# Poisson-Lie structures as shifted Poisson structures

## TL;DR

This paper explores the relationship between classical limits of quantum groups, such as Poisson-Lie structures, and the broader framework of shifted Poisson structures, providing new insights into their geometric and algebraic properties.

## Contribution

It introduces a conceptual framework linking multiplicative Poisson structures to shifted Poisson structures and proposes a notion of symplectic realization, with examples from Manin pairs and triples.

## Key findings

- Relates classical quantum group limits to shifted Poisson structures
- Proposes a symplectic realization concept for shifted Poisson structures
- Provides examples using Manin pairs and triples

## Abstract

Classical limits of quantum groups give rise to multiplicative Poisson structures such as Poisson-Lie and quasi-Poisson structures. We relate them to the notion of a shifted Poisson structure which gives a conceptual framework for understanding classical (dynamical) $r$-matrices, quasi-Poisson groupoids and so on. We also propose a notion of a symplectic realization of shifted Poisson structures and show that Manin pairs and Manin triples give examples of such.