Kontsevich Deformation Quantization on Lie Algebras

Nabiha Ben Amar; Mouna Chaabouni; Mabrouka Hfaiedh

Displaying similar documents to “Kontsevich Deformation Quantization on Lie Algebras”

The structure of split regular Hom-Poisson algebras

María J. Aragón Periñán, Antonio J. Calderón Martín (2016)

Colloquium Mathematicae

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We introduce the class of split regular Hom-Poisson algebras formed by those Hom-Poisson algebras whose underlying Hom-Lie algebras are split and regular. This class is the natural extension of the ones of split Hom-Lie algebras and of split Poisson algebras. We show that the structure theorems for split Poisson algebras can be extended to the more general setting of split regular Hom-Poisson algebras. That is, we prove that an arbitrary split regular Hom-Poisson algebra is of the form...

The Dixmier-Moeglin equivalence and a Gel’fand-Kirillov problem for Poisson polynomial algebras

K. R. Goodearl, S. Launois (2011)

Bulletin de la Société Mathématique de France

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The structure of Poisson polynomial algebras of the type obtained as semiclassical limits of quantized coordinate rings is investigated. Sufficient conditions for a rational Poisson action of a torus on such an algebra to leave only finitely many Poisson prime ideals invariant are obtained. Combined with previous work of the first-named author, this establishes the Poisson Dixmier-Moeglin equivalence for large classes of Poisson polynomial rings, including semiclassical limits of quantum...

Mean field limit for the one dimensional Vlasov-Poisson equation

Maxime Hauray (2012-2013)

Séminaire Laurent Schwartz — EDP et applications

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We consider systems of $N$ particles in dimension one, driven by pair Coulombian or gravitational interactions. When the number of particles goes to infinity in the so called mean field scaling, we formally expect convergence towards the Vlasov-Poisson equation. Actually a rigorous proof of that convergence was given by Trocheris in [Tro86]. Here we shall give a simpler proof of this result, and explain why it implies the so-called “Propagation of molecular chaos”. More precisely, both...

On the global regularity of subcritical Euler–Poisson equations with pressure

Eitan Tadmor, Dongming Wei (2008)

Journal of the European Mathematical Society

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We prove that the one-dimensional Euler–Poisson system driven by the Poisson forcing together with the usual $γ$ -law pressure, $γ \geq 1$ , admits global solutions for a large class of initial data. Thus, the Poisson forcing regularizes the generic finite-time breakdown in the $2 \times 2$ $p$ -system. Global regularity is shown to depend on whether or not the initial configuration of the Riemann invariants and density crosses an intrinsic critical threshold.

Quantization of Drinfeld Zastava in type $A$

Michael Finkelberg, Leonid Rybnikov (2014)

Journal of the European Mathematical Society

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Drinfeld Zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of the affine Lie algebra ${\hat{𝔰𝔩}}_{n}$ . We introduce an affine, reduced, irreducible, normal quiver variety $Z$ which maps to the Zastava space bijectively at the level of complex points. The natural Poisson structure on the Zastava space can be described on $Z$ in terms of Hamiltonian reduction of a certain Poisson subvariety of the dual space of a (nonsemisimple) Lie algebra. The...

A note on n-ary Poisson brackets

Michor, Peter W., Vaisman, Izu

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An $n$ -ary Poisson bracket (or generalized Poisson bracket) on the manifold $M$ is a skew-symmetric $n$ -linear bracket ${, \dots,}$ of functions which is a derivation in each argument and satisfies the generalized Jacobi identity of order $n$ , i.e., $\sum_{σ \in S_{2 n - 1}} (sign σ) {{f_{σ_{1}}, \dots, f_{σ_{n}}}, f_{σ_{n + 1}}, \dots, f_{σ_{2 n - 1}}} = 0,$ $S_{2 n - 1}$ being the symmetric group. The notion of generalized Poisson bracket was introduced by et al. in [J. Phys. A, Math. Gen. 29, No. 7, L151–L157 (1996; Zbl 0912.53019) and J. Phys. A, Math. Gen. 30, No. 18, L607–L616 (1997; Zbl 0932.37056)]....

Cauchy-Poisson transform and polynomial inequalities

Mirosław Baran (2009)

Annales Polonici Mathematici

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We apply the Cauchy-Poisson transform to prove some multivariate polynomial inequalities. In particular, we show that if the pluricomplex Green function of a fat compact set E in $ℝ^{N}$ is Hölder continuous then E admits a Szegö type inequality with weight function $d i s t {(x, \partial E)}^{- (1 - κ)}$ with a positive κ. This can be viewed as a (nontrivial) generalization of the classical result for the interval E = [-1,1] ⊂ ℝ.

Universal lifting theorem and quasi-Poisson groupoids

David Inglesias-Ponte, Camille Laurent-Gengoux, Ping Xu (2012)

Journal of the European Mathematical Society

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We prove the universal lifting theorem: for an $α$ -simply connected and $α$ -connected Lie groupoid $Γ$ with Lie algebroid $A$ , the graded Lie algebra of multi-differentials on $A$ is isomorphic to that of multiplicative multi-vector fields on $Γ$ . As a consequence, we obtain the integration theorem for a quasi-Lie bialgebroid, which generalizes various integration theorems in the literature in special cases. The second goal of the paper is the study of basic properties of quasi-Poisson groupoids....

