Shifted Poisson structures on higher Chevalley-Eilenberg algebras

Area of Science:

• Algebraic Topology
• Mathematical Physics
• Differential Geometry

Background:

• Commutative differential graded algebras are fundamental in algebraic topology and mathematical physics.
• Poisson structures and their generalizations (n-shifted Poisson structures) are crucial for understanding classical and quantum systems.
• Lie algebras and Lie 2-algebras provide frameworks for describing symmetries in various physical theories.

Purpose of the Study:

• To develop a novel graphical calculus for determining n-shifted Poisson structures.
• To analyze these structures on finitely generated semi-free commutative differential graded algebras.
• To generalize existing results from Lie algebras to Lie 2-algebras.

Main Methods:

• Development of a graphical calculus tailored for n-shifted Poisson structures.
• Application of the calculus to the Chevalley-Eilenberg algebra of Lie algebras and Lie 2-algebras.
• Comparison and extension of Safronov's results for n=1 and n=2 shifted Poisson structures.

Main Results:

• The graphical calculus successfully determines n-shifted Poisson structures on the specified algebras.
• For ordinary Lie algebras, the (n=1) and (n=2) shifted Poisson structures correspond to quasi-Lie bialgebra structures and invariant symmetric tensors, respectively.
• Generalization to Lie 2-algebras yields n-shifted Poisson structures for n in {1, 2, 3, 4}, interpreted as semi-classical data of higher quantum groups.

Conclusions:

• The developed graphical calculus offers a powerful tool for studying n-shifted Poisson structures.
• The findings extend the understanding of Poisson structures to higher algebraic structures like Lie 2-algebras.
• This work provides a bridge between algebraic structures and the semi-classical data of higher quantum groups.