Finite noncommutative geometries related to $F_p[x]$

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 $H_{\mathrm{dR}}^0({\mathbb{F}}_p[x];m)={\mathbb{F}}_p[g_d]$ if and only if $\mathrm{Tr}(m)\ne 0$, 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 (${\mu }_M$ 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 $\Omega (A_d;m)$. 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 $\frac{1}{d}{\sum }_{k|d}\varphi (k)p^{\frac{d}{k}}$ ($\varphi$ 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, $A_2$ 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.

25 pages ams latex no figures

References in corpus (2)

• Connections on central bimodules
• Cross Product Quantisation, Nonabelian Cohomology And Twisting Of Hopf Algebras