Related papers: Noncommutative geometry and a class of completely …
We lay the foundations for a general approach to nonassociative spectral geometry as an extension of Connes' noncommutative geometry by explaining how to construct finite-dimensional, discrete spectral geometries with exceptional symmetry,…
This article is an introduction to our recent work in harmonic analysis associated with semigroups of operators, in the effort of finding a noncommutative Calder\'on-Zygmund theory for von Neumann algebras. The classical CZ theory has been…
We investigate boundary conditions for the nonlinear sigma model on the compact symmetric space $G/H$, where $H \subset G$ is the subgroup fixed by an involution $\sigma$ of $G$. The Poisson brackets and the classical local conserved…
Nowadays, noncommutative geometry is a growing domain of mathematics, which can appear as a promising framework for modern physics. Quantum field theories on "noncommutative spaces" are indeed much investigated, and suffer from a new type…
Several sets of quaternionic functions are described and studied with respect to hy-perholomorphy, addition and (non commutative) multiplication, on open sets of H, then Hamil-ton 4-manifolds analogous to Riemann surfaces, for H instead of…
The conditions under which a general two-dimensional non-linear sigma model is classically integrable are given. These requirements are found by demanding that the equations of motion of the theory are expressible as a zero curvature…
We construct a model unifying general relativity and quantum mechanics in a broader structure of noncommutative geometry. The geometry in question is that of a transformation groupoid given by the action of a finite group G on a space E. We…
Several generalizations of a commutative ring that is a graded complete intersection are proposed for a noncommutative graded $k$-algebra; these notions are justified by examples from noncommutative invariant theory.
Classical spectral theory gives a complete description of a single normal operator, but it fails for noncommuting operators, where no canonical joint spectrum or simultaneous diagonalization exists. Existing approaches provide only partial…
We illustrate an isomorphic representation of the observable algebra for quantum mechanics in terms of the functions on the projective Hilbert space, and its Hilbert space analog, with a noncommutative product in terms of explicit…
We describe $\sigma$-matching, interchangeable and, as a consequence, totally compatible products on some classes of associative algebras, including unital algebras, the semigroup algebras of rectangular bands, algebras with enough…
We find a class of algebras A satisfying the following property: for every nontrivial noncommutative polynomial, the linear span of all its values in A equals A. This class includes the algebras of all bounded and all compact operators on…
A classification of integrable two-component systems of non-evolutionary partial differential equations that are analogous to the Camassa-Holm equation is carried out via the perturbative symmetry approach. Independently, a classification…
A unital $C^*$-algebra is called $N$-subhomogeneous if its irreducible representations are finite dimensional with dimension at most $N$. We extend this notion to operator systems, replacing irreducible representations by boundary…
We introduce non-linear $\sigma$-models in the framework of noncommutative geometry with special emphasis on models defined on the noncommutative torus. We choose as target spaces the two point space and the circle and illustrate some…
In the paper we describe the C*-algebras of noncommutative spherical tight frames over some C*-algebras and then apply to study the noncommutative version of the universal classifying space.
We give a geometric classification of $n$-dimensional nilpotent, commutative nilpotent and anticommutative nilpotent algebras. We prove that the corresponding geometric varieties are irreducible, find their dimensions and describe explicit…
We give a survey on higher invariants in noncommutative geometry and their applications to differential geometry and topology.
We investigate non-commutative gauge theories in homogeneous spaces G/H. We construct such theories by adding cubic terms to IIB matrix model which contain the structure constants of G. The isometry of a homogeneous space, G must be a…
The aim of the present survey paper is to provide an accessible introduction to a new chapter of representation theory - harmonic analysis for noncommutative groups with infinite-dimensional dual space. I omitted detailed proofs but tried…