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

Format:

Book

Author/Creator:

Golasiński, Marek.

Contributor:

Gómez Ruiz, Francisco.

Series:

RSME Springer Series, 2509-8896 ; 9

Language:

English

Subjects (All):

Geometry, Differential.

Differential Geometry.

Local Subjects:

Differential Geometry.

Physical Description:

1 online resource (342 pages)

Edition:

1st ed. 2023.

Place of Publication:

Cham : Springer Nature Switzerland : Imprint: Springer, 2023.

Summary:

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.

Contents:

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))

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-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.

Other Format:

Print version: Golasiński, Marek Grassmann and Stiefel Varieties over Composition Algebras

ISBN:

3-031-36405-8