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

  • M. E. Bassett
  • S. MajidEmail author
Article
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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.

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 

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of MathematicsQueen Mary University of LondonLondonUK

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