2024/10/04 by Julian J. Holstein, Manuel Rivera, Holstein, Julian +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Geometry and complex manifolds #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2410.03604
openalex publication_date 2024/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that Koszul duality between differential graded categories and pointed curved coalgebras interchanges smooth and proper Calabi-Yau structures. This result is a generalization and conceptual explanation of the following two applications. For a finite-dimensional Lie algebra a smooth Calabi-Yau structure on the universal enveloping algebra is equivalent to a proper Calabi-Yau structure on the Chevalley-Eilenberg chain coalgebra, which exists if and only if Poincare duality is satisfied. For a topological space X having the homotopy type of a finite complex we show an oriented Poincare duality structure (with local coefficients) on X is equivalent to a proper Calabi-Yau structure on the dg coalgebra of chains on X and to a smooth Calabi-Yau structure on the dg algebra of chains on the based loop space of X.