Warped Products and Yang-Mills equations on non commutative spaces

This paper presents a non self-dual solution of the Yang-Mills equations on a non commutative version of the classical $R^4_q\backslash\{0\}$, so generalizing the classical meron solution first introduced by de Alfaro, Fubini and Furlan in 1976. The basic tool for that is a generalization to non commutative spaces of the classical notion of warped products between metric spaces.Comment: 18 page

Examples of Hodge Laplacians on quantum spheres

Using a non canonical braiding over the 3d left covariant calculus we present a family of Hodge operators on the quantum SU(2) and its homogeneous quantum two-sphere.Comment: 7 pages, evolving the subject of a talk at the conference "FuninGeo" 2011, Ischia (Italy

Calculi, Hodge operators and Laplacians on a quantum Hopf fibration

We describe Laplacian operators on the quantum group SUq (2) equipped with the four dimensional bicovariant differential calculus of Woronowicz as well as on the quantum homogeneous space S2q with the restricted left covariant three dimensional differential calculus. This is done by giving a family of Hodge dualities on both the exterior algebras of SUq (2) and S2q . We also study gauged Laplacian operators acting on sections of line bundles over the quantum sphere.Comment: v3, one reference corrected, one reference added. 31 page

Derivation based differential calculi for noncommutative algebras deforming a class of three dimensional spaces

We equip a family of algebras whose noncommutativity is of Lie type with a derivation based differential calculus obtained, upon suitably using both inner and outer derivations, as a reduction of a redundant calculus over the Moyal four dimensional space.Comment: 18 page

Gauged Laplacians on quantum Hopf bundles

We study gauged Laplacian operators on line bundles on a quantum 2-dimensional sphere. Symmetry under the (co)-action of a quantum group allows for their complete diagonalization. These operators describe `excitations moving on the quantum sphere' in the field of a magnetic monopole. The energies are not invariant under the exchange monopole/antimonopole, that is under inverting the direction of the magnetic field. There are potential applications to models of quantum Hall effect.Comment: v2: latex; 32 pages. Papers re-organized; no major changes, several minor ones. Commun. Math. Phys. In pres

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

Linear Algebra and Analytic Geometry for Physical Sciences

This book originates from a collection of lecture notes that the first author prepared at the University of Trieste with Michela Brundu, over a span of fifteen years, together with the more recent one written by the second author. The notes were meant for undergraduate classes on linear algebra, geometry and more generally basic mathematical physics delivered to physics and engineering students, as well as mathematics students in Italy, Germany and Luxembourg. The book is mainly intended to be a self-contained introduction to the theory of finite-dimensional vector spaces and linear transformations (matrices) with their spectral analysis both on Euclidean and Hermitian spaces, to affine Euclidean geometry as well as to quadratic forms and conic sections. Many topics are introduced and motivated by examples, mostly from physics. They show how a definition is natural and how the main theorems and results are first of all plausible before a proof is given. Following this approach, the book presents a number of examples and exercises, which are meant as a central part in the development of the theory. They are all completely solved and intended both to guide the student to appreciate the relevant formal structures and to give in several cases a proof and a discussion, within a geometric formalism, of results from physics, notably from mechanics (including celestial) and electromagnetism