Shifted Poisson structures on higher Chevalley-Eilenberg algebras
Cameron Kemp, Robert Laugwitz, Alexander Schenkel
TL;DR
The paper develops a graphical calculus to determine $n$-shifted Poisson structures on finitely generated semi-free CDGAs and applies it to Chevalley–Eilenberg algebras of Lie $N$-algebras. For ordinary Lie algebras, it recovers Safronov’s results: $(n=1)$-shifted Poisson structures correspond to quasi-Lie bialgebras and $(n=2)$-shifted structures to invariant symmetric tensors. In the Lie $2$-algebra case, it shows that $n$-shifted Poisson structures exist for $n eq 1$, specifically $n ooxed{1,2,3,4}$, with finite data describing the structures for $n=2,3,4$, and an infinite tower for $n=1$. The work also provides explicit examples (abelian, string, and shifted cotangent Lie $2$-algebras) and interprets these structures as semi-classical data for a notion of higher quantum groups, connecting to $L_ ext{infty}$-quasi-bialgebra data and potential higher Tannakian reconstructions.
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'.