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Finite noncommutative geometries related to \(\mathbb{F}_p[x]\). (English) Zbl 1437.81043

Summary: 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 \(\operatorname{Trace}(m) \neq 0\), 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 \(A_d = C_{d} \otimes_\chi A_1\) is 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, \(A_1\cong \mathbb{F}_p(\mathbb{Z}/p\mathbb{Z})\), 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.

MSC:

81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
58B32 Geometry of quantum groups
46L87 Noncommutative differential geometry
16T05 Hopf algebras and their applications
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