2019/08/31 by R. A. Kycia, Radosław Antoni Kycia · 1 citation
Mathematics · Physics and Astronomy · #Algebra over a field #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #Differential operator #Eigenvalues and eigenvectors #Harmonic oscillator #Homotopy #Homotopy and Cohomology in Algebraic Topology #Lemma (botany) #Mathematics #Operator (biology) #Physics #Pure mathematics #Quantum mechanics #math-ph #math.DG #math.MP #msc:58A12 #msc:58Z05
paper · pdf · doi:10.1007/s00025-020-01247-8
published as Results Math (2020) 75:122 · 14 pages, 4 figures
openalex publication_date 2020/07/11 · arxiv created 2020/07/12 · arxiv updated 2020/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The paper focuses on various properties and applications of the homotopy operator, which occurs in the Poincaré lemma. In the first part, an abstract operator calculus is constructed, where the exterior derivative is an abstract derivative and the homotopy operator plays the role of an abstract integral. This operator calculus can be used to formulate abstract differential equations. An example of the eigenvalue problem that resembles the fermionic quantum harmonic oscillator is presented. The second part presents the dual complex to the Dolbeault bicomplex generated by the homotopy operator on complex manifolds.