As is well-known, the $(2N-1)$-quantum sphere $S^{2N-1}_q$ is defined to be the invariant subalgebra of $SU_q(N)$ under the coaction $\Delta_R = (id \otimes \pi) \circ \Delta$, where $\Delta$ is the comultiplication of $SU_q(N)$, and $\pi: SU_q(N) \to U_q(N-1)$ is the Hopf algebra surjection defined by setting, for $i,j \neq 1$, $\pi(u^i_1)=\pi(u^1_j)=0$, $\pi(u^1_1)$ = det$_q^{-1}$, and $\pi (u^i_j) = u^{i-1}_{j-1}$. (Recall that the invariant subalgebra of a $H$-coaction $\Delta_R$ on vector space $V$ is the subspace of all elements $v$ for which $\Delta_R(v) = v \otimes 1$.) An oft quoted result is that $S^{2N-1}_q$ is generated, as an algebra, by the elements $u^i, S(u^1_i)$, for $i=1, \ldots N$. Now it is clear that these elements are invariant, but it is far from clear (at least to me) that generate all the invariant subspace. Can anyone see why? The usual references given are in Russian and, even at that, are unavailable on the web.

  • $\begingroup$ Does anyone know where this result was first proved? $\endgroup$ Mar 7, 2011 at 17:35

1 Answer 1


This is shown in the book Quantum Groups and Their Representations, by Klimyk and Schmudgen. The result you ask for is Proposition 63 in Chapter 11. I'd expand more upon this but I have to give a talk shortly, and anyway it's all there in the book.

Edit: I guess I should say that the point here is the representation theory. Classically, we can think of the algebra of functions on the sphere as the representation of $SU(N)$ induced by the trivial representation of the subgroup $U(N-1)$. Then Frobenius Reciprocity tells you which representations of $SU(N)$ occur as summands in the function algebra, and hence which matrix coefficients generate it. These things all transfer over to the quantum setting because they are phrased in terms of representations.


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