We rst exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its in nitesimal length element ds. Its unitary representations correspond to Riemannian metrics and Spin structure while ds is the Dirac propagator ds = | = D 1 where D is the Dirac operator. We extend these simple relations to the non commutative case using Tomita's involut...

We construct the Georgi-Glashow Lagrangian for gauge group SU q (n). Breaking this symmetry spontaneously gives q-dependent masses of gauge field and vacuum manifold. It turned out that the vacuum manifold is param-eterized by the non-commutative quantities. We showed that the monopole solutions exist in this model, which is indicated by the presence of the BPS states.

We construct integrable models on flag manifold by using the symplectic structure explicitly given in the Bruhat coordinatization of flag manifold. They are non-commutative integrable and some of the conserved quantities are given by the Casimir invariants. We quantize the systems using the coherent state path integral technique and find the exact expression for the propagator for some special ...

It was known that the quaternion group and the octic group could not be generated by the squares of any group [5, pp. 193-194]. A natural question is which groups are generated by the squares of some groups. Clearly, groups of odd order and simple groups are generated by their own squares. In this paper, we show in a concrete manner that abelian groups are generated by the squares of some group...

The study of m-primary irreducible ideals in a commutative graded connected Noetherean algebra over a field is in principal equivalent to the study of the corresponding quotient algebras. Such algebras are Poincaré duality algebras. The prototype of such an algebra (apart from the cosmetic difference of being graded commutative rather than commutative graded) is the cohomology with field coeffi...

We construct a canonical noncommutative spectral triple for every oriented closed Riemannian manifold, which represents the fundamental class in the twisted K-homology of the manifold. This so-called “projective spectral triple” is Morita equivalent to the well-known commutative spin spectral triple provided that the manifold is spin-c. We give an explicit local formula for the twisted Chern ch...

Around 1980 Connes extended the notions of geometry to the noncommutative setting. Since then non-commutative geometry has turned into a very active area of mathematical research. As a first non-trivial example of a noncommutative manifold Connes discussed subalgebras of rotation algebras, the socalled non-commutative tori. In the last two decades researchers have unrevealed the relevance of no...

We show that the algebra A of a commutative unital spectral triple (A,H,D) satisfying several additional conditions, slightly stronger than those proposed by Connes, is the algebra of smooth functions on a compact spin manifold.

Using new spaces of tracial non-commutative smooth functions, we formulate a free probabilistic analog the Wasserstein manifold on $\mathbb{R}^d$ (the formal Riemannian probability densities $\mathbb{R}^d$), and use it to s

The unit sphere S can be identified with the unitary group SU(2). Under this identification the unit sphere can be considered as a non-commutative Lie group. The commutation relations for the vector fields of the corresponding Lie algebra define a 2-step sub-Riemannian manifold. We study sub-Riemannian geodesics on this sub-Riemannian manifold making use of the Hamiltonian formalism and solving...