On some approaches towards non-commutative algebraic geometry

On The non-commutative Riemannian geometry of GL_q(n)

A recently proposed definition of a linear connection in non-commutative geometry, based on a generalized permutation, is used to construct linear connections on GL_q(n). Restrictions on the generalized permutation arising from the stability of linear connections under involution are discussed. Candidates for generalized permutation on GL_q(n) are found. It is shown that, for a given generalized permutation, there exists one and only one associated linear connection. Properties of the linear connection are discussed, in particular its bicovariance, torsion and commutative limit.

交换多项式环上斜PBW扩张的光滑几何 I

Smooth geometry of skew PBW extensions over commutative polynomial rings I

研究交换多项式环上斜PBW扩张的微分光滑性，针对一个和两个不定元情形。

Braided Commutative Geometry and Drinfel'd Twist Deformations

In this thesis we give obstructions for Drinfel'd twist deformation quantization on several classes of symplectic manifolds. Motivated from this quantization procedure, we further construct a noncommutative Cartan calculus on any braided commutative algebra, as well as an equivariant Levi-Civita covariant derivative for any non-degenerate equivariant metric. This generalizes and unifies the Cartan calculus on a smooth manifold and the Cartan calculus on twist star product algebras. We prove that the Drinfel'd functor leads to equivalence classes in braided commutative geometry and commutes wit…

Riemann-Roch Theorem and Index Theorem in Non-commutative Geometry

In this lecture we review apprearance of the Riemann-Roch Theorem in classical function theory, Algebraic topology, in theory of pseudo-differential operators and finally in noncommutative geometry. We show also it usefulness in many problem of quantization and harmonic analysis on Lie groups.

Gravity coupled with matter and foundation of non-commutative geometry

We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length element $ds$. Its unitary representations correspond to Riemannian metrics and Spin structure while $ds$ is the Dirac propagator $ds = \ts \!\!$---$\!\! \ts = D^{-1}$ where $D$ is the Dirac operator. We extend these simple relations to the non commutative case using Tomita's involution $J$. We then write a spectral action, the trace of a function of the length element in Planck units, which when applied to the non commutative geometry of the Standa…

Symmetry Reduction of States II: A non-commutative Positivstellensatz for CPn

We give a non-commutative Positivstellensatz for CP^n: The (commutative) *-algebra of polynomials on the real algebraic set CP^n with the pointwise product can be realized by phase space reduction as the U(1)-invariant polynomials on C^{1+n}, restricted to the real (2n+1)-sphere inside C^{1+n}, and Schmüdgen's Positivstellensatz gives an algebraic description of the real-valued U(1)-invariant polynomials on CP^n that are strictly pointwise positive on the sphere. In analogy to this commutative case, we consider a non-commutative *-algebra of polynomials on C^{1+n}, the Weyl algebra, and give a…

The Chern-Simons Action in Non-Commutative Geometry

A general definition of Chern-Simons actions in non-commutative geometry is proposed and illustrated in several examples. These are based on ``space-times'' which are products of even-dimensional, Riemannian spin manifolds by a discrete (two-point) set. If the *algebras of operators describing the non-commutative spaces are generated by functions over such ``space-times'' with values in certain Clifford algebras the Chern-Simons actions turn out to be the actions of topological gravity on the even-dimensional spin manifolds. By contrasting the space of field configurations in these examples in…

Vectors and covectors in non-commutative setting

Following the guidelines of classical differential geometry the `building material' for the tensor calculus in non-commutative geometry is suggested. The algebraic account of moduli of vectors and covectors is carried out.

A Lorentzian version of the non-commutative geometry of the standard model of particle physics

A formulation of the non-commutative geometry for the standard model of particle physics with a Lorentzian signature metric is presented. The elimination of the fermion doubling in the Lorentzian case is achieved by a modification of Connes' internal space geometry so that it has signature 6 (mod 8) rather than 0. The fermionic part of the Connes-Chamseddine spectral action can be formulated, and it is shown that it allows an extension with right-handed neutrinos and the correct mass terms for the see-saw mechanism of neutrino mass generation.