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Abstract. We show that tensoriality constraints in noncommutative Riemannian geometry in the 2-dimensional bicrossproduct model quantum spacetime algebra [x, t] = λx drastically reduce the moduli of possible metrics g up to normalisation to a single real parameter which we interpret as a time in the past from which all timelike geodesics emerge and a corresponding time in the future at which they all converge. Our analysis also implies a reduction of moduli in n-dimensions and we study the suggested spherically symmetric classical geometry in n = 4 in detail, identifying two 1-parameter subcases where the Einstein tensor matches that of a perfect fluid for (a) positive pressure, zero density and (b) negative pressure and positive density with ratio w Q = − 1 2 . The classical geometry is conformally flat and its geodesics motivate new coordinates which we extend to the quantum case as a new description of the quantum spacetime model as a quadratic algebra. The noncommutative Riemannian geometry is fully solved for n = 2 and includes the quantum Levi-Civita connection and a second, nonperturbative, Levi-Civita connection which blows up as λ → 0. We also propose a 'quantum Einstein tensor' which is identically zero for the main part of the moduli space of connections (as classically in 2D). However, when the quantum Ricci tensor and metric are viewed as deformations of their classical counterparts there would be an O(λ 2 ) correction to the classical Einstein tensor and an O(λ) correction to the classical metric.

a b s t r a c tWe develop the formalism for noncommutative differential geometry and Riemmannian geometry to take full account of the * -algebra structure on the (possibly noncommutative) coordinate ring and the bimodule structure on the differential forms. We show that * -compatible bimodule connections lead to braid operators σ in some generality (going beyond the quantum group case) and we develop their role in the exterior algebra.We study metrics in the form of Hermitian structures on Hilbert * -modules and metric compatibility in both the usual form and a cotorsion form. We show that the theory works well for the quantum group C q [SU 2 ] with its three-dimensional calculus, finding for each point of a three-parameter space of covariant metrics a unique 'Levi-Civita' connection deforming the classical one and characterised by zero torsion, metric preservation and * -compatibility. Allowing torsion, we find a unique connection with a classical limit that is metric preserving and * -compatible and for which σ obeys the braid relations. It projects to a unique 'Levi-Civita' connection on the quantum sphere. The theory also works for finite groups, and in particular for the permutation group S 3 , where we find somewhat similar results.

We semiclassicalise the standard notion of differential calculus in noncommutative geometry on algebras and quantum groups. We show in the symplectic case that the infinitesimal data for a differential calculus is a symplectic connection, and interpret its curvature as lowest order nonassociativity of the exterior algebra. Semiclassicalisation of the noncommutative torus provides an example with zero curvature. In the Poisson-Lie group case we study left-covariant infinitesimal data in terms of partially defined preconnections. We show that the moduli space of bicovariant infinitesimal data for quasitriangular Poisson-Lie groups has a canonical reference point which is flat in the triangular case. Using a theorem of Kostant, we completely determine the moduli space when the Lie algebra is simple: the canonical preconnection is the unique point other than in the case of sl n , n > 2, when the moduli space is 1-dimensional. We relate the canonical preconnection to Drinfeld twists and thereby quantise it to a super coquasiHopf exterior algebra. We also discuss links with Fedosov quantisation.

We discuss combining physical experiments with machine computations and introduce a form of analogue-digital (AD) Turing machine. We examine in detail a case study where an experimental procedure based on Newtonian kinematics is combined with a class of Turing machines. Three forms of AD machine are studied, in which physical parameters can be set exactly and approximately. Using non-uniform complexity theory, and some probability, we prove theorems that show that these machines can compute more than classical Turing machines.

We study noncommutative bundles and Riemannian geometry at the semiclassical level of first order in a deformation parameter λ, using a functorial approach. The data for quantisation of the cotangent bundle is known to be a Poisson structure and Poisson preconnection and we now show that this data defines to a functor Q from the monoidal category of classical vector bundles equipped with connections to the monodial category of bimodules equipped with bimodule connections over the quantised algebra. We adapt this functor to quantise the wedge product of the exterior algebra and in the Riemannian case, the metric and the Levi-Civita connection. Full metric compatibility requires vanishing of an obstruction in the classical data, expressed in terms of a generalised Ricci 2-form R, without which our quantum Levi-Civita connection is still the best possible. We apply the theory to the Schwarzschild black-hole and to Riemann surfaces as examples, as well as verifying our results on the 2D bicrossproduct model quantum spacetime. The quantized Schwarzschild black-hole in particular has features similar to those encountered in q-deformed models, notably the necessity of nonassociativity of any rotationally invariant quantum differential calculus of classical dimensions.2000 Mathematics Subject Classification. Primary 81R50, 58B32, 83C57.

We compute the quantum double, braiding, and other canonical Hopf algebra constructions for the bicrossproduct Hopf algebra H associated to the factorization of a finite group X into two subgroups. The representations of the quantum double are described by a notion of bicrossed bimodules, generalising the cross modules of Whitehead. We also show that basis-preserving self-duality structures for the bicrossproduct Hopf algebras are in one᎐one correspondence with factorreversing group isomorphisms. The example ‫ޚ‬ ‫ޚ‬ is given in detail. We show 6 6 Ž .Ž . further that the quantum double D H is the twisting of D X by a nontrivial quantum cocycle. ᮊ

Abstract. We show that several standard associative quantizations in mathematical physics can be expressed as cochain module-algebra twists in the spirit of Moyal products at least to O( 3 ), but to achieve this we twist not by a 2-cocycle but by a 2-cochain. This implies a hidden nonassociavitity not visible in the algebra itself but present in its deeper noncommutative differential geometry, a phenomenon first seen in our previous work on semiclassicalisation of differential structures. The quantisations are induced by a classical group covariance and include: enveloping algebras U (g) as quantisations of g * , a Fedosov-type quantisation of the sphere S 2 under a Lorentz group covariance, the Mackey quantisation of homogeneous spaces, and the standard quantum groups Cq [G]. We also consider the differential quantisation of R n for a given symplectic connection as part of our semiclassical analysis and we outline a proposal for the Dirac operator.