Grassmann and Stiefel Varieties over Composition Algebras (RSME Springer Series, 9)

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