2009/01/15 by Marcos M. Alexandrino, Alexandrino, Marcos M., Renato G. Bettiol +1 · 1 citation
Mathematics · #17Bxx #22C05 #22Exx #22F05 #53C12 #57R30 #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.0901.2374
openalex publication_date 2009/01/15 · openalex created_date 2022/09/09 · openalex updated_date 2026/07/28
The main purpose of these lecture notes is to provide a concise introduction\nto Lie groups, Lie algebras, and isometric and adjoint actions, aiming mostly\nat advanced undergraduate and graduate students. In addition, the connection\nbetween such classic theories and the research area of the first author is\nexplored. Namely, generalizations to isoparametric submanifolds, polar actions\nand singular Riemannian foliations with sections (s.r.f.s.) are mentioned. The\nfirst chapters cover basic concepts, giving results on adjoint representation,\nclosed subgroups, bi-invariant metrics, Killing forms and splitting in simple\nideals. In the following chapters, proper and isometric actions are recalled\ntogether with adjoint action and foliations, mostly concerning the Weyl group,\nnormal slices and Dynkin diagrams. A special focus is given to maximal tori and\nroots of compact Lie groups, exploring its connection with isoparametric\nsubmanifolds and polar actions. Furthermore, in the last chapter, a survey on\nrecent research results on s.r.f.s. is given.\n In this revised version, more details about fiber bundles, proper and\nisometric actions are explored, and further exercises and examples were added.\nIt also features new sections with examples of singular Riemannian foliations\nconstructed with surgery and suspension of homomorphisms. This is still a\npreliminary version and we expect to improve it in the future. We would be\ngrateful for any kind of suggestions.\n