Poisson algebras associated to quasi-Hopf algebras

We define admissible quasi-Hopf quantized universal enveloping (QHQUE) algebras by ?-adic valuation conditions. We show that any QHQUE algebra is twist-equivalent to an admissible one. We prove a related statement: any associator is twist-equivalent to a Lie associator. We attach a quantized formal series algebra to each admissible QHQUE algebra and study the resulting Poisson algebras.     

Braiding structures on formal Poisson groups and classical solutions of the QYBE

If is a quasitriangular Lie bialgebra, the formal Poisson group can be given a braiding structure. This was achieved by Weinstein and Xu using purely geometrical means, and independently by the authors by means of quantum groups. In this paper we compare these two approaches. First, we show that the braidings they produce share several similar properties (in particular, the construction is functorial); secondly, in the simplest case (G=SL₂) they do coincide. The question then rises of whether they are always the same this is positively answered in a separate paper.     

Quantization of Γ-Lie bialgebras

We introduce the notion of Γ-Lie bialgebras, where Γ is a group. These objects give rise to cocommutative co-Poisson bialgebras, for which we construct quantization functors. This enlarges the class of co-Poisson algebras for which a quantization is known. Our result relies on our earlier work, where we showed that twists of Lie bialgebras can be quantized; we complement this work by studying the behavior of this quantization under compositions of twists.     

Formality theorem for Lie bialgebras and quantization of twists and coboundary r-matrices

Let (g,δ?) be a Lie bialgebra. Let (U?(g),Δ?) a quantization of (g,δ?) through Etingof-Kazhdan functor. We prove the existence of a L_∞-morphism between the Lie algebra C(g)=Λ(g) and the tensor algebra (without unit) T₊U=T₊(U?(g)[−1]) with Lie algebra structure given by the Gerstenhaber bracket. When s is a twist for (g,δ), we deduce from the formality morphism the existence of a quantum twist F. When (g,δ,r) is a coboundary Lie bialgebra, we get the existence of a quantization R of r.     

Calcul d'un Invariant de Star-Produit Fermé sur une Variété Symplectique

Let M be a symplectic manifold over $ℝ. In [CFS] the authors construct an invariant ϕ in the cyclic cohomology of M for any closed star-product. They compute this invariant in the de Rham complex of M when M=T ^* V. We generalize this result by computing the image of ϕ in the de Rham complex for any symplectic manifold and any star-product and we show how this invariant is related to the general classification of Kontsevich. The proof uses the Riemann–Roch theorem for periodic cyclic chains of Nest–Tsygan.

---

Calcul d'un Invariant de Star-Produit Fermé sur une Variété Symplectique

---

Received: 30 November 1998 / Accepted: 15 February 1999     

Formality Conjectures and Deformation

Let M be a Poisson manifold. Kontsevich proved that star products exist on M and he gave a classification. To relate his classification with other classifications, one could try to extend the Connes–Flato–Sternheimer invariant to a general Poisson manifold. We show how generalization of this invariant is related to the formality conjecture for chains. Finally, we show how to prove those conjectures step by step. Our approach, different from Tamarkin's, will give explicit formulas but doesn't yet solve the general conjecture.     

A Formality Theorem for Poisson Manifolds

Letters in Mathematical Physics - Let M be a differential manifold. Using different methods, Kontsevich and Tamarkin have proved a formality theorem, which states the existence of a Lie...     

Lifts of - and -morphisms to -morphisms

Let be the Hochschild complex of cochains on and let be the space of multivector fields on . In this paper we prove that given any -structure (i.e. Gerstenhaber algebra up to homotopy structure) on , and any -morphism (i.e. morphism of a commutative, associative algebra up to homotopy) between and , there exists a -morphism between and that restricts to . We also show that any -morphism (i.e. morphism of a Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a -morphism, using Tamarkin's method for any -structure on . We also show that any two of such -morphisms are homotopic.

     

Weak quantization of Poisson structures

In this paper we prove that any Poisson structure on a sheaf of Lie algebroids admits a weak deformation quantization, and give a sufficient condition for such a Poisson structure to admit an actual deformation quantization. We also answer the corresponding classification problems. In the complex symplectic case, we recover in particular some results of Nest-Tsygan and Polesello-Schapira.