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The categorified Grothendiek-Riemann-Roch Theorem Scherotzke, Sarah ; ; et al in Compositio Mathematica (in press) Detailed reference viewed: 91 (0 UL)Classification of universal formality maps for quantizations of Lie bialgebras Merkulov, Sergei ; in Compositio Mathematica (2020), 156(10), 2111-2148 We introduce an endofunctor D in the category of augmented props with the property that for any representation of a prop P in a vector space V the associated prop DP admits an induced representation on ... [more ▼] We introduce an endofunctor D in the category of augmented props with the property that for any representation of a prop P in a vector space V the associated prop DP admits an induced representation on the graded commutative tensor algebra S(V) given in terms of polydifferential operators. Applying this functor to the prop LieB of Lie bialgebras we show that universal formality maps for quantizations of Lie bialgebras are in in 1-1 correspondence with prop morphisms from the minimal resolution AssB_infty of the prop of associative bialgebras to the polydifferential prop DLieB_infty satisfying certain boundary conditions. We prove that the set of such formality morphisms (having an extra property of being Lie connected) is non-empty. The latter result is used in turn to give a short proof of the formality theorem for universal quantizations of arbitrary Lie bialgebras which says that for any Drinfeld associator there is an associated Lie_infty quasi-isomorphism between the Lie_infty algebras controlling, respectively, deformations of the standard bialgebra structure in S(V) and deformations of any given Lie bialgebra structure in V. We study the deformation complex of an arbitrary universal formality morphism and show that it is quasi-isomorphic (up to one class corresponding to the standard rescaling automorphism of the properad LieB) to the oriented graph complex GC or 3 studied earlier in \cite{Wi2}. This result gives a complete classification of the set of gauge equivalence classes of universal Lie connected formality maps --- it is a torsor over the Grothendieck-Teichm\"uller group GRT and can hence can be identified with the set of Drinfeld associators. [less ▲] Detailed reference viewed: 101 (0 UL)Galois level and congruence ideal for p-adic families of finite slope Siegel modular forms Conti, Andrea in Compositio Mathematica (2019), 155(4), 776-831 We consider families of Siegel eigenforms of genus 2 and nite slope, de ned as local pieces of an eigenvariety and equipped with a suitable integral structure. Under some assumptions on the residual image ... [more ▼] We consider families of Siegel eigenforms of genus 2 and nite slope, de ned as local pieces of an eigenvariety and equipped with a suitable integral structure. Under some assumptions on the residual image, we show that the image of the Galois representation associated with a family is big, in the sense that a Lie algebra attached to it contains a congruence subalgebra of non-zero level. We call Galois level of the family the largest such level. We show that it is trivial when the residual representation has full image. When the residual representation is a symmetric cube, the zero locus de ned by the Galois level of the family admits an automorphic description: it is the locus of points that arise from overconvergent eigenforms for GL2, via a p-adic Langlands lift attached to the symmetric cube representation. Our proof goes via the comparison of the Galois level with a \fortuitous" congruence ideal, that describes the zero- and one-dimensional subvarieties of symmetric cube type appearing in the family. We show that some of the p-adic lifts are interpolated by a morphism of rigid analytic spaces from an eigencurve for GL2 to an eigenvariety for GSp4. The remaining lifts appear as isolated points on the eigenvariety. [less ▲] Detailed reference viewed: 45 (1 UL)l-independence for compatible systems of (mod l) representations Hui, Chun Yin in Compositio Mathematica (2015), 151(7), 1215-1241 Detailed reference viewed: 99 (8 UL)Automorphisms of quantum and classical Poisson algebras ; Poncin, Norbert in Compositio Mathematica (2004), 140(2), 511-527 Detailed reference viewed: 126 (9 UL) |
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