Timeline for The Irreducible Corepresentations of the eight-dimensional Kac-Paljutkin Quantum Group
Current License: CC BY-SA 3.0
27 events
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| Mar 25, 2021 at 8:55 | comment | added | JP McCarthy | @user839372 you possibly need some kind of degeneracy assumption like $a^*a=0$ exactly when $a=0$. The assumption of it being C* does this. Example 3 from above shows the necessity of this I think. | |
| Mar 25, 2021 at 8:53 | comment | added | user167952 | @JP McCarthy Thanks! This completely solves my problem. | |
| Mar 25, 2021 at 8:51 | comment | added | JP McCarthy | @user839372 forgot to tag you. | |
| Mar 25, 2021 at 8:16 | comment | added | JP McCarthy | They all have Haar states: researchgate.net/publication/… | |
| Mar 25, 2021 at 8:08 | comment | added | user167952 | @JP McCarthy I like the algebraic compact quantum group approach. Any idea how you would show that these Hopf*-algebras have integrals? | |
| Mar 25, 2021 at 2:26 | comment | added | JP McCarthy | 3. I wonder can a full set of unitary irreducible corepresentation matrices form a block diagonal that is a fundamental representation for the finite quantum group so that it would be compact matrix and thus Woronowicz. I think that should work too | |
| Mar 25, 2021 at 2:25 | comment | added | JP McCarthy | My understanding is that all algebras of functions on finite quantum groups are Woronowicz. 1. It should be possible, perhaps using matrix coefficients of irreducible corepresentations, to mirror the proof that compact matrix quantum groups are Woronowicz to prove directly that they satisfy Woronowicz cancellation. 2. They are Hopf*algebras with an integral so algebras of regular functions on a compact quantum group. Trivial closure is Woronowicz. | |
| Mar 25, 2021 at 2:22 | comment | added | JP McCarthy | @user839372 Yes. T. Banica and J. Bichon, Quantum groups acting on 4 points, J. Reine Angew. Math. 626 (2009), 74-114 shows that it is a quantum subgroup of the compact matrix quantum group $S_4^+$ and thus Woronowicz. If you promise to see this jpmccarthymaths.com/2021/03/19/… is a very rough draft with issues, an explicit fundamental representation is on p. 21 | |
| Mar 24, 2021 at 23:22 | comment | added | user167952 | @JPMcCarthy Is this a quantum group in the sense of Woronowicz? Thus a compact C*-algebraic one? If so, how can one see this? | |
| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
replaced http://math.stackexchange.com/ with https://math.stackexchange.com/
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| Jun 25, 2015 at 12:53 | comment | added | JP McCarthy | I have since found an explicit answer to this question. The matrix elements of these five corepresentations are on p.147 of books.google.ie/…. | |
| Aug 20, 2014 at 13:53 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
i've fixed mistakes on the big formula.
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| Aug 20, 2014 at 13:36 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
I've changed the fontsize of the big formula for an easy global view
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| S Sep 12, 2013 at 21:52 | history | suggested | Sebastien Palcoux | CC BY-SA 3.0 |
I corrected $\Delta(e_3)$ and $\Delta(e_4)$
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| Sep 12, 2013 at 20:55 | review | Suggested edits | |||
| S Sep 12, 2013 at 21:52 | |||||
| Jul 17, 2013 at 11:47 | comment | added | JP McCarthy | @YemonChoi I must do the sum later on... if the algebra of functions on an abelian group is cocommutative and on a non-abelian group non-cocommutative, then it sounds like a good candidate. Interestingly I was thinking about one dimensional coreps as another candidate but I understand that there are infinite dimensional non-abelian groups with one dimensional representations only. | |
| Jul 16, 2013 at 16:13 | comment | added | Yemon Choi | If you want abelian quantum groups to generalize the notion of abelian group I suspect cocommutative is the right thing. But I am not an expert! (This is related to one of my pet peeves, namely that what people call quantum groups are really the co-ordinate rings of quantum groups, just as Cstar algebras are not really NC topological spaces, but the algebras of functions on NC top spaces.) | |
| Jul 16, 2013 at 12:21 | comment | added | JP McCarthy | @YemonChoi This is an issue that my supervisor has raised. As quantum groups, the algebra of functions on an (finite) abelian group is commutative and all of the irreps are one dimensional. However the algebra of functions on a non-abelian group is also commutative! My supervisor asks, what is an appropriate definition of an abelian quantum group? Are you suggesting that cocummutative is the right answer? And does cocummutative indeed imply that all of the coreps are one dimensional? | |
| S Jul 15, 2013 at 23:57 | history | bounty ended | JP McCarthy | ||
| S Jul 15, 2013 at 23:57 | history | notice removed | JP McCarthy | ||
| Jul 10, 2013 at 10:51 | vote | accept | JP McCarthy | ||
| S Jul 10, 2013 at 10:51 | history | bounty started | JP McCarthy | ||
| S Jul 10, 2013 at 10:51 | history | notice added | JP McCarthy | Authoritative reference needed | |
| Jun 30, 2013 at 16:11 | answer | added | UwF | timeline score: 7 | |
| Jun 29, 2013 at 21:40 | comment | added | Yemon Choi | (The point being that the KP example is neither commutative nor cocommutative) | |
| Jun 29, 2013 at 21:21 | comment | added | Yemon Choi | Hang on, if all your coreps are one dimensional, wouldn't this make your example cocommutative? | |
| Jun 29, 2013 at 13:41 | history | asked | JP McCarthy | CC BY-SA 3.0 |