# Shifted Poisson structures on higher Chevalley–Eilenberg algebras

## TL;DR

The paper introduces a graphical method to study shifted Poisson structures on algebras related to Lie algebras and extends known results to higher structures.

## Contribution

A new graphical calculus is developed to determine n-shifted Poisson structures on semi-free commutative differential graded algebras.

## Key findings

- The graphical calculus recovers Safronov’s results for ordinary Lie algebras with n=1 and n=2.
- The method is generalized to Lie 2-algebras, yielding shifted Poisson structures for n=1,2,3,4.
- These structures are interpreted as semi-classical data for 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 \documentclass[12pt]{minimal}
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\begin{document}$$(n=1)$$\end{document}(n=1)- and \documentclass[12pt]{minimal}
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\begin{document}$$(n=2)$$\end{document}(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 \documentclass[12pt]{minimal}
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\begin{document}$$n\in \{1,2,3,4\}$$\end{document}n∈{1,2,3,4} shifted Poisson structures in this case, which we interpret as semi-classical data of ‘higher quantum groups’.