Shifted Poisson structures on higher Chevalley–Eilenberg algebras

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) (n = 1) - and (n=2) (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\ n ∈ 1, 2, 3, 4 shifted Poisson structures in this case, which we interpret as semi-classical data of ‘higher quantum groups’.