Timeline for What's the relation between half-twists, star structures and bar involutions on Hopf algebras?
Current License: CC BY-SA 4.0
16 events
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| Nov 13, 2018 at 20:19 | history | edited | LSpice | CC BY-SA 4.0 |
Added name of Snyder and Tingley paper
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Apr 27, 2016 at 22:19 | comment | added | Manuel Bärenz | @DavidHill, I revisited this question and I actually copied from Snyder and Tingley incorrectly. Conjugation by X actually sends $K$ to $K^{-1}$. I still don't understand well what the bar involution is. It's surprising now that in the *-structure for $U_qsu(1,1)$, $q$ should be real. | |
| Apr 27, 2016 at 22:16 | history | edited | Manuel Bärenz | CC BY-SA 3.0 |
added 829 characters in body; edited title
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| Sep 4, 2015 at 15:20 | comment | added | Manuel Bärenz | @DavidHill, I've clarified the situation a bit. | |
| Sep 4, 2015 at 15:19 | history | edited | Manuel Bärenz | CC BY-SA 3.0 |
Clarified real forms of U_qSL(2)_C
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| Sep 3, 2015 at 17:24 | comment | added | David Hill | Let us continue this discussion in chat. | |
| Sep 3, 2015 at 17:23 | comment | added | Manuel Bärenz | @DavidHill, I understand, that would mean that all complex numbers "in use" have to be generated by $q$. That might well be a solution. | |
| Sep 3, 2015 at 17:19 | comment | added | David Hill | 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. | |
| Sep 3, 2015 at 17:11 | comment | added | Manuel Bärenz | @DavidHill, I'm actually not sure. I guess rather that $q$ is a fixed root of unity (which will probably rule out one of the real forms from my above comment). How is it related to the bar involution? | |
| Sep 3, 2015 at 15:14 | comment | added | David Hill | 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? | |
| Sep 3, 2015 at 13:46 | comment | added | Manuel Bärenz | @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 $K^* = K^{-1}$. | |
| Sep 3, 2015 at 12:18 | comment | added | Manuel Bärenz | @DavidHill, can you give a reference to the bar involution? I'll have a look! | |
| Sep 3, 2015 at 3:46 | comment | added | David Hill | Are you sure both send $K\mapsto K$? My first thought was that they differ by the bar involution, but then one should be $K\mapsto K^{-1}$. | |
| Sep 2, 2015 at 19:09 | history | edited | Manuel Bärenz | CC BY-SA 3.0 |
added 43 characters in body
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| Sep 2, 2015 at 18:21 | history | asked | Manuel Bärenz | CC BY-SA 3.0 |