Timeline for A representation similar to coadjoint representation?
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| Feb 18, 2016 at 8:24 | comment | added | Alexander Chervov | @whitejet it would be enough prove existence or NON-existence of Hilbert space representation for non-integer "R" for the algebra in question. Unfortunately it seems it does NOT exist :( Berezin-Toeplitz does not give any new information - because for integer "R" it is "obvious" that there is finite-dim irrep, just basic rep-theory for su(2). | |
| Feb 17, 2016 at 0:06 | comment | added | whitejet | @AlexanderChervov Just viewed your question, is deriving the star product with Berezin symbol good enough for defining $C^*$-algebra to you? A short computation can be found in sec. 2.2 arxiv:1310.8345. | |
| Feb 16, 2016 at 23:47 | comment | added | whitejet | @SebastianGoette I agree with you for$so(3)$, its coadjoint representation is equivalent to its adjoint representation, which is equivalent to the standard representation. What confuses me is when $d>3$, due to unmatched number of generators, $g^*$ is not dual to $g$, which includes all generators of $so(d)$. | |
| Feb 16, 2016 at 23:32 | comment | added | whitejet | @AlexanderChervov To a physicist, fuzzy sphere means a noncommutative sphere embeded in $R^3$, where the noncommutative coordinate $(X,Y,Z)$ corresponds to the matrix representation of three generators of $so(3)$. This is the result of a sphere in deformation quantization, which is probably familiar to you. The matrix product agrees with star product in the large N limit. | |
| Feb 16, 2016 at 21:15 | comment | added | Sebastian Goette | Ok, I realise that the (co) adjoint representation of $so(3)$ is isomorphic to the standard representation (cross product is the keyword). Anyway, $g^*$ is the standard representation. In the formula $xg^*x^{-1}$, the left $x$ acts on the column piece of $g^*=\left(\begin{matrix}&&&*\\&&&:\\&&&*\\*&\cdot\cdot&*\end{matrix}\right)$, and the right $x^{-1}=x^t$ acts on the row piece. Regarding references - most books on compact Lie groups should say something on it. | |
| Feb 16, 2016 at 20:42 | comment | added | user21574 | mathoverflow.net/questions/133458/… | |
| Feb 16, 2016 at 20:25 | history | edited | whitejet | CC BY-SA 3.0 |
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| Feb 16, 2016 at 7:31 | comment | added | Alexander Chervov | What is " This is the well-known fuzzy sphere" ? See question: mathoverflow.net/questions/231072/… | |
| Feb 16, 2016 at 5:00 | review | First posts | |||
| Feb 16, 2016 at 6:31 | |||||
| Feb 16, 2016 at 4:58 | history | asked | whitejet | CC BY-SA 3.0 |