Phase Space Reduction of Star Products on Cotangent Bundles

In this paper we construct star products on Marsden-Weinstein reduced spaces in case both the original phase space and the reduced phase space are (symplectomorphic to) cotangent bundles. Under the assumption that the original cotangent bundle $T^*Q$ carries a symplectique structure of form $\omega_{B_0}=\omega_0 + \pi^*B_0$ with $B_0$ a closed two-form on $Q$, is equipped by the cotangent lift of a proper and free Lie group action on $Q$ and by an invariant star product that admits a $G$-equivariant quantum momentum map, we show that the reduced phase space inherits from $T^*Q$ a star product. Moreover, we provide a concrete description of the resulting star product in terms of the initial star product on $T^*Q$ and prove that our reduction scheme is independent of the characteristic class of the initial star product. Unlike other existing reduction schemes we are thus able to reduce not only strongly invariant star products. Furthermore in this article, we establish a relation between the characteristic class of the original star product and the characteristic class of the reduced star product and provide a classification up to $G$-equivalence of those star products on $(T^*Q,\omega_{B_0})$, which are invariant with respect to a lifted Lie group action. Finally, we investigate the question under which circumstances `quantization commutes with reduction' and show that in our examples non-trivial restrictions arise

Morita base change in Hopf-cyclic (co)homology

In this paper, we establish the invariance of cyclic (co)homology of left Hopf algebroids under the change of Morita equivalent base algebras. The classical result on Morita invariance for cyclic homology of associative algebras appears as a special example of this theory. In our main application we consider the Morita equivalence between the algebra of complex-valued smooth functions on the classical 2-torus and the coordinate algebra of the noncommutative 2-torus with rational parameter. We then construct a Morita base change left Hopf algebroid over this noncommutative 2-torus and show that its cyclic (co)homology can be computed by means of the homology of the Lie algebroid of vector fields on the classical 2-torus.Comment: Final version to appear in Lett. Math. Phy

A manifesto for the study of ancient Mediterranean maritime networks

In this one-off, extended Project Gallery article, the participants of a recent workshop jointly present a manifesto for the study of ancient Mediterranean maritime connectivity. Reviewing the advantages and perils of network modelling, they advance conceptual and methodological frameworks for the productive study of seaborne connectivity. They show how progressive research methods can overcome some of the problems encountered when working with uneven datasets spanning large geographical regions and long periods of time. The manifesto suggests research directions that could better inform our interpretations of human connections, both within and beyond the Mediterranean

On the Hochschild (co)homology of quantum homogeneous spaces

The recent result of Brown and Zhang establishing Poincaré duality in the Hochschild (co)homology of a large class of Hopf algebras is extended to right coideal subalgebras over which the Hopf algebra is faithfully flat, and applied to the standard Podleś quantum 2-sphere