Related papers: Laplacians on $ q $-deformations of compact semisi…

We consider a Laplacian on periodic discrete graphs. Its spectrum consists of a finite number of bands. In a class of periodic 1-forms, i.e., functions defined on edges of the periodic graph, we introduce a subclass of minimal forms with a…

We establish two results concerning the Quantum Limits (QLs) of some sub-Laplacians. First, under a commutativity assumption on the vector fields involved in the definition of the sub- Laplacian, we prove that it is possible to split any QL…

We establish the $L^p$-$L^q$-boundedness of subelliptic pseudo-differential operators on a compact Lie group $G$. Effectively, we deal with the $L^p$-$L^q$-bounds for operators in the sub-Riemmanian setting because the subelliptic classes…

For a (unital) $C^*$-algebra $\cla$, we construct a $C^*$-algebraic discrete quantum group (DQG) $\clq_{\rm aut}(\cla)$, coacting on $\cla$, which is a quantum generalization of ${\text Aut}(\cla)$ in the framework of discrete quantum…

Laplacian operators are classical objects that are fundamental in both pure and applied mathematics and are becoming increasingly prominent in modern computational and data science fields such as applied and computational topology and…

Let $L$ denote a right-invariant sub-Laplacian on an exponential, hence solvable Lie group $G$, endowed with a left-invariant Haar measure. Depending on the structure of $G$, and possibly also that of $L$, $L$ may admit differentiable…

All bicovariant first order differential calculi on the quantum group GLq(3,C) are determined. There are two distinct one-parameter families of calculi. In terms of a suitable basis of 1-forms the commutation relations can be expressed with…

We study the algebra of differential operators on non-compact simply connected harmonic manifolds and provide sufficient conditions for them to have a radial fundamental solution and be surjective on the space of smooth function.…

We introduce and study a number of invariants of locally compact quantum groups defined by their scaling and modular groups and the spectrum of their modular elements. Focusing mainly on compact quantum groups we consider the question…

Let $k$ be a field and let $C$ be a small category. A $k$-linear representation of $C$, or a $kC$-module, is a functor from $C$ to the category of finite dimensional vector spaces over $k$. When the category $C$ is more general than a…

We consider Laplacians on $\Z^2$-periodic discrete graphs. The following results are obtained: 1) The Floquet-Bloch decomposition is constructed and basic properties are derived. 2) The estimates of the Lebesgue measure of the spectrum in…

We study *-differential calculi over compact quantum groups in the sense of S.L. Woronowicz. Our principal results are the construction of a Hodge operator commuting with the Laplacian, the derivation of a corresponding Hodge decomposition…

We consider different sub-Laplacians on a sub-Riemannian manifold $M$. Namely, we compare different natural choices for such operators, and give conditions under which they coincide. One of these operators is a sub-Laplacian we constructed…

We consider a magnetic Laplacian with periodic magnetic potentials on periodic discrete graphs. Its spectrum consists of a finite number of bands, where degenerate bands are eigenvalues of infinite multiplicity. We obtain a specific…

We extend discrete calculus for arbitrary ($p$-form) fields on embedded lattices to abstract discrete geometries based on combinatorial complexes. We then provide a general definition of discrete Laplacian using both the primal cellular…

For the q-deformation G_q, 0<q<1, of any simply connected simple compact Lie group G we construct an equivariant spectral triple which is an isospectral deformation of that defined by the Dirac operator D on G. Our quantum Dirac operator…

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…

This paper provides the foundations of quantum Clifford analysis in $q$-commutative variables with symmetric difference operators. We consider a $q$-Dirac operator on the quantum Euclidean space that factorizes the $U_q(\frak{o})$-invariant…

We give a construction of a Dirac operator on a quantum group based on any simple Lie algebra of classical type. The Dirac operator is an element in the vector space $U_q(\g) \otimes \mathrm{cl}_q(\g)$ where the second tensor factor is a…

We present a thorough study of the theoretical properties and devise efficient algorithms for the \emph{persistent Laplacian}, an extension of the standard combinatorial Laplacian to the setting of pairs (or, in more generality, sequences)…