Related papers: Affine noncommutative geometry
The sphere $S^{N-1}_\mathbb R$ has a half-liberated analogue $S^{N-1}_{\mathbb R,*}$, and a free analogue $S^{N-1}_{\mathbb R,+}$. This is a presentation of the construction and main properties of these noncommutative spheres,…
We develop an approach to noncommutative algebraic geometry ``in the perturbative regime" around ordinary commutative geometry. Let R be a noncommutative algebra and A=R/[R,R] its commutativization. We describe what should be the formal…
We make an attempt to develop "noncommutative algebraic geometry" in which noncommutative affine schemes are in one-to-one correspondence with associative algebras. In the first part we discuss various aspects of smoothness in affine…
This is a survey article on the currently very active research area of free (=non-commutative) real algebra and geometry. We first review some of the important results from the commutative theory, and then explain similarities and…
We discuss generalizations of the notion of i) the group of unitary elements of a (real or complex) finite dimensional C*-algebra, ii) gauge transformations and iii) (real) automorphisms, in the framework of compact quantum group theory and…
We describe basic concepts of noncommutative geometry and a general construction extending the familiar duality between ordinary spaces and commutative algebras to a duality between Quotient spaces and Noncommutative algebras. Basic tools…
Divided into three parts, the first marks out enormous geometric issues with the notion of quasi-freenss of an algebra and seeks to replace this notion of formal smoothness with an approximation by means of a minimal unital commutative…
We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of $\Rb^n$. They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global…
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions (possibly in…
We discuss the notion of matrix model, $\pi:C(X)\to M_K(C(T))$, for algebraic submanifolds of the free complex sphere, $X\subset S^{N-1}_{\mathbb C,+}$. When $K\in\mathbb N$ is fixed there is a universal such model, which factorizes as…
The real sphere $S^{N-1}_\mathbb R$ appears as increasing union, over $d\in\{1,...,N\}$, of its "polygonal" versions $S^{N-1,d-1}_\mathbb R=\{x\in S^{N-1}_\mathbb R|x_{i_0}... x_{i_d}=0,\forall i_0,...,i_d\ {\rm distinct}\}$. Motivated by…
Under very strong axioms, there is precisely one real noncommutative geometry between the classical one and the free one, namely the half-classical one, coming from the relations $abc=cba$. We discuss here the complex analogues of this…
This paper develops the tools of formal algebraic geometry in the setting of noncommutative manifolds, roughly ringed spaces locally modeled on the free associative algebra. We define a notion of noncommutative coordinate system, which is a…
In [1], Connes presented axioms governing noncommutative geometry. He went on to claim that when specialised to the commutative case, these axioms recover spin or spin^c geometry depending on whether the geometry is ''real'' or not. We…
We propose and discuss how basic notions (quadratic modules, positive elements, semialgebraic sets, Archimedean orderings) and results (Positivstellensaetze) from real algebraic geometry can be generalized to noncommutative $*$-algebras. A…
The formalism of non-commutative geometry of A. Connes is used to construct models in particle physics. The physical space-time is taken to be a product of a continuous four-manifold by a discrete set of points. The treatment of Connes is…
A very first step to develop non-commutative algebraic geometry is the arithmetic of polynomials in non-commuting variables over a commutative field, that is, the study of elements in free associative algebras. This investigation is…
This paper introduces arithmetic geometry for polynomial identity algebras using non-commutative (formal) deformation theory. Since formal deformation theory is inherently local the arithmetic and geometric results that follow give local…
The classical concept of affine locally symmetric spaces allows a generalization for various geometric structures on a smooth manifold. We remind the notion of symmetry for parabolic geometries and we summarize the known facts for…
A noncommutative algebra $A$, called an algebraic noncommutative geometry, is defined, with a parameter $\epsilon$ in the centre. When $\epsilon$ is set to zero, the commutative algebra $A^0$ of algebraic functions on an algebraic manifold…