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 $S^7 \to S^4$. 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 $A(S^7_q)$ is a non trivial quantum principal bundle over $A(S^4_q)$ with structure quantum group $A(SU_q(2))$.Comment: 27 pages. Latex. v2 several substantial changes and improvements; to appear in CM

Quantum Principal Bundles and Instantons

See theses, p.

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 $4D_+$ differential calculus on $SU_q(2)$ is described.Comment: 20 pages, no figures. Minor corrections in the example in Section 4

Noncommutative families of instantons

We construct $\theta$-deformations of the classical groups SL(2,H) and Sp(2). Coacting on the basic instanton on a noncommutative four-sphere $S^4_\theta$, we construct a noncommutative family of instantons of charge 1. The family is parametrized by the quantum quotient of $SL_\theta(2,H)$ by $Sp_\theta(2)$.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 $SU(2)$-principal bundle over the sphere $S^{4}_\theta$ an equivariant splitting of the Atiyah sequence recovers the instanton connection. An infinitesimal action of the braided conformal Lie algebra $so_\theta(5,1)$ yields a five parameter family of splittings. On the principal $SO_\theta(2n,\mathbb{R})$-bundle of orthonormal frames over the sphere $S^{2n}_\theta$, the splitting of the sequence leads to the Levi-Civita connection for the `round' metric on the $S^{2n}_\theta$. The corresponding Riemannian geometry of $S^{2n}_\theta$ 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 $\mathrm{GL}(n) \to \mathrm{GL}(n)/P= \mathbf{P}^{n-1}(\mathbb{C})$ to the Levi subgroup $G_0$ inside the maximal parabolic subgroup $P$ of $\mathrm{GL}(n)$. We characterize reductions in the sheaf theoretic setting.Comment: 25 page