Associative rings and algebras Definition

Ribbon Hopf algebra

Builds on Quasitriangular Hopf algebra Hopf algebra

Definition

Let $(H, R)$ be a quasitriangular Hopf algebra over a field $k$ , with coproduct $\Delta$ , counit $\varepsilon$ and antipode $S$ . Write $R = \sum_i a_i \otimes b_i$ , let $R_{21} = \sum_i b_i \otimes a_i$ , and let $u = \sum_i S(b_i)\, a_i$ be the Drinfeld element. A ribbon element of $H$ is a central invertible element $v \in H$ such that:

$v^2 = u\, S(u)$ .
$S(v) = v$ .
$\varepsilon(v) = 1$ .
$\Delta(v) = (R_{21} R)^{-1} (v \otimes v)$ .

A ribbon Hopf algebra is a quasitriangular Hopf algebra together with a ribbon element.

Remarks

Example: let $H = k[G]$ be the group algebra of a group $G$ with $R = 1 \otimes 1$ . Then $u = S(1) \cdot 1 = 1$ , and $v = 1$ is central and invertible. Clauses 1 to 3 read $1 = 1$ , and clause 4 reads $1 \otimes 1 = (1 \otimes 1)^{-1}(1 \otimes 1)$ . Non-example: in the same algebra with $\operatorname{char} k \ne 2$ , the element $v = -1$ satisfies clauses 1 and 2 but fails clause 3, since $\varepsilon(-1) = -1$ . The Drinfeld element $u$ is invertible and satisfies $S^2(x) = u x u^{-1}$ for all $x \in H$ . Some authors use $v^{-1}$ in place of $v$ , which changes clause 4 to $\Delta(v) = (v \otimes v) R_{21} R$ .

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Tag	`5SRDQB4NG7`
URI	`m2://5SRDQB4NG7`
Kind	definition
Writing	W0 stub
MSC	16T05, 18M15, 57K16
Defines	Ribbon Hopf algebra
Uses	Quasitriangular Hopf algebra, Hopf algebra
Changed	2 November 2026

Ribbon Hopf algebra

Definition

Remarks

Reading path

Further reading

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