Riemannian Geometry and General Relativity Reframed as a Generalized Lie Algebra

Quantum Theory (QT) and the Standard Model (SM) are expressible in Lie algebra frameworks while General Relativity (GR) is framed in the non-linear differential equations of Riemannian Geometry (RG), a very different framework that makes their union difficult. We show that RG can be reframed as a NonCommutative Algebra (NCA) that is a generalization of a Lie algebra (LA) where ``structure functions'' of position (X) generalize the LA structure constants. Such a NCA becomes an (approximate) LA in small regions of space-time. We begin with an Abelian algebra of n Hermitian operators X$^{\mathrm{\mu }}$ ($\mu \quad =$ 0, 1, .. n-1) with representations on a Hilbert space whose eigenvalues represent independent variables such as space-time. We define operators D$^{\mathrm{\mu }}$ that by definition translate the corresponding eigenvalues of X$^{\mathrm{\mu }}$ each by a distance ds as dX$^{\mathrm{\lambda }}$(ds) $=$ exp(a ds $\eta _{\mathrm{\mu \thinspace }}$D$^{\mathrm{\mu }})$ X$^{\mathrm{\lambda \thinspace }}$exp(-a ds $\eta_{\mathrm{\nu \thinspace }}$D$^{\mathrm{\nu }}$ ) - X$^{\mathrm{\lambda }} \quad =$ ds $\eta_{\mathrm{\mu \thinspace }}$[ D$^{\mathrm{\mu }}$, X$^{\mathrm{\thinspace \lambda }}$]/a $+$ ho where a is a constant and $\eta_{\mathrm{\mu }}$ is a unit vector for the translation. We define the functions g$^{\mathrm{\mu \nu }}$(X) $=$ [D$^{\mathrm{\mu }}$, X$^{\mathrm{\nu }}$]/a and show that ds$^{\mathrm{2}}$ $=$ g$_{\mathrm{\mu \nu }}$(X) dX$^{\mathrm{\mu }}_{\mathrm{\thinspace }}$dX$^{\mathrm{\nu }}$ proving that g$_{\mathrm{\mu \nu }}$(X) is the metric for the space taken in the position diagonal representation where D$^{\mathrm{\thinspace \mu }} \quad =$ a g$^{\mathrm{\mu \upsilon }}$(y) ($\partial $/$\partial $y$^{\mathrm{\nu }}) \quad +$ A$^{\mathrm{\mu }}$ (y) thereby defining [D$^{\mathrm{\thinspace \mu }}$, D$^{\mathrm{\thinspace \nu }}$]. Integration with QT gives a $=$ iž. Details and predictions are discussed.