Instantons on the Quantum 4-Spheres S^4_q

Details Instantons on the Quantum 4-Spheres S^4_q Instantons on the Quantum 4-Spheres S^4_q

2000-12-13

arxiv.org/abs/math/0012103v2

Commun.Math.Phys. 221 (2001) 161-168

Mathematics

Quantum Algebra

Minor changes. To appear in CMP

Scientific paper

10.1007/PL00005572

We introduce noncommutative algebras $A_q$ of quantum 4-spheres $S^4_q$, with $q\in\IR$, defined via a suspension of the quantum group $SU_q(2)$, and a quantum instanton bundle described by a selfadjoint idempotent $e\in \Mat_4(A_q)$, $e^2=e=e^*$. Contrary to what happens for the classical case or for the noncommutative instanton constructed in Connes-Landi, the first Chern-Connes class $ch_1(e)$ does not vanish thus signaling a dimension drop. The second Chern-Connes class $ch_2(e)$ does not vanish as well and the couple $(ch_1(e), ch_2(e))$ defines a cycle in the $(b,B)$ bicomplex of cyclic homology.

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