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Suppose $H$ is a quasi-Hopf algebra with non-trivial evaluation $\alpha$. I cannot find any sources about grouplike elements in $H$. What I mean by "grouplike" is $$ \Delta(g) = g\otimes g \quad\text{and}\quad\epsilon(g)=1 .$$ Obviously in a Hopf algebra this would imply $S(g)=g^{-1}$, but in quasi-Hopf we only get $$ \alpha = S(g)\alpha g.$$

So, can we use some trick to deduce an answer to the question in the title, or is it in general only true for quasi-Hopf algebras with $\alpha =1$ (or invertible, ofc)?

(I tried cooking up a counterexample, but my qHA-Fu is still weak, and most examples are still terribly complicated to me)

VividD
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Jo Mo
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1 Answers1

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Too long for a comment:

Given a ordinary Hopf algebra $H$ the grouplike elements of $H$ correspond to the one-dimensional comodules of $H$. As the category of $H$-comodules is a monoidal category, the tensor product induces a group structure one the one-dimensional comodules. This group is isomorphic to $G(H)$.

A quasi-Hopf algebra $H$ is in general not an coassociative coalgebra and as such, comodules are not defined. From this perspective you don't expect grouplike elements (i.e. $\Delta(g)=g\otimes g$) to hold any important significance. There are some reasonable analogous of comodules over quasi-Hopf algebras in the literature, perhaps this is a good place to start.

Mathematician 42
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