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)