Ribbon Hopf algebra

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

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

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

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

Here

${\displaystyle A}$ is a vector space

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

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

${\displaystyle \eta }$ is the unit operator ${\displaystyle \eta :\mathbb{ℂ}\to A}$

${\displaystyle \epsilon }$ is the co-unit operator ${\displaystyle \epsilon :A\to \mathbb{ℂ}}$

${\displaystyle S}$ is the antipode ${\displaystyle S:A\to A}$

${\displaystyle {ℛ}}$ is a universal R matrix

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

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

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

• Altschuler, D.; Coste, A. (1992). "Quasi-quantum groups, knots, three-manifolds and topological field theory". Commun. Math. Phys. 150 (1): 83–107. doi:10.1007/bf02096567. Bibcode: 1992CMaPh.150...83A. 
• Chari, V. C.; Pressley, A. (1994). A Guide to Quantum Groups. Cambridge University Press. ISBN 0-521-55884-0. https://archive.org/details/guidetoquantumgr0000char. 
• Drinfeld, Vladimir (1989). "Quasi-Hopf algebras". Leningrad Math J. 1: 1419–1457. 
• Majid, Shahn (1995). Foundations of Quantum Group Theory. Cambridge University Press. 

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