Ribbon Hopf algebra

{{Short description|Algebraic structure}}

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

:$\backslash nu^\{2\}=uS(u),\; \backslash ;\; S(\backslash nu)=\backslash nu,\; \backslash ;\; \backslash varepsilon\; (\backslash nu)=1$

:$\backslash Delta\; (\backslash nu)=(\backslash mathcal\{R\}\_\{21\}\backslash mathcal\{R\}\_\{12\})^\{-1\}(\backslash nu\; \backslash otimes\; \backslash nu\; )$

where $u=\backslash nabla(S\backslash otimes\; \backslash text\{id\})(\backslash mathcal\{R\}\_\{21\})$. Note that the element u exists for any quasitriangular Hopf algebra, and

$uS(u)$ must always be central and satisfies $S(uS(u))=uS(u),\; \backslash varepsilon(uS(u))=1,\; \backslash Delta(uS(u))\; =$

(\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

:$A$ is a vector space

:$\backslash nabla$ is the multiplication map $\backslash nabla:A\; \backslash otimes\; A\; \backslash rightarrow\; A$

:$\backslash Delta$ is the co-product map $\backslash Delta:\; A\; \backslash rightarrow\; A\; \backslash otimes\; A$

:$\backslash eta$ is the unit operator $\backslash eta:\backslash mathbb\{C\}\; \backslash rightarrow\; A$

:$\backslash varepsilon$ is the co-unit operator $\backslash varepsilon:\; A\; \backslash rightarrow\; \backslash mathbb\{C\}$

:$S$ is the antipode $S:\; A\backslash rightarrow\; A$

:$\backslash mathcal\{R\}$ is a universal R matrix

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

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.