Related papers: Noncommutative versions of some classical biration…
Spatial noncommutativity is similar and can even be related to the non-Abelian nature of multiple D-branes. But they have so far seemed independent of each other. Reflecting this decoupling, the algebra of matrix valued fields on…
We consider the coherent state approach to non-commutativity, and we derive from it an effective quantum scalar field theory. We show how the non-commutativity can be taken in account by a suitable modification of the Klein-Gordon product,…
This paper introduces a notion of integrality that is suitable for non-commutative varieties. It is compatible with the usual notion of integrality for schemes. The function field and generic point of a non-commutative integral space are…
Motivated by Iyama's higher representation theory, we introduce $n$-translation quivers and $n$-translation algebras. The classical $\mathbb Z Q$ construction of the translation quiver is generalized to construct an $(n+1)$-translation…
Veneroni maps are a class of birational transformations of projective spaces. This class contains the classical Cremona transformation of the plane, the cubo-cubic transformation of the space and the quatro-quartic transformation of…
Given a brane tiling, that is, a bipartite graph on a torus, we can associate with it a singular 3-Calabi-Yau variety. In this paper we study its commutative and non-commutative crepant resolutions. We give an explicit toric description of…
A noncommutative algebra corresponding to the classical catenoid is introduced together with a differential calculus of derivations. We prove that there exists a unique metric and torsion-free connection that is compatible with the complex…
Starting from the concept of the universal exterior algebra in non-commutative differential geometry we construct differential forms on the quantum phase-space of an arbitrary system. They bear the same natural relationship to quantum…
We introduce the notion of the moduli stack of relations of a quiver. When the quiver with relations is derived-equivalent to an algebraic variety, the corresponding compact moduli scheme can be viewed as a compact moduli of noncommutative…
We introduce and study an analog of the Kelvin transformation connected with the Vladimirov-Taibleson operator acting on real- or complex-valued functions on a space $K^n$ over a non-Archimedean local field $K$.
A new trigonometric degeneration of the Sklyanin algebra is found and the functional realization of its representations in space of polynomials in one variable is studied. A further contraction gives the standard quantum algebra…
Let R be a noetherian connected graded domain of Gelfand-Kirillov dimension 3 over an uncountable algebraically closed field. Suppose that the graded quotient ring of R is a skew-Laurent ring over a field; we say that R is a birationally…
Our experience shows that dealing with noncommutative objects one should not imitate the classical commutative mathematics, but follow "the way it is" starting with basics. In this paper we consider mainly two such problems: noncommutative…
Recently, some problems have been found in the definition of the partial derivative in the case of the presence of both explicit and implicit functional dependencies in the classical analysis. In this talk we investigate the influence of…
This article describes recent applications of algebraic geometry to noncommutative algebra. These techniques have been particularly successful in describing graded algebras of small dimension.
Classical differential geometry can be encoded in spectral data, such as Connes' spectral triples, involving supersymmetry algebras. In this paper, we formulate non-commutative geometry in terms of supersymmetric spectral data. This leads…
Classical and quantum mechanics for an extended Heisenberg algebra with canonical commutation relations for position and momentum coordinates are considered. In this approach additional noncommutativity is removed from the algebra by linear…
The central structure in various versions of noncommutative geometry is a differential calculus on an associative algebra. This is an analogue of the calculus of differential forms on a manifold. In this short review we collect examples of…
In a minimalistic view, the use of noncommutative coordinates can be seen just as a way to better express non-local interactions of a special kind: 1-particle solutions (wavefunctions) of the equation of motion in the presence of an…
In the present paper by Frobenius algebra Y we mean a finite dimensional algebra possessing an associative and invertible (nondegenerate) form a scalar product, referred to as the Frobenius structure. The nondegenerate form has an inverse.…