Giovanni Landi, Chiara Pagani, Cesare Reina
A Hopf Bundle over a Quantum Four-Sphere from the Symplectic Group
Preprint series: ESI preprints

MSC:

58B30 Noncommutative differential geometry and topology, See also {46L30, 46L87, 46L89}

16W30 Coalgebras, bialgebras, Hopf algebras, See also {57T05,

19D55 $K$-theory and homology; cyclic homology and cohomology, See also {18G60}

Abstract: We construct a quantum version of the $SU(2)$ Hopf bundle
$S^7 \rightarrow S^4$. 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 of coinvariants is the algebra
$A(S^4_q)$. The algebra
$A(S^7_q)$ turns out to be a $A(SU_q(2))$ faithfully flat
Hopf-Galois extension
over $A(S^4_q)$, a notion which extends to noncommutative geometry the one of a
principal bundle in differential geometry; it is also not cleft, i.e. not trivial.
Keywords: Noncommutative geometry, Quantum groups, Quantum spheres, Instanton quantum bundles