Ribbon Hopf algebra explained

A ribbon Hopf algebra

(A,\nabla, η ,\Delta,\varepsilon,S, l{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, \varepsilon (\nu) =1 \Delta (\nu) = ( l{R} _{21} l{R} _{12} ) ^{-1} (\nu ⊗ \nu )

where

u = \nabla(S ⊗ id )( l{R} _{21} ) . Note that the element u exists for any quasitriangular Hopf algebra, anduS(u) must always be central and satisfies S(uS(u)) = uS(u), \varepsilon(uS(u)) =1 , \Delta(uS(u)) = ( l{R} _{21} l{R} _{12} ) ^{-2} (uS(u) ⊗ 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\nabla is the multiplication map \nabla:A ⊗ A → A \Delta is the co-product map \Delta: A → A ⊗ A η is the unit operator η : C → A \varepsilon is the co-unit operator \varepsilon: A → C S is the antipode S: A → A l{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

• Altschuler . D. . Coste . A. . Quasi-quantum groups, knots, three-manifolds and topological field theory . . 150 . 1992 . 1 . 83–107 . hep-th/9202047 . 10.1007/bf02096567. 1992CMaPh.150...83A .
• Book: Chari . V. C. . Pressley . A. . A Guide to Quantum Groups . registration . Cambridge University Press . 1994 . 0-521-55884-0 .
• Vladimir Drinfeld . Vladimir . Drinfeld . Quasi-Hopf algebras . Leningrad Math J. . 1 . 1989 . 1419–1457 .
• Book: Majid, Shahn . Foundations of Quantum Group Theory . Cambridge University Press . 1995 .