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 $(n=1)$- 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'.

v2: 28 pages. Final version accepted for publication in Letters in Mathematical Physics

References in corpus (8)

• Shifted Poisson Structures and Deformation Quantization
• Shifted Poisson and symplectic structures on derived N-stacks
• Lie 2-bialgebras
• 4-d Chern-Simons Theory: Higher Gauge Symmetry and Holographic Aspects
• Infinitesimal 2-braidings and differential crossed modules
• Higher Gauge Theory
• Infinitesimal 2-braidings from 2-shifted Poisson structures
• Deformation Quantization via Categorical Factorization Homology