Finite noncommutative geometries related to $F_p[x]$

Abstract

It is known that irreducible noncommutative differential structures over $\Bbb F_p[x]$ are classified by irreducible monics $m$. We show that the cohomology $H_{\rm dR}^0(\Bbb F_p[x]; m)=\Bbb F_p[g_d]$ if and only if ${\rm Tr}(m)\ne 0$, where $g_d=x^{p^d}-x$ and $d$ is the degree of $m$. This implies that there are ${p-1\over pd}\sum_{k|d, p\nmid k}\mu_M(k)p^{d\over k}$ such noncommutative differential structures ($\mu_M$ the M\"obius function). Motivated by killing this zero'th cohomology, we consider the directed system of finite-dimensional Hopf algebras $A_d=\Bbb F_p[x]/(g_d)$ as well as their inherited bicovariant differential calculi $\Omega(A_d;m)$. We show that $A_d=C_d\otimes_\chi A_1$ a cocycle extension where $C_d=A_d^\psi$ is the subalgebra of elements fixed under $\psi(x)=x+1$. We also have a Frobenius-fixed subalgebra $B_d$ of dimension $\frac{1}{d} \sum_{k | d} \phi(k) p^\frac{d}{k}$ ($\phi$ the Euler totient function), generalising Boolean algebras when $p=2$. As special cases, $A_1\cong \Bbb F_p(\Bbb Z/p\Bbb Z)$, the algebra of functions on the finite group $\Bbb Z/p\Bbb Z$, and we show dually that $\Bbb F_p\Bbb Z/p\Bbb Z\cong\Bbb F_p[L]/(L^p)$ for a `Lie algebra' generator $L$ with $e^L$ group-like, using a truncated exponential. By contrast, $A_2$ over $\Bbb F_2$ is a cocycle modification of $\Bbb F_2((\Bbb Z/2\Bbb Z)^2)$ 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.