Symmetric invariant cocycles on the duals of q-deformations

We prove that for q not a nontrivial root of unity any symmetric invariant 2-cocycle for a completion of Uq(g) is the coboundary of a central element. Equivalently, a Drinfeld twist relating the coproducts on completions of Uq(g) and U(g) is unique up to coboundary of a central element. As an application we show that the spectral triple we defined in an earlier paper for the q-deformation of a simply connected semisimple compact Lie group G does not depend on any choices up to unitary equivalence.Comment: 18 pages; minor changes, to appear in AI

Deformation of C*-algebras by cocycles on locally compact quantum groups

Given a C*-algebra A with a left action of a locally compact quantum group G on it and a unitary 2-cocycle Omega on \hat G, we define a deformation A_Omega of A. The construction behaves well under certain additional technical assumptions on Omega, the most important of which is regularity, meaning that C_0(G)_Omega\rtimes G is isomorphic to the algebra of compact operators on some Hilbert space. In particular, then A_\Omega is stably isomorphic to the iterated twisted crossed product \hat G^{op}\ltimes_\Omega G\ltimes A. Also, in good situations, the C*-algebra A_\Omega carries a left action of the deformed quantum group G_\Omega and we have an isomorphism G_\Omega\ltimes A_\Omega\cong G\ltimes A. When G is a genuine locally compact group, we show that the action of G on C_0(G)_Omega=C*_r(\hat G;Omega) is always integrable. Stronger assumptions of properness and saturation of the action imply regularity. As an example, we make a preliminary analysis of the cocycles on the duals of some solvable Lie groups recently constructed by Bieliavsky et al., and discuss the relation of our construction to that of Bieliavsky and Gayral.Comment: 29 pages; new version emphasizes the role of 'quantization maps', improvements in exposition, a few more examples and reference

On second cohomology of duals of compact groups

We show that for any compact connected group G the second cohomology group defined by unitary invariant 2-cocycles on \hat G is canonically isomorphic to H^2(\hat{Z(G)};T). This implies that the group of autoequivalences of the C*-tensor category Rep G is isomorphic to H^2(\hat{Z(G)};T)\rtimes\Out(G). We also show that a compact connected group G is completely determined by Rep G. More generally, extending a result of Etingof-Gelaki and Izumi-Kosaki we describe all pairs of compact separable monoidally equivalent groups. The proofs rely on the theory of ergodic actions of compact groups developed by Landstad and Wassermann and on its algebraic counterpart developed by Etingof and Gelaki for the classification of triangular semisimple Hopf algebras. In two appendices we give a self-contained account of amenability of tensor categories, fusion rings and discrete quantum groups, and prove an analogue of Radford's theorem on minimal Hopf subalgebras of quasitriangular Hopf algebras for compact quantum groups.Comment: 22 pages, AMS-LaTeX; minor changes, remark 1.2 expanded to explain unitarity of an element u, final (hopefully) versio

Pseudo-two-girdles c-axis fabric patterns in a quartz-feldspar mylonite (Costabona granodiorite, Canigo massif)

An example of the relationship that exist between the preferred crystaliografic orientation of quartz grains and the attitude of the mylonite foliation of quartz-feldspar mylonites is described. These rocks are the result of the inhomogeneous deformation under low-grade metamorphic conditions of a late Hercynian granodiorite, intruded into the gneisses of the slopes of the Canig massif (Eastern Pyrenees). The Costabona mylonites have a quartz c-axis fabric in pseudo-twogirdles symmetrical with respect to the mylonite foliation and perpendicular to the shearband systems which produce an extensional crenulation of the mylonite foliation

Autoequivalences of the tensor category of Uq(g)-modules

We prove that for q\in\C* not a nontrivial root of unity the cohomology group defined by invariant 2-cocycles in a completion of Uq(g) is isomorphic to H^2(P/Q;\T), where P and Q are the weight and root lattices of g. This implies that the group of autoequivalences of the tensor category of Uq(g)-modules is the semidirect product of H^2(P/Q;\T) and the automorphism group of the based root datum of g. For q=1 we also obtain similar results for all compact connected separable groups.Comment: 5 pages; minor corrections; corollary about Drinfeld twists adde