Connexions contravariantes sur les groupes de Lie-Poisson

TL;DR

This paper studies Riemannian Poisson-Lie groups satisfying Hawkins conditions, establishing a classification link with Lie bialgebra structures on Milnor algebras, and provides explicit classifications in specific low-dimensional cases.

Contribution

It establishes an equivalence between classifying Hawkins-satisfying Riemannian Poisson-Lie groups and certain Lie bialgebra structures, with explicit results in low dimensions.

Findings

Classified Riemannian Poisson-Lie groups satisfying Hawkins conditions in linear, Heisenberg, and low-dimensional cases.

Established the equivalence between geometric classification and algebraic Lie bialgebra structures.

Identified open problems for the general case beyond low dimensions.

Abstract

This work is devoted to the study of a class of Poisson-Lie groups endowed with left invariant metrics. The triples $(G,\pi,<,>)$ are considered, where $G$ is a simply connected Lie group, ?$\pi$ is a multiplicative Poisson tensor and $<,>$ is a left invariant riemannian metric such that Hawkins conditions are satisfied. Hawkins conditions are necessary conditions for the deformation of the graded algebra of differential forms of a riemannian manifold. These conditions come from the deformation of the noncommutative spectral triple describing the manifold. The main result of this thesis is the equivalence between, on one hand, the geometric problem of classifying riemannian Poisson-Lie groups that satisfy the conditions of Hawkins and, secondly, the problem of classifying algebraic structures of Lie bialgebras on Milnor algebras that satisfy certain conditions. Exploiting the fact that the structures of Lie bialgebras on Milnor algebras, in certain situations, can be calculated, we determine riemannian Poisson-Lie groups that satisfy Hawkins in the linear case, in the case of Heisenberg in the triangular case and in low dimensions (up to dimension 5). The general case remains an open problem.