Author:
Landi Giovanni,Pagani Chiara
Abstract
AbstractWe study two classes of quantum spheres and hyperboloids, one class consisting of homogeneous spaces, which are $$*$$
∗
-quantum spaces for the quantum orthogonal group $$\mathcal {O}(SO_q(3))$$
O
(
S
O
q
(
3
)
)
. We construct line bundles over the quantum homogeneous space associated with the quantum subgroup SO(2) of $$SO_q(3)$$
S
O
q
(
3
)
. The line bundles are associated to the quantum principal bundle via representations of SO(2) and are described dually by finitely-generated projective modules $$\mathcal {E}_n$$
E
n
of rank 1 and of degree computed to be an even integer $$-2n$$
-
2
n
. The corresponding idempotents, that represent classes in the K-theory of the base space, are explicitly worked out and are paired with two suitable Fredhom modules that compute the rank and the degree of the bundles. For q real, we show how to diagonalise the action (on the base space algebra) of the Casimir operator of the Hopf algebra $${\mathcal {U}_{q^{1/2}}(sl_2)}$$
U
q
1
/
2
(
s
l
2
)
which is dual to $$\mathcal {O}(SO_q(3))$$
O
(
S
O
q
(
3
)
)
.
Publisher
Springer Science and Business Media LLC
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