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We investigate a quantum geometric space in the context of what could beconsidered an emerging effective theory from Quantum Gravity. Specifically weconsider a two-parameter class of twisted Poincar\'e algebras, from whichLie-algebraic noncommutativities of the translations are derived as well asassociative star-products, deformed Riemannian geometries, Lie-algebraictwisted Minkowski spaces and quantum effects that arise as noncommutativities.Starting from a universal differential algebra of forms based on the abovementioned Lie-algebraic noncommutativities of the translations, we constructthe noncommutative differential forms and Inner and Outer derivations, whichare the noncommutative equivalents of the vector fields in the case ofcommutative differential geometry. Having established the essentials of thisformalism we construct a bimodule, required to be central under the action ofthe Inner derivations in order to have well defined contractions and from wherethe algebraic dependence of its coefficients is derived. This again thendefines the noncommutative equivalent of the geometrical line-element incommutative differential geometry. We stress, however, that even though thecomponents of the twisted metric are by construction symmetric in their algebravaluation, this is not so for their inverse and thus to construct it we madeuse of Gel'fand's theory of quasi-determinants, which is conceptuallystraightforward but computationally becoming quite complicate beyond an algebraof 3 generators. The consequences of the noncommutativity of the Lie-algebratwisted geometry are further discussed.

Gravity from Quantum Spacetime by Twisted Deformation of the Quantum Poincar\'e Group