Multiplicative deformations of spectrale triples associated to left invariant metrics on Lie groups

TL;DR

This paper investigates conditions under which certain geometric structures on Lie groups can be deformed non-commutatively, classifying these structures in low dimensions by translating the problem into algebraic terms.

Contribution

It translates the geometric classification problem into an algebraic one and provides a complete classification up to dimension four.

Findings

Classified triples $(G, ext{π}, ext{prs})$ satisfying Hawkins's conditions in low dimensions.

Established the equivalence between geometric and algebraic classification problems.

Provided explicit lists of such triples up to four dimensions.

Abstract

We study the triple $(G,\pi,\prs)$ where $G$ is a connected and simply connected Lie group, $\pi$ and $\prs$ are, respectively, a multiplicative Poisson tensor and a left invariant Riemannian metric on $G$ such that the necessary conditions, introduced by Hawkins, to the existence of a non commutative deformation (in the direction of $\pi$) of the spectrale triple associated to $\prs$ are satisfied. We show that the geometric problem of the classification of such triple $(G,\pi,\prs)$ is equivalent to an algebraic one. We solve this algebraic problem in low dimensions and we give the list of all $(G,\pi,\prs)$ satisfying Hawkins's conditions, up to dimension four.