In a project of quantization, I come up with a finite dimensional representation of $so(d)$ that I wish to find some decent references for it. I guess it could have been studied thoroughly in representation theory field with a proper name addressed. I understand for classic Lie algebra, we can certainly compute the character for such representation and address it with the Dynkin coefficients, but I'm more looking for references of related study that orbits as symbolic manifold which can be quantized with such representation.

My interested representation of $so(d)$ is as following. Consider the generators of $so(d+1)$ denoted $M_{\mu \nu}$ for $\mu,\nu=1 \dots d+1$, and these generators are divided into two sets, $g=\{M_{i,j}| i,j=1\dots d\}$ and $g^*=\{M_{i,d+1}| i=1\dots d\}$, the representation $$\pi:(g^*,g)\rightarrow g^*$$ acts by conjugation $$\pi(x)g^*=xg^*x^{-1}\ \mathrm{for}\ x\in g.$$

When $d=3$, this is exactly the coadjoint representation of $so(3)$. According to Kirillov's orbit method, the coadjoint orbit assembles a sympletic manifold as a sphere, and an unitary irreducible representation corresponding to the orbit can be found as a proper quantization of the sphere, which is just an unirrep of $su(2)$ in this case. This is the well-known fuzzy sphere. However, you see immediately $g^*$ doesn't have enough generators as in $g$ when $d>3$, so it can't be coadjoint to $so(d)$. I'm wondering if there's a proper name for the representation of $so(d)$ from $so(d+1)$ as described, and any decent references related? I'm also interesting to know if the orbit method applies here for higher dimensional sphere $S^{d-1}$ with orbits on $g^*$.

Please forgive my phrasing and let me know if the question isn't clear to you. I'm not a mathematician.

PS for $d=3$, we have $<g,g^*>=Tr(M_{ij}\cdot M_{k4})=\epsilon_{ijk}\frac{1}{3}(2j_1+1)(2j_2+1)(j_1+j_2+1)(j_2-j_1)$ in the irrep of $so(4)$ with Dynkin coefficients $(2j_1,2j_2)$, or equivalently $so(3)\otimes so(3)$ in spin $j_1$ and spin $j_2$ representations correspondingly. Hence, $g^*$ is dual to $g$, and the conjugation action on $g^*$ is the coadjoint action.

The coadjoint orbit and the corresponding unirrep can be found in Kirillov's book, Ch.3 sec.1 Lectures on the Orbit Method. (Sorry, I can't find an online resource for it.)

  • $\begingroup$ What is " This is the well-known fuzzy sphere" ? See question: mathoverflow.net/questions/231072/… $\endgroup$ – Alexander Chervov Feb 16 '16 at 7:31
  • $\begingroup$ mathoverflow.net/questions/133458/… $\endgroup$ – user21574 Feb 16 '16 at 20:42
  • $\begingroup$ 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. $\endgroup$ – Sebastian Goette Feb 16 '16 at 21:15
  • $\begingroup$ @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. $\endgroup$ – whitejet Feb 16 '16 at 23:32
  • $\begingroup$ @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 $so(d)$. $\endgroup$ – whitejet Feb 16 '16 at 23:47

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