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

Abstract

It is known that irreducible noncommutative differential structures over $\mathbb {F}_{p}[x]$ are classified by irreducible monics m. We show that the cohomology $H_{\text {dR}}^{0}(\mathbb {F}_{p}[x]; m)=\mathbb {F}_{p}[g_{d}]$ if 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 ${\frac {p-1}{pd}}{\sum }_{k|d, p\nmid k}\mu _{M}(k)p^{\frac {d}{k}}$ 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 $A_{d}=\mathbb {F}_{p}[x]/(g_{d})$ as well as their inherited bicovariant differential calculi Ω(A_d;m). We show that A_d = C_d ⊗_χA₁ is a cocycle extension where $C_{d}=A_{d}^{\psi }$ is the subalgebra of elements fixed under ψ(x) = x + 1. We also have a Frobenius-fixed subalgebra B_d of dimension $\frac {1}{d} {\sum }_{k | d} \phi (k) p^{\frac {d}{k}}$ (ϕ the Euler totient function), generalising Boolean algebras when p = 2. As special cases, $A_{1}\cong \mathbb {F}_{p}(\mathbb {Z}/p\mathbb {Z})$, the algebra of functions on the finite group $\mathbb {Z}/p\mathbb {Z}$, and we show dually that $\mathbb {F}_{p}\mathbb {Z}/p\mathbb {Z}\cong \mathbb {F}_{p}[L]/(L^{p})$ for a ‘Lie algebra’ generator L with e^L group-like, using a truncated exponential. By contrast, A₂ over $\mathbb {F}_{2}$ is a cocycle modification of $\mathbb {F}_{2}((\mathbb {Z}/2\mathbb {Z})^{2})$ 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.