Dunkl operator and quantization of $\mathbb{Z}_2$-singularity (2009)
Let $(X,\omega)$ be a symplectic orbifold which is locally like the quotient of a $\mathbb{Z}_2$ action on $\reals^n$. Let $A^{((\hbar))}_X$ be a deformation quantization of $X$ constructed via the...
Deformations of linear Poisson orbifolds (2008)
Halbout, Gilles, Oudom, Jean-Michel, Tang, Xiang
Let $\Gamma$ be a finite group acting faithfully and linearly on a vector space $V$. Let $T(V)$ ($S(V)$) be the tensor (symmetric) algebra associated to $V$ which has a natural $\Gamma$ action. We...
Le $X$ be a $C^\infty$-manifold and $\g$ be a finite dimensional Lie algebra acting freely on $X$. Let $r \in \ve^2(\g)$ be such that $Z=[r,r] \in \ve^3(\g)^\g$. In this paper we prove that every...
Deformation quantization of weak Poisson structures (2007)
Calaque, Damien, Halbout, Gilles
In this paper we prove that any weak Poisson structure (in the general context of Lie algebroids) admits a weak deformation quantization. We also give a sufficient condition for an actual Poisson...
Weak quantization of Poisson structures (2007)
Calaque, Damien, Halbout, Gilles
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...
Quantization of Poisson-Hopf stacks associated with group Lie bialgebras (2007)
Let $G$ be a Poisson Lie group and $\g$ its Lie bialgebra. Suppose that $\g$ is a group Lie bialgebra. This means that there is an action of a discrete group $\Gamma$ on $G$ deforming the Poisson...
Formality theorems for Hochschild chains in the Lie algebroid setting (2006)
Calaque, Damien, Dolgushev, Vasiliy, Halbout, Gilles
In this paper we prove Lie algebroid versions of Tsygan's formality conjecture for Hochschild chains both in the smooth and holomorphic settings. In the holomorphic setting our result implies a...
Coboundary Lie bialgebras and commutative subalgebras of universal enveloping algebras (2006)
Enriquez, Benjamin, Halbout, Gilles
We solve a functional version of the problem of twist quantization of a coboundary Lie bialgebra (g,r,Z). We derive from this the following results: (a) the formal Poisson manifolds g^* and G^* are...
Noncommutative Poisson structures on Orbifolds (2006)
In this paper, we compute the Gerstenhaber bracket on the Hochschild cohomology of $C^\infty(M)\rtimes \Gamma$. Using this computation, we classify all the noncommutative Poisson structures on...
Lift of $C_\infty$ and $L_\infty$ morphisms to $G_\infty$ morphisms (2006)
Ginot, Grégory, Halbout, Gilles
Let $\g_2$ be the Hochschild complex of cochains on $C^\infty(\RM^n)$ and $\g_1$ be the space of multivector fields on $\RM^n$. In this paper we prove that given any $G_\infty$-structure ({\rm i.e.}...
Lift of $C_\infty$ and $L_\infty$ morphisms to $G_\infty$ morphisms (2006)
Ginot, Grégory, Halbout, Gilles
Let $\g_2$ be the Hochschild complex of cochains on $C^\infty(\RM^n)$ and $\g_1$ be the space of multivector fields on $\RM^n$. In this paper we prove that given any $G_\infty$-structure ({\rm i.e.}...
Formality theorem for Lie bialgebras and quantization of coboundary r-matrices (2005)
Let $(g,\delta_\hbar)$ be a Lie bialgebra. Let $(U_\hbar(g),\Delta_\hbar)$ a quantization of $(g,\delta_\hbar)$ through Etingof-Kazhdan functor. We prove the existence of a $L_\infty$-morphism...
Formality theorems for Hochschild chains in the Lie algebroid setting (2005)
Calaque, Damien, Dolgushev, Vasiliy, Halbout, Gilles
In this paper we prove Lie algebroid versions of Tsygan's formality conjecture for Hochschild chains both in the smooth and holomorphic settings. In the holomorphic setting our result implies a...
Formalite $G_\infty$ adaptee et star-representations sur des sous-varietes coisotropes (2005)
Bordemann, Martin, Ginot, Gregory, Halbout, Gilles, Herbig, Hans-Christian, Waldmann, Stefan
Let X be a Poisson manifold and C a coisotropic submanifold and let I be the vanishing ideal of C. In this work we want to construct a star product * on X such that I[[lambda]] is a left ideal for *....
Lift of $C_\infty$ and $L_\infty$ morphisms to $G_\infty$ morphisms (2003)
Ginot, Grégory, Halbout, Gilles
Let $\g_2$ be the Hochschild complex of cochains on $C^\infty(\RM^n)$ and $\g_1$ be the space of multivector fields on $\RM^n$. In this paper we prove that given any $G_\infty$-structure ({\rm i.e.}...
Lift of $C_\infty$ and $L_\infty$ morphisms to $G_\infty$ morphisms (2003)
Ginot, Grégory, Halbout, Gilles
Let $\g_2$ be the Hochschild complex of cochains on $C^\infty(\RM^n)$ and $\g_1$ be the space of multivector fields on $\RM^n$. In this paper we prove that given any $G_\infty$-structure ({\rm i.e.}...
Lift of $C\_\infty$ and $L\_\infty$ morphisms to $G\_\infty$ morphisms (2003)
Ginot, Grégory, Halbout, Gilles
Let $\g\_2$ be the Hochschild complex of cochains on $C^\infty(\RM^n)$ and $\g\_1$ be the space of multivector fields on $\RM^n$. In this paper we prove that given any $G\_\infty$-structure ({\rm...
Uniqueness of braidings of quasitriangular Lie bialgebras and lifts of classical r-Matrices (2003)
Enriquez, Benjamin, Gavarini, Fabio, Halbout, Gilles
It is known that any quantization of a quasitriangular Lie bialgebra g gives rise to a braiding on the dual Poisson-Lie formal group G*. We show that this braiding always coincides with the...
Braiding structures on formal Poisson groups and classical solutions of the QYBE (2002)
Gavarini, Fabio, Halbout, Gilles
If g is a quasitriangular Lie bialgebra, one can asks what is the geometrical meaning of its r-matrix. A first answer was given in a paper by Weinstein and Xu, using purely geometrical means:...
Gavarini, Fabio, Halbout, Gilles
Let g be a quasitriangular Lie bialgebra over a field k of characteristic zero, and let g^* be its dual Lie bialgebra. We prove that the formal Poisson group F[[g^*]] is a braided Hopf algebra. More...
Sur la classification des déformations des variétés de Poisson. (1999)
Halbout, Gilles, Cartier, Pierre.
Tese (Doutorado)-- Université Louis Pasteur, 02/02/1999.
Gavarini, Fabio, Halbout, Gilles
Let g be a quasitriangular Lie bialgebra over a field K of characteristic zero, and let g^* be its dual Lie bialgebra. We prove that the formal Poisson group K[[g^*]] is a braided Hopf algebra, thus...