Finite noncommutative geometries related to $F_p[x]$

TL;DR

This paper classifies and constructs finite noncommutative geometries over finite fields, analyzing their cohomology, algebraic structures, and geometric properties, revealing new connections to group algebras and Boolean algebras.

Contribution

It introduces a classification of noncommutative differential structures over finite fields and explores their algebraic and geometric properties, including cohomology and Fourier theory.

Findings

Cohomology $H_{ m dR}^0(F_p[x]; m)$ is $F_p[g_d]$ if ${ m Tr}(m) e 0$.

Finite-dimensional Hopf algebras $A_d$ are cocycle extensions of $A_1$.

Explicit structures for $A_1$, $A_2$, and their geometric properties are computed.

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.