Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0403334

(Bi)modules, morphismes et réduction des star-produits: le cas symplectique, feuilletages et obstructions

Martin Bordemann
View original source ↗Read source document ↗Export metadata ↓

Abstract

(Bi)modules, morphisms and reduction of star-products are studied in a framework of multidifferential operators along maps: morphisms deform Poisson maps and representations on functions spaces deform coisotropic maps. If a star-product is representable on a coisotropic submanifold, is is equivalent to a star-product for which the vanishing ideal is a left ideal. If the reduced phase space exists, a star-product with suitable Deligne class is representable and the reduced algebra is the commutant of this module (hence a bimodule). Obstructions to representability to third order are related to the Atiyah-Molino class of the foliation of the coisotropic submanifold, and the same kind of obstructions occurs for the quantisation of a Poisson map between symplectic manifolds. For vanishing Atiyah-Molino class, the representation and morphism problem is solvable.

Explore related subjects

math.QAmath.SG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Martin Bordemann. 2004-03-20. (Bi)modules, morphismes et réduction des star-produits: le cas symplectique, feuilletages et obstructions. https://arxiv.org/abs/math/0403334

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

An infinitesimal deformation of the post-Lie and post-Hopf algebra correspondence

We describe infinitesimal deformations of post-Lie algebras and post-Hopf algebras and prove that the universal enveloping algebra and primitive elements adjunction is compatible with the infinitesimal structure. In characteristic zero, this restricts to an equivalence for connected cocommutative infinitesimal post-Hopf algebras, extending the Cartier--Milnor--Moore theorem. We parametrize all infinitesimal post-Lie structures on $\mathfrak{sl}(2)$ and exhibit a second-order obstruction. From a geometric point of view, infinitesimal post-Lie algebras emerge from flat connections with covariantly constant torsion and an additional compatible symmetric $(1,2)$-tensor. This leads to explicit examples of infinitesimal post-Lie structures based on Heisenberg Lie algebras. On the Hopf side, we classify infinitesimal structures above the trivial action, prove rigidity for group algebras, and describe the structures on Sweedler's Hopf algebra. The subadjacent construction induces a Hopf algebra deformation with fixed coproduct. Finally, we prove that the quadratic operad of infinitesimal post-Lie algebras is Koszul, using a filtered distributive law between the operads of Lie algebras and bi-magmas.

math.QA↗
arXiv2026

Galois-modular obstructions to the categorification of near-group fusion rings

In this article we establish new obstructions to the categorification of near-group fusion rings by combining the values of irreducible characters of the Grothendieck ring with the Galois symmetry of the induced unit object $I(1)$ in the Drinfeld center. As a result, we show that a near-group ring of type $G+m$, where $G$ is an abelian group and both $|G|$ and $m$ are odd, can only categorify if $m=|G|$.

math.QA↗
arXiv2026

Reflective algebras over factorizable Hopf algebras

The reflective algebra $(R_H(A),K_{\mathrm{ref}}(A))$, introduced by Laugwitz, Walton and Yakimov, is a quasitriangular left $H$-comodule algebra constructed from a quasitriangular Hopf algebra $(H,R)$ and a left $H$-comodule algebra $A$. We study the structure and factorizability of finite-dimensional reflective algebras. When $(H,R)$ is factorizable, we show that $(R_H(A),K_{\mathrm{ref}}(A))$ is isomorphic, as a quasitriangular left $H$-comodule algebra, to $A\otimes H$ equipped with the trivial left $H$-coaction on $A$ and the universal $K$-matrix induced by $R_{21}R$. In this case, we also classify all universal $K$-matrices on $R_H(A)$. Finally, we prove that $(R_H(A),K_{\mathrm{ref}}(A))$ is factorizable if and only if $(H,R)$ is factorizable and $A$ is simple as an algebra.

math.QA↗