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A star structure on a Hopf algebra is an antilinear antiautomorphism squaring to 1 and satisfying some further axioms. A Hopf algebra with a star structure is then a star algebra and a Hopf algebra in a compatible way.

Recently, there's been research into half-twists on Hopf algebra. See for example the question 180˚ vs 360˚ Twists in String Diagrams for Ribbon Categories. Half-twists, or half-ribbon elements are implemented by an element $X$ such that $X^{-2}$ is a twist, or ribbon element and $(X^{-1} \otimes X^{-1}) \Delta X$ is the $R$-matrix.

Looking at the example of $U_qsl(2)_\mathbb{C}$ as the complexification of $U_qsu(1,1)$ with generators $K,E,F$ and real $q$, there is a star structure that satisfies $\Delta \circ * = (* \otimes *) \circ \Delta$. It has $K^* = K$, $E^* = -F$ and $F^* = -E$ (see e.g. Majid's book, pages 90 and 91, with different conventions). But also if we define $C_X(h) := X^{-1}hX$, then $C_X(K) = K^{-1}$, $C_X(E) = -F$ and $C_X(F) = -E$ according to Snyder and Tingley - The half-twist for $U_q(\mathfrak g)$ representations.

Surely this is no coincidence? But yet I can't find a mention of star structures in their article. It can't be that $C_X$ is exactly the star, since the star is antilinear and $C_X$ is linear, so what is going on here?

David Hill suggests in the comments that this is related to the bar involution (to which I can't find the original references since I don't have a copy of Lusztig's book), which seems to be an antilinear map generated by $\overline{E} = E$, $\overline{F} = F$ and $\overline{K} = K^{-1}$.

What is the precise relation between star structures, half-twists and the bar involution?

It seems like in this example, one can compose two to get the third, but is this a general phenomenon for all *-Hopf algebras or at least those of the form $U_q\mathfrak{g}$? And what happens if there are several star structures? Why is one of them preferred, and why is it the one where $q$ is real, so $\overline{q} = q$? Or are there several bar involutions as well?

For example, there are two other star structures on $U_qsl(2)_\mathbb{C}$:

One arises as the complexification of $U_qsl(2,\mathbb{R})$ with complex, root of unity $q$. It has $K^* = K^{-1}$ and $E^* = -E$.

A further one arises from $U_qsu(2)$, with $K^* = K$ and $E^* = F$. If $q$ is not real, but a root of unity, the star is a co-antihomomorphism: $\Delta \circ * = (* \otimes *) \circ \Delta^{\text{op}}$

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  • $\begingroup$ Are you sure both send $K\mapsto K$? My first thought was that they differ by the bar involution, but then one should be $K\mapsto K^{-1}$. $\endgroup$
    – David Hill
    Sep 3, 2015 at 3:46
  • $\begingroup$ @DavidHill, can you give a reference to the bar involution? I'll have a look! $\endgroup$
    – Manuel Bärenz
    Sep 3, 2015 at 12:18
  • $\begingroup$ @DavidHill, it seems like there are different star structures, each one corresponding to a different real form ($U_qSU(2)$, $U_qSL(1,1)$ and $U_qSL(2,\mathbb{R})$) and I believe there is one star structure where $K^* = K^{-1}$. $\endgroup$
    – Manuel Bärenz
    Sep 3, 2015 at 13:46
  • $\begingroup$ Maybe I am misunderstanding what you mean by $U_qSL(2)_{\mathbb{C}}$. Is this the quantum group over $\mathbb{C}(q)$, or $U_qSL(2)\otimes_{\mathbb{Q}(q)}\mathbb{C}$ where $\mathbb{C}$ is regarded as a $\mathbb{Q}(q)$ module with $q$ acting as a scalar? $\endgroup$
    – David Hill
    Sep 3, 2015 at 15:14
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    $\begingroup$ The bar involution is an algebra automorphism defined by $\overline{E}=E$, $\overline{F}=F$ and $\overline{q}=q^{-1}$. The relation $[E,F]=(K^2-K^{-2})/(q-q^{-1})$ implies that $\overline{K}=K^{-1}$ since $\overline{[E,F]}=[E,F]$. I guess if ``the star is antilinear'' means take complex conjugates of coefficients, then, for $q$ a root of unity, this would be compatible. $\endgroup$
    – David Hill
    Sep 3, 2015 at 17:19

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