Grassmann and Stiefel Varieties over Composition Algebras / / by Marek Golasiński, Francisco Gómez Ruiz

• Autore: Golasiński Marek
• Pubblicazione: Cham : , : Springer Nature Switzerland : , : Imprint : Springer, , 2023
• Edizione: 1st ed. 2023.
• Descrizione fisica: 1 online resource (342 pages)
• Disciplina: 514.34
• Soggetto topico: Geometry, Differential; Differential Geometry; Varietats de Grassmann; Varietats de Stiefel; Varietats algebraiques
• Soggetto genere / forma: Llibres electrònics
• Altri autori: Gómez RuizFrancisco
• Nota di contenuto: Intro -- Preface -- About the Book -- Contents -- 1 Algebraic Preliminaries -- 1.1 K-Algebras with Involutions: Composition K-Algebras -- 1.2 Generalized Frobenius-Hurwitz's Theorem -- 1.3 Matrices over K-Algebras -- Hermitian and Symmetric Matrices -- Trace -- Inner Products on M(A) -- General Linear Group -- Unitary Matrices -- Gram-Schmidt Orthonormalization Process -- Orientation on Vector Spaces -- Diagonalization of Hermitian Matrices -- Rank of a Matrix -- Idempotent Matrices -- 1.4 Background on Algebraic Geometry -- 1.5 Natural, Order and Zariski Topologies on M(A) -- The Natural Topology on M(A) -- The Order Topology on M(A) -- The Zariski Topology on M(A) -- 2 Exceptional Groups G2(K) and F4(K) -- 2.1 Cross Products and the Exceptional Group G2(K) -- Cross Product in Ln -- Properties -- The Group Aut(Ln,n-1) -- Cross Product in K3 -- Cross Product in C(K)n -- Cross Product in C(K)3 -- Cross Product in K7 -- The Group G2(K) -- Automorphisms of A -- Action of G2(K) on S6(K) -- Jordan Multiplication -- Automorphisms of M(A) -- 2.2 Automorphisms of Hermn(O(K)) -- The Canonical 3-form on the C(K)-Vector Space Sksymn(C(K)) -- Particular Case of n=3 -- 2.3 The Exceptional Group F4(K) -- Homogeneous Polynomials on Herm3(O(K)) -- Trace and Characteristic Coefficients -- Action of F4(K) on PK(Herm3(O(K)) -- Some Observations -- Further Observations -- A Canonical Inclusion of U3(H(K)) into F4(K) -- 3 Stiefel, Grassmann Manifolds and Generalizations -- 3.1 Stiefel Varieties -- 3.2 Grassmannians -- 3.3 Flag Varieties -- 4 More Classical Matrix Varieties -- 4.1 i-Grassmannians and i-Stiefel Varieties -- 4.2 i-Flag Varieties -- 5 Algebraic Generalizations of Matrix Varieties -- 5.1 Varieties of Idempotent Matrices -- Stiefel Varieties -- Tangent to Stiefel Varieties -- Normal to Stiefel varieties -- Grassmann Varieties -- The Stiefel Map. ; Grassmannians as Homogeneous Spaces -- Tangent and Normal to Grassmannian Varieties -- Grassmann Varieties over K-Octonions -- Particular Cases -- A Canonical Decomposition for Matrices in Herm3(O(K)) -- Properties -- A Polynomial Inclusion S8(K) -3mu(F4(K))E11 -- Flag Varieties -- Stiefel Maps over Flag Varieties -- Flag Varieties as Homogeneous Spaces -- i-Grassmann and i-Stiefel Varieties -- i-Stiefel Maps -- i-Grassmannians as Homogeneous Spaces -- i-Flag Varieties -- i-Stiefel Maps -- i-Flags Varieties as Homogeneous Spaces -- The Shuffle Product -- Idempotent Maps Representing the Tangent Bundles to (A), V(A), Idem(A) and G(A) -- 5.2 Atlas on Varieties of Matrices -- Some Zariski Closed Subsets of M(A) -- Atlas in Idem-,r(A) -- Another Atlas in Idem-,r(A) -- Atlas in G-,r(A) -- Another Atlas in G-,r(A) -- Hermitian Metric on G(C(K)) -- 6 Curvature, Geodesics and Distance on Matrix Varieties -- 6.1 The Stiefel Submersion -- 6.2 Curvatures -- 6.3 Ricci Tensor and Einstein Structures -- 6.4 Geodesics of G(A) and Idem(A) -- 6.5 Volume of Gn,r(A) -- 6.6 Riemannian Geometry of the Cayley Plane -- 6.7 Ricci Tensor and Einstein Structure of the Cayley Plane -- 6.8 Volume of the Cayley Plane -- A Definitions and Notations -- A.1 Multiplication Table in O(K) -- A.2 Matrices -- A.3 A,m,n-Operations -- A.4 Hermitian and Skew-Hermitian Matrices -- A.5 Trace -- A.6 Inner Product on M(A) -- A.7 Rank of a Matrix -- A.8 Groups of Matrices -- A.9 Classical Notations -- A.10 Group Monomorphisms -- A.11 Stiefel Varieties -- A.12 Non-compact Stiefel Varieties -- A.13 Grassmann Varieties (Classical) -- A.14 Grassmann Varieties -- A.15 Stiefel Maps -- A.16 Stiefel Varieties as Homogenous Spaces -- A.17 Grassmann Varieties as Homogenous Spaces -- A.18 Flag Varieties (Classical) -- A.19 Flag Varieties -- A.20 Stiefel Maps Over Flag Varieties. ; A.21 i-Grassmann Varieties (Classical) -- A.22 i-Grassmann Varieties -- A.23 i-Stiefel Varieties -- A.24 i-Stiefel Maps -- A.25 i-Stiefel Varieties as Homogenous Spaces -- A.26 i-Grassmann Varieties as Homogenous Spaces -- A.27 i-Flag Varieties -- A.28 Dimensions -- A.29 Tangents and Normals -- References -- Index.
• Sommario/riassunto: This monograph deals with matrix manifolds, i.e., manifolds for which there is a natural representation of their elements as matrix arrays. Classical matrix manifolds (Stiefel, Grassmann and flag manifolds) are studied in a more general setting. It provides tools to investigate matrix varieties over Pythagorean formally real fields. The presentation of the book is reasonably self-contained. It contains a number of nontrivial results on matrix manifolds useful for people working not only in differential geometry and Riemannian geometry but in other areas of mathematics as well. It is also designed to be readable by a graduate student who has taken introductory courses in algebraic and differential geometry.
• Titolo autorizzato: Grassmann and Stiefel Varieties over Composition Algebras
• ISBN: 3-031-36405-8
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910739476003321
• Lo trovi qui: Univ. Federico II
• Serie: RSME Springer Series, . 2509-8896 ; ; 9