, Volume 263, Issue 1, pp 65-88

A Hopf Bundle Over a Quantum Four-Sphere from the Symplectic Group

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Abstract

We construct a quantum version of the SU(2) Hopf bundle S 7S 4. The quantum sphere S 7 q arises from the symplectic group Sp q (2) and a quantum 4-sphere S 4 q is obtained via a suitable self-adjoint idempotent p whose entries generate the algebra A(S 4 q ) of polynomial functions over it. This projection determines a deformation of an (anti-)instanton bundle over the classical sphere S 4. We compute the fundamental K-homology class of S 4 q and pair it with the class of p in the K-theory getting the value −1 for the topological charge. There is a right coaction of SU q (2) on S 7 q such that the algebra A(S 7 q ) is a non-trivial quantum principal bundle over A(S 4 q ) with structure quantum group A(SU q (2)).

Communicated by L. Takhtajan