Shifted Poisson structures on higher Chevalley–Eilenberg algebras

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∈{1,2,3,4} shifted Poisson structures in this case, which we interpret as semi-classical data of ‘higher quantum groups’.