Poisson's equation and characterizations of reflexivity of Banach spaces

Vladimir P. Fonf, Michael Lin, Przemysław Wojtaszczyk (2011)

Colloquium Mathematicae

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Let X be a Banach space with a basis. We prove that X is reflexive if and only if every power-bounded linear operator T satisfies Browder’s equality $x \in X : s u p_{n} | | \sum_{k = 1}^{n} T^{k} x | | < \infty$ = (I-T)X $.$ We then deduce that X (with a basis) is reflexive if and only if every strongly continuous bounded semigroup $T_{t} : t \geq 0$ with generator A satisfies $A X = x \in X : s u p_{s > 0} | | \int_{0}^{s} T_{t} x d t | | < \infty$ . The range (I-T)X (respectively, AX for continuous time) is the space of x ∈ X for which Poisson’s equation (I-T)y = x (Ay = x in continuous time) has a solution y ∈ X; the above equalities...

Canonical Poisson-Nijenhuis structures on higher order tangent bundles

P. M. Kouotchop Wamba (2014)

Annales Polonici Mathematici

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Let M be a smooth manifold of dimension m>0, and denote by $S_{c a n}$ the canonical Nijenhuis tensor on TM. Let Π be a Poisson bivector on M and $Π^{T}$ the complete lift of Π on TM. In a previous paper, we have shown that $(T M, Π^{T}, S_{c a n})$ is a Poisson-Nijenhuis manifold. Recently, the higher order tangent lifts of Poisson manifolds from M to $T^{r} M$ have been studied and some properties were given. Furthermore, the canonical Nijenhuis tensors on $T^{A} M$ are described by A. Cabras and I. Kolář [Arch. Math. (Brno) 38 (2002),...

Мономиальные $n$ -лиевы алгебры.

А.П. Пожидаев (1998)

Algebra i Logika

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Prolongation of Poisson $2$ -form on Weil bundles

Norbert Mahoungou Moukala, Basile Guy Richard Bossoto (2016)

Archivum Mathematicum

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In this paper, $M$ denotes a smooth manifold of dimension $n$ , $A$ a Weil algebra and $M^{A}$ the associated Weil bundle. When $(M, ω_{M})$ is a Poisson manifold with $2$ -form $ω_{M}$ , we construct the $2$ -Poisson form $ω_{M^{A}}^{A}$ , prolongation on $M^{A}$ of the $2$ -Poisson form $ω_{M}$ . We give a necessary and sufficient condition for that $M^{A}$ be an $A$ -Poisson manifold.

Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras

Johannes Huebschmann (1998)

Annales de l'institut Fourier

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For any Lie-Rinehart algebra $(A, L)$ , B(atalin)-V(ilkovisky) algebra structures $\partial$ on the exterior $A$ -algebra $Λ_{A} L$ correspond bijectively to right $(A, L)$ -module structures on $A$ ; likewise, generators for the Gerstenhaber algebra $Λ_{A} L$ correspond bijectively to right $(A, L)$ -connections on $A$ . When $L$ is projective as an $A$ -module, given a B-V algebra structure $\partial$ on $Λ_{A} L$ , the homology of the B-V algebra $(Λ_{A} L, \partial)$ coincides with the homology of $L$ with coefficients in $A$ with reference to the right $(A, L)$ -module structure determined...

One-parameter contractions of Lie-Poisson brackets

Oksana Yakimova (2014)

Journal of the European Mathematical Society

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We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra $𝒜 = 𝕂 [𝔸^{n}]$ is said to be of Kostant type, if its centre $Z (𝒜)$ is freely generated by homogeneous polynomials $F_{1}, ..., F_{r}$ such that they give Kostant’s regularity criterion on $𝔸^{n} (d_{x} F_{i}$ are linear independent if and only if the Poisson tensor has the maximal rank at $x$ ). If the initial Poisson algebra is of Kostant type and $F_{i}$ satisfy a certain degree-equality, then the contraction...

Involutivity of truncated microsupports

Masaki Kashiwara, Térésa Monteiro Fernandes, Pierre Schapira (2003)

Bulletin de la Société Mathématique de France

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Using a result of J.-M. Bony, we prove the weak involutivity of truncated microsupports. More precisely, given a sheaf $F$ on a real manifold and $k \in ℤ$ , if two functions vanish on ${SS}_{k} (F)$ , then so does their Poisson bracket.

Subalgebra of $L_{1} (G)$ associated with Laplacian on a Lie group

A. Hulanicki (1974)

Colloquium Mathematicae

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$\mathcal{L}$ -homomorphisms of Lie algebras

Aleksander A. Lashkhi (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Si studiano gli omomorfismi reticolari ( $\mathcal{L}$ -omomorfismi) di algebre di Lie sopra anelli commutativi con unità. Le algebre di Lie sopra un campo e le $p$ -algebre di Lie non ammettono $\mathcal{L}$ -omomorfismi propri. Si assegnano condizioni necessarie e sufficienti affinchè un'algebra di Lie periodica o mista possieda un « $\mathcal{L}$ -omomorfismo su una catena di lunghezza $n$ .

The evolution and Poisson kernels on nilpotent meta-abelian groups

Richard Penney, Roman Urban (2013)

Studia Mathematica

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Let S be a semidirect product S = N⋊ A where N is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and A is isomorphic to $ℝ^{k}$ , k>1. We consider a class of second order left-invariant differential operators on S of the form $ℒ_{α} = L^{a} + Δ_{α}$ , where $α \in ℝ^{k}$ , and for each $a \in ℝ^{k}, L^{a}$ is left-invariant second order differential operator on N and $Δ_{α} = Δ - ⟨ α, \nabla ⟩$ , where Δ is the usual Laplacian on $ℝ^{k}$ . Using some probabilistic techniques (e.g., skew-product formulas for diffusions on S and N respectively)...