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 (μM 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(Ad;m). We show that Ad = CdχA1 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 Bd 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 eL 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.

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Bassett, M.E., Majid, S. Finite Noncommutative Geometries Related to \(\mathbb {F}_{p}[x]\). Algebr Represent Theor 23, 251–274 (2020). https://doi.org/10.1007/s10468-018-09846-4

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Keywords

  • Noncommutative geometry
  • Finite field
  • Prime number
  • Hopf algebra
  • Quantum group
  • Bimodule Riemannian geometry
  • Galois extension
  • Cocycle
  • Boolean algebra

Mathematics Subject Classification (2010)

  • Primary 81R50
  • 58B32
  • 46L87