Finite Noncommutative Geometries Related to $\mathbb {F}_{p}[x]$

Abstract

It is known that irreducible noncommutative differential structures over Fp[x] are classified by irreducible monics m. We show that the coho-and only if Trace(m) = 0, where g d = x p d −x and d is the degree of m. This implies that there are p−1 pd k|d,p∤k µ M (k)p d k such noncommutative differential structures (µ M the Möbius function). Motivated by killing this zero'th cohomology, we consider the directed system of finite-dimensional Hopf algebras A d = Fp[x]/(g d ) as well as their inherited bicovariant differential calculi Ω(A d ; m). We show that A d = C d ⊗χ A 1 a cocycle extension where C d = A ψ d is the subalgebra of elements fixed under ψ(x) = x + 1. We also have a Frobenius-fixed subalgebra B d of dimension 1 d k|d φ(k)p d k (φ the Euler totient function), generalising Boolean algebras when p = 2. As special cases, A 1 ∼ = Fp(Z/pZ), the algebra of functions on the finite group Z/pZ, and we show dually that FpZ/pZ ∼ = Fp[L]/(L p ) for a 'Lie algebra' generator L with e L group-like, using a truncated exponential. By contrast, A 2 over F 2 is a cocycle modification of F 2 ((Z/2Z) 2 ) and is a 1-dimensional extension of the Boolean algebra on 3 elements. In both cases we compute the Fourier theory, the invariant metrics and the Levi-Civita connections within bimodule noncommutative geometry.2010 Mathematics Subject Classification. Primary 81R50, 58B32, 46L87.

Levi-Civita connections for a class of spectral triples

“…Thus, we have proved (6) for all L of the above form. But since the maps ζ E⊗AE,E , Φ g , V g (2) and P sym are all right A-linear, we can conclude that (6) (2) is nondegenerate as a map from E ⊗ sym A E to (E ⊗ sym A E) * . Proof: Let us start by claiming that (E ⊗ sym A E) * can be identified with the bimodule {φ ∈ (E ⊗ A E) * : φ•(1−P sym ) = 0}.…”

“…Suppose E is centered as an A-A-bimodule and also that E is finitely generated and projective as a right A module. Let g be a pseudo-Riemannian bilinear metric on E. Then the map V g (2) :…”

“…Proof: Let us start by proving that the map V g (2) is onto. Since E is a finitely generated projective module over A, we can use the isomorphism of (…”

“…The bilinearity of V g (2) is easy to check and hence we omit its proof. This completes the proof of the proposition.…”

“…Here, the symbols P sym and ⊗ sym A are as in Subsection 2.3. It should also be mentioned that there is a very different approach by S. Majid, E. Beggs and their co-authors ( [28], [3], [4], [2] etc. and references therein).…”