LEADER 06763nam 22006135 450 001 9910739476003321 005 20240710090249.0 010 $a3-031-36405-8 024 7 $a10.1007/978-3-031-36405-1 035 $a(MiAaPQ)EBC30703002 035 $a(Au-PeEL)EBL30703002 035 $a(DE-He213)978-3-031-36405-1 035 $a(PPN)272266469 035 $a(CKB)27991710400041 035 $a(EXLCZ)9927991710400041 100 $a20230816d2023 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aGrassmann and Stiefel Varieties over Composition Algebras /$fby Marek Golasi?ski, Francisco Gómez Ruiz 205 $a1st ed. 2023. 210 1$aCham :$cSpringer Nature Switzerland :$cImprint: Springer,$d2023. 215 $a1 online resource (342 pages) 225 1 $aRSME Springer Series,$x2509-8896 ;$v9 311 08$aPrint version: Golasi?ski, Marek Grassmann and Stiefel Varieties over Composition Algebras Cham : Springer International Publishing AG,c2023 9783031364044 327 $aIntro -- 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. 327 $aGrassmannians 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. 327 $aA.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. 330 $aThis monograph deals with matrix manifolds, i.e., manifolds for which there is a natural representation of their elements as matrix arrays. Classical matrix manifolds (Stiefel, Grassmann and flag manifolds) are studied in a more general setting. It provides tools to investigate matrix varieties over Pythagorean formally real fields. The presentation of the book is reasonably self-contained. It contains a number of nontrivial results on matrix manifolds useful for people working not only in differential geometry and Riemannian geometry but in other areas of mathematics as well. It is also designed to be readable by a graduate student who has taken introductory courses in algebraic and differential geometry. 410 0$aRSME Springer Series,$x2509-8896 ;$v9 606 $aGeometry, Differential 606 $aDifferential Geometry 606 $aVarietats de Grassmann$2thub 606 $aVarietats de Stiefel$2thub 606 $aVarietats algebraiques$2thub 608 $aLlibres electrònics$2thub 615 0$aGeometry, Differential. 615 14$aDifferential Geometry. 615 7$aVarietats de Grassmann 615 7$aVarietats de Stiefel 615 7$aVarietats algebraiques 676 $a514.34 676 $a514.34 700 $aGolasi?ski$b Marek$0721227 701 $aGómez Ruiz$b Francisco$01424192 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910739476003321 996 $aGrassmann and Stiefel Varieties over Composition Algebras$93553230 997 $aUNINA