361 research outputs found
A 4-sphere with non central radius and its instanton sheaf
We build an SU(2)-Hopf bundle over a quantum toric four-sphere whose radius
is non central. The construction is carried out using local methods in terms of
sheaves of Hopf-Galois extensions. The associated instanton bundle is presented
and endowed with a connection with anti-selfdual curvature.Comment: minor changes, appendix section extended. To appear in Letters in
Mathematical Physics. 22 pages, no figure
A Hopf bundle over a quantum four-sphere from the symplectic group
We construct a quantum version of the SU(2) Hopf bundle . The
quantum sphere arises from the symplectic group and a quantum
4-sphere is obtained via a suitable self-adjoint idempotent whose
entries generate the algebra of polynomial functions over it. This
projection determines a deformation of an (anti-)instanton bundle over the
classical sphere . We compute the fundamental -homology class of
and pair it with the class of in the -theory getting the value
-1 for the topological charge. There is a right coaction of on
such that the algebra is a non trivial quantum principal
bundle over with structure quantum group .Comment: 27 pages. Latex. v2 several substantial changes and improvements; to
appear in CM
The quantum Cartan algebra associated to a bicovariant differential calculus
We associate to any (suitable) bicovariant differential calculus on a quantum
group a Cartan Hopf algebra which has a left, respectively right,
representation in terms of left, respectively right, Cartan calculus operators.
The example of the Hopf algebra associated to the differential calculus
on is described.Comment: 20 pages, no figures. Minor corrections in the example in Section 4
Noncommutative families of instantons
We construct -deformations of the classical groups SL(2,H) and Sp(2).
Coacting on the basic instanton on a noncommutative four-sphere ,
we construct a noncommutative family of instantons of charge 1. The family is
parametrized by the quantum quotient of by .Comment: v2: Minor changes; computation of the pairing at the end of Sect. 5.1
improve
Atiyah sequences of braided Lie algebras and their splittings
Associated with an equivariant noncommutative principal bundle we give an
Atiyah sequence of braided derivations whose splittings give connections on the
bundle. Vertical braided derivations act as infinitesimal gauge transformations
on connections. For the -principal bundle over the sphere
an equivariant splitting of the Atiyah sequence recovers the instanton
connection. An infinitesimal action of the braided conformal Lie algebra
yields a five parameter family of splittings. On the principal
-bundle of orthonormal frames over the sphere
, the splitting of the sequence leads to the Levi-Civita
connection for the `round' metric on the . The corresponding
Riemannian geometry of is worked out.Comment: 34 pages, no figure
Reduction of Quantum Principal Bundles over non affine bases
In this paper we develop the theory of reduction of quantum principal bundles
over projective bases. We show how the sheaf theoretic approach can be
effectively applied to certain relevant examples as the Klein model for the
projective spaces; in particular we study in the algebraic setting the
reduction of the principal bundle to the Levi subgroup inside the maximal
parabolic subgroup of . We characterize reductions in the
sheaf theoretic setting.Comment: 25 page
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