A deformation of the algebra of diffeomorphisms is constructed for canonically deformed spaces with constant deformation parameter θ. The algebraic relations remain the same, whereas the comultiplication rule (Leibniz rule) is different from the undeformed one. Based on this deformed algebra a covariant tensor calculus is constructed and all the concepts like metric, covariant derivatives, curvature and torsion can be defined on the deformed space as well. The construction of these geometric quantities is presented in detail. This leads to an action invariant under the deformed diffeomorphism algebra and can be interpreted as a θ-deformed Einstein-Hilbert action. The metric or the vierbein field will be the dynamical variable as they are in the undeformed theory. The action and all relevant quantities are expanded up to second order in θ.

When the vacuum Einstein equations are cast in the form of hamiltonian evolution equations, the initial data lie in the cotangent bundle of the manifold MΣ of riemannian metrics on a Cauchy hypersurface Σ. As in every lagrangian field theory with symmetries, the initial data must satisfy constraints. But, unlike those of gauge theories, the constraints of general relativity do not arise as momenta of any hamiltonian group action. In this paper, we show that the bracket relations among the constraints of general relativity are identical to the bracket relations in the Lie algebroid of a groupoid consisting of diffeomorphisms between space-like hypersurfaces in spacetimes.A direct connection is still missing between the constraints themselves, whose definition is closely related to the Einstein equations, and our groupoid, in which the Einstein equations play no role at all. We discuss some of the difficulties involved in making such a connection.2010 Mathematics Subject Classification. 83C05 (Primary); 58H05 (Secondary).See Section 4 below for historical remarks and references. 5 Note that we are using two meanings of "momentum" in this discussion, first in a slightly extended version of the usual "mass times velocity", and second in the sense used in the theory of hamiltonian actions. In the latter sense, the term "moment" is often used instead. (See the footnote on p. 133 of [33]) for some remarks on the two nomenclatures.) 6 The definition of groupoid is reviewed briefly in Section A.2. We refer to [32] for a full treatment of Lie algebroids and Lie groupoids.

Polyhedral realizations of crystal bases for quantum algebras of classical affine types J. Math. Phys. 54, 053511 (2013); 10.1063/1.4805584 Loop realizations of quantum affine algebrasCovariance of a quantum space with respect to a quantum enveloping algebra ties the deformation of the multiplication of the space algebra to the deformation of the coproduct of the enveloping algebra. Since the deformation of the coproduct is governed by a Drinfeld twist, the same twist naturally defines a covariant star product on the commutative space. However, this product is in general not associative and does not yield the quantum space. It is shown that there are certain Drinfeld twists which realize the associative product of the quantum plane, quantum Euclidean four-space, and quantum Minkowski space. These twists are unique up to a central two-coboundary. The appropriate formal deformation of real structures of the quantum spaces is also expressed by these twists.