The half-twist forU_q(g) representations

Abstract

We introduce the notion of a half-ribbon Hopf algebra, which is a Hopf algebra H along with a distinguished element t ∈ H such that (H, R, C) is a ribbon Hopf algebra, where R = (t −1 ⊗ t −1 ) (t) and C = t −2 . The element t is closely related to the topological "half-twist", which twists a ribbon by 180 degrees. We construct a functor from a topological category of ribbons with half-twists to the category of representations of any half-ribbon Hopf algebra. We show that U q (g) is a (topological) half-ribbon Hopf algebra, but that t −2 is not the standard ribbon element. For U q (sl 2 ), we show that there is no half-ribbon element t such that t −2 is the standard ribbon element. We then discuss how ribbon elements can be modified, and some consequences of these modifications.

Z/2Z-extensions of Hopf algebra module categories by their base categories

“…R-matrices of this form occur in the theory of quantum groups 5 [21,23] and are studied in [29] as part of the definition of a 'half-ribbon Hopf algebra'.…”

An introduction to categorifying quantum knot invariants

“…/ q h ; iC2h ; i . This is the inverse of the ribbon element constructed by Snyder and Tingley in [40]; we must take inverse because Snyder and Tingley use the opposite choice of coproduct from ours. See Theorem 4.6 of that paper for a proof that this is a ribbon element.…”

On webs in quantum type C

“…To expand on the latter, the ribbon Hopf algebra structure produces a group-like element := −1 ∈ ( ) (here is an element satisfying 2 = ( )) and for Γ ∈ Rep( ( )) the pivotal isomorphism Γ Γ * * is given by ↦ → ( ↦ → ( )) for ∈ Γ and ∈ Γ * . This element is also relevant as it determines the 2021/02/15 10:23 [46] for complete details. To summarize, they show that there exists a "half-ribbon" element ∈ ( ) so that all ribbon elements are given by ( ) −2 .…”

“…We follow the convention, as in[46], that the ribbon element acts as the negative full twist.2021/02/15 10:23Downloaded from https://www.cambridge.org/core. 08 Mar 2021 at 03:52:43, subject to the Cambridge Core terms of use.…”