$q$-invariance of quantum quaternion spheres

Abstract

The $C^*$-algebra of continuous functions on the quantum quaternion sphere $H_q^{2n}$ can be identified with the quotient algebra $C(SP_q(2n)/SP_q(2n-2))$. In commutative case i.e. for $q=1$, the topological space $SP(2n)/SP(2n-2)$ is homeomorphic to the odd dimensional sphere $S^{4n-1}$. In this paper, we prove the noncommutative analogue of this result. Using homogeneous $C^*$-extension theory, we prove that the $C^*$-algebra $C(H_q^{2n})$ is isomorphic to the $C^*$-algebra $C(S_q^{4n-1})$. This further implies that for different values of $q \in [0,1)$, the $C^*$-algebras underlying the noncommutative space $H_q^{2n}$ are isomorphic.