Related papers: Quantum groups, differential calculi and the eigen…

The problem of formulating a correct notion of Laplacian on compact quantum groups (CQGs) has long been recognized as both fundamental and nontrivial. Existing constructions typically rely on selecting a specific first-order differential…

We study $N^2-1$ dimensional left-covariant differential calculi on the quantum group $SL_q(N)$ for which the generators of the quantum Lie algebras annihilate the quantum trace. In this way we obtain one distinguished calculus on $SL_q(2)$…

Let $M^n$ be a compact orientable smooth Riemannian submanifold of dimension $n\geq 3$ in $\mathbb R^d$. We construct a family of deformed Hodge Laplacians $\Delta_t^*$, $t>0$, acting on differential forms and defined through the extrinsic…

We consider the fractional powers of singular (point-like) perturbations of the Laplacian, and the singular perturbations of fractional powers of the Laplacian, and we compare such two constructions focusing on their perturbative structure…

The Novikov-Shubin invariants for a non-compact Riemannian manifold M can be defined in terms of the large time decay of the heat operator of the Laplacian on square integrable p-forms on M. For the (2n+1)-dimensional Heisenberg group H,…

The main aim of this paper is to introduce some examples of non-compact locally compact quantum groups to a non-specialized audience. The major importance of these examples is their simplicity. Other examples as the quantum E(2) group of…

A $(p,q)$-double form on a Riemannian manifold $(M,g)$ can be considered simultaneously as a vector-valued differential $p$-form over $M$ or alternatively as a vector-valued $q$-form. Accordingly, the usual Hodge-de Rham Laplacian on…

We give a pedagogical introduction to the differential calculus on quantum groups by stressing at all stages the connection with the classical case ($q \rightarrow 1$ limit). The Lie derivative and the contraction operator on forms and…

A review of recent developments in the quantum differential calculus. The quantum group $GL_q(n)$ is treated by considering it as a particular quantum space. Functions on $SL_q(n)$ are defined as a subclass of functions on $GL_q(n)$. The…

We investigate inhomogeneous quantum groups G built from a quantum group H and translations. The corresponding commutation relations contain inhomogeneous terms. Under certain conditions (which are satisfied in our study of quantum Poincare…

In this memoir we extend the theory of global pseudo-differential operators to the setting of arbitrary sub-Riemannian structures on a compact Lie group. More precisely, given a compact Lie group $G$, and the sub-Laplacian $\mathcal{L}$…

We consider several differential operators on compact almost-complex, almost-Hermitian and almost-K\"ahler manifolds. We discuss Hodge Theory for these operators and a possible cohomological interpretation. We compare the associated spaces…

Explicit construction of the second order left differential calculi on the quantum group and its subgroups are obtained with the property of the natural reduction: the differential calculus on the quantum group $GL_q(2,C)$ has to contain…

The topic of this thesis is the development of a versatile and geometrically motivated differential calculus on non-commutative or quantum spaces, providing powerful but easy-to-use mathematical tools for applications in physics and related…

We establish a quantum Galois correspondence for compact Lie groups of automorphisms acting on a simple vertex operator algebra.

We examine the problem of defining Lagrangian and Hamiltonian mechanics for a particle moving on a quantum plane $Q_{q,p}$. For Lagrangian mechanics, we first define a tangent quantum plane $TQ_{q,p}$ spanned by noncommuting particle…

A three-dimensional $q$-Lie algebra of $SU_q(2)$ is realized in terms of first- and second-order differential operators. Starting from the $q$-Lie algebra one has constructed a left-covariant differential calculus on the quantum group. The…

We give some sharp lower bounds of the first eigenvalue for the Hodge Laplacian acting on differential forms on the boundary of a Riemannian manifold. We also give some sharp estimates for the first nonzero Steklov eigenvalue for…

Let A be a Hopf algebra and $Gamma$ be a bicovariant first order differential calculus over A. It is known that there are three possibilities to construct a differential Hopf algebra $Gamma^wedge$ that contains $Gamma$ as its first order…

We exploit the connection between quantum dot Dirac operators and $\overline\partial$-Robin Laplacians. First, we find a graphical relation between their smallest positive eigenvalues, which allows us to deduce a recipe for translating…