Journal of Lie Theory

Isomorphisms of Quantum Spheres
D’Andrea, Francesco1, 2
1Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli Federico II, via Ciantia, 80126 Napoli, Italy
2INFN – Sezione di Napoli, via Cintia, 80126 Napoli, Italy
Journal of Lie Theory, Online first, pp. 1-10

For $n\in \mathbb{N}$ and $q\in [0,1\mathclose [$, the Vaksman–Soibelman quantum sphere $S^{2n+1}_q$ is described by an associative *-algebra $\mathcal{A}(S^{2n+1}_q)$ deforming the algebra of polynomial functions on the $2n+1$ dimensional unit sphere. Its C*-enveloping algebra is known to be independent of the deformation parameter $q$. In contrast to what happens in the C*-algebraic setting, we show here that, for all $q,q^{\prime }$ in the above range, $\mathcal{A}(S^{2n+1}_q)$ is isomorphic to $\mathcal{A}(S^{2n+1}_{q^{\prime }})$ if and only if $q=q^{\prime }$. From a geometric point of view, this means that quantum spheres corresponding to different values of the deformation parameter are isomorphic as noncommutative topological spaces but not as noncommutative algebraic varieties.

Received:
Accepted:
DOI: 10.5802/jolt.1432
Classification: 16T20, 20G42
Keywords: Compact quantum groups, quantum homogeneous spaces, $q$-deformations, Vaksman–Soibelman quantum spheres

D’Andrea, Francesco  1 , 2

1 Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli Federico II, via Ciantia, 80126 Napoli, Italy
2 INFN – Sezione di Napoli, via Cintia, 80126 Napoli, Italy 
D’Andrea, Francesco. Isomorphisms of Quantum Spheres. Journal of Lie Theory, Online first, pp. 1-10
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