TY - JOUR
T1 - The Isospectral Dirac Operator on the 4-dimensional Orthogonal Quantum Sphere
JF - Comm. Math. Phys. 279 (2008) 77-116
Y1 - 2008
A1 - Francesco D'Andrea
A1 - Ludwik Dabrowski
A1 - Giovanni Landi
AB - Equivariance under the action of Uq(so(5)) is used to compute the left regular and (chiral) spinorial representations of the algebra of the quantum Euclidean 4-sphere S^4_q. These representations are the constituents of a spectral triple on this sphere with a Dirac operator which is isospectral to the canonical one of the spin structure of the round undeformed four-sphere and which gives metric dimension four for the noncommutative geometry. Non-triviality of the geometry is proved by pairing the associated Fredholm module with an `instanton\\\' projection. A real structure which satisfies all required properties modulo a suitable ideal of `infinitesimals\\\' is also introduced.
UR - http://hdl.handle.net/1963/2567
U1 - 1553
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Instanton algebras and quantum 4-spheres
JF - Differential Geom. Appl. 16 (2002) 277-284
Y1 - 2002
A1 - Ludwik Dabrowski
A1 - Giovanni Landi
AB - We study some generalized instanton algebras which are required to describe `instantonic complex rank 2 bundles\\\'. The spaces on which the bundles are defined are not prescribed from the beginning but rather are obtained from some natural requirements on the instantons. They turn out to be quantum 4-spheres $S^4_q$, with $q\\\\in\\\\IC$, and the instantons are described by self-adjoint idempotents e. We shall also clarify some issues related to the vanishing of the first Chern-Connes class $ch_1(e)$ and on the use of the second Chern-Connes class $ch_2(e)$ as a volume form.
PB - Elsevier
UR - http://hdl.handle.net/1963/3134
U1 - 1199
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Instantons on the Quantum 4-Spheres S^4_q
JF - Comm. Math. Phys. 221 (2001) 161-168
Y1 - 2001
A1 - Ludwik Dabrowski
A1 - Giovanni Landi
A1 - Tetsuya Masuda
AB - 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.
PB - Springer
UR - http://hdl.handle.net/1963/3135
U1 - 1198
U2 - Mathematics
U3 - Mathematical Physics
ER -