Shifted Poisson structures on higher Chevalley-Eilenberg algebras

TL;DR

This paper introduces a graphical calculus to classify shifted Poisson structures on Chevalley-Eilenberg algebras, extending known results from Lie algebras to Lie 2-algebras and linking to higher quantum groups.

Contribution

It develops a new graphical calculus for shifted Poisson structures and generalizes classifications from Lie algebras to Lie 2-algebras, revealing connections to higher quantum groups.

Findings

Classifies n-shifted Poisson structures on Chevalley-Eilenberg algebras.

Recovers known results for Lie algebras as special cases.

Identifies new structures for Lie 2-algebras related to higher quantum groups.

Abstract

This paper develops a graphical calculus to determine the $n$-shifted Poisson structures on finitely generated semi-free commutative differential graded algebras. When applied to the Chevalley-Eilenberg algebra of an ordinary Lie algebra, we recover Safronov's result that the $(n=1)$- and $(n=2)$-shifted Poisson structures in this case are given by quasi-Lie bialgebra structures and, respectively, invariant symmetric tensors. We generalize these results to the Chevalley-Eilenberg algebra of a Lie $2$-algebra and obtain $n\in\{1,2,3,4\}$ shifted Poisson structures in this case, which we interpret as semi-classical data of `higher quantum groups'.