otimes \nu )$

where $u=\mathrm{\nabla }(S\otimes \text{id})({ℛ}_{21})$. Note that the element u exists for any quasitriangular Hopf algebra, and $uS(u)$ must always be central and satisfies $S(uS(u))=uS(u),\epsilon (uS(u))=1,\mathrm{\Delta }(uS(u))=({ℛ}_{21}{ℛ}_{12}{)}^{-2}(uS(u)\otimes uS(u))$, so that all that is required is that it have a central square root with the above properties.

Here

$A$ is a vector space

$\mathrm{\nabla }$ is the multiplication map $\mathrm{\nabla }:A\otimes A\to A$

$\mathrm{\Delta }$ is the co-product map $\mathrm{\Delta }:A\to A\otimes A$

$\eta$ is the unit operator $\eta :ℂ\to A$

$\epsilon$ is the co-unit operator $\epsilon :A\to ℂ$

$S$ is the antipode $S:A\to A$

$ℛ$ is a universal R matrix

We assume that the underlying field $K$ is $ℂ$

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

See also

• Quasitriangular Hopf algebra
• Quasi-triangular quasi-Hopf algebra

References

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