It is open the possibility of imposing requisites to the quantisation of Spectral Triples in such a way that a critical dimension D=26 appears. From [1] it is known that commutative spectral triples contain the Einstein Hilbert action, which is extracted by using the Wodziski residue over D/ |D/ |, being D/ a Dirac operator. The theorem was initially enunciated [1] with a complicated proportion...

Let R be a commutative ring. The cup-length of R is defined by the greatest number n such that there exist x1, . . . , xn ∈ R \ R with x1 · · · xn , 0. We denote the cup-length of R by cup(R). In particular, for a space X and a commutative ring A, the cup-length of X with the coefficient A, is defined by cup(H̃(X; A)). We denote it by cupA(X). It is well-known that cupA(X) is a lower bound for t...

In this paper we analyze the obstructions to the existence of global action-angle variables for regular non-commutative integrable systems (NCI systems) on Poisson manifolds. In contrast with local action-angle variables, which exist as soon as the fibers of the momentum map of such an integrable system are compact, global action-angle variables rarely exist. This fact was first observed and an...

We exhibit a series of discrete spectral triples converging to the canonical spectral triple of a finite dimensional manifold. Thus we bypass the non-go theorem of Gökeler and Schücker. Sort of counterexample. In [2] Connes gave the most general example of non commutative manifolds defined from finite spectral triples, and such discrete manifolds were classified independently by Krajewsky [5, 6...

This paper is devoted to a conjecture concerning the deformation quantization. This conjecture implies that arbitrary smooth Poisson manifold can be formally quantized, and the equivalence class of the resulting algebra is canonically defined. In other terms, it means that non-commutative geometry, in the formal approximation to the commutative geometry of smooth spaces, is described by the sem...

B-series originated from the work of John Butcher in the 1960s as a tool to analyze numerical integration of differential equations, in particular Runge–Kutta methods. Connections to renormalization have been established in recent years. The algebraic structure of classical Runge–Kutta methods is described by the Connes–Kreimer Hopf algebra. Lie–Butcher theory is a generalization of B-series ai...

Invariant structures on homogeneous manifolds are of fundamental importance in differential geometry. Recall that an affinor structure F (i.e., a tensor field F of type (1,1)) on a homogeneous manifold G/H is called invariant (with respect to G) if for any g ∈ G we have dτ(g)◦F = F ◦dτ(g), where τ(g)(xH)= (gx)H . An important place among homogeneous manifolds is occupied by homogeneousΦ-spaces ...

A ring R is said to be semi-commutative if whenever a, b ∈ such that ab = 0, then aRb 0. In this article, we introduce the concepts of g−semi-commutative rings and g−N−semi-commutative several results concerning these two concepts. Let a G-graded g supp(R, G). Then with aRgb Also, − N−semi-commutative for any N(R) ⋂ Ann(a), bRg ⊆ Ann(a). We an example which N-semi-commutative some G) but itself...

For a group $G$ and $\omega\in Z^{3}(G, \text{U}(1))$, an $\omega$-anomalous action on C*-algebra $B$ is $\text{U}(1)$-linear monoidal functor between 2-groups $\text{2-Gr}(G, \text{U}(1), \omega)\rightarrow \underline{\text{Aut}}(B)$, where the latter denotes 2-group of $*$-automorphisms $B$. The class $[\omega]\in H^{3}(G, \text{U}(1))$ called anomaly action. We show for every $n\ge 2$ finite...