The Bernstein-Gelfand-Gelfand (BGG) Construction: Algebra, Geometry, and Analysis; Part I

These are lecture notes for the first part of a short course that I taught jointly with Kaibo Hu as part of the thematic program “Differential Complexes: Theory, Discretization, and Applications” at the Erwin Schrödinger Institute (ESI) in Vienna. We give an introduction to differential forms and the construction of Berstein-Gelfand-Gelfand (BGG) complexes on open domains in R^n, focusing on the role of representation theory of semisimple Lie groups and Lie algebras in the construction.