Noncommutative Riemannian geometry from quantum spacetime generated by twisted Poincaré group

Abstract

We investigate a quantum geometric space in the context of what could be considered an emerging effective theory from quantum gravity. Specifically we consider a two-parameter class of twisted Poincaré algebras, from which Lie-algebraic noncommutativities of the translations are derived as well as associative star-products, deformed Riemannian geometries, Lie-algebraic twisted Minkowski spaces, and quantum effects that arise as noncommutativities. Starting from a universal differential algebra of forms based on the above-mentioned Lie-algebraic noncommutativities of the translations, we construct the noncommutative differential forms and inner and outer derivations, which are the noncommutative equivalents of the vector fields in the case of commutative differential geometry. Having established the essentials of this formalism, we construct a bimodule, which is required to be central under the action of the inner derivations in order to have well-defined contractions and from where the algebraic dependence of its coefficients is derived. This again then defines the noncommutative equivalent of the geometrical line-element in commutative differential geometry. We stress, however, that even though the components of the twisted metric are by construction symmetric in their algebra valuation, it is not so for their inverse, and thus to construct it, we made use of Gel’fand’s theory of quasi-determinants, which is conceptually straightforward but computationally quite complicated beyond an algebra of 3 generators. The consequences of the noncommutativity of the Lie-algebra twisted geometry are further discussed.