Grassmann and Stiefel Varieties over Composition Algebras

ISBN-13: 9783031364044 / Angielski / Twarda / 2023

Marek Golasiński;Francisco Gómez Ruiz

Opis

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.

Spis treści

Chapter 1 Algebraic preliminaries

The main goal of this chapter is to introduce some notations and

terminologies. We assume that the reader is more or less familiar

with the basic concepts of algebraic geometry and linear algebra.

 

Chapter 2 Exceptional groups $G_2(K)$ and $F_4(K)$

This chapter takes up the systematic study of a generalization

of the exceptional compact Lie groups $G_2$ and $F_4$ on groups $G_2(K)$

and $F_4(K)$ provided $K$ is Pythagorean formally real field. The main result stated in Theorem 2.48 says that any

Hermitian $3 \times 3$-matrix $A \in \mbox_3(\mathbb(K))$ can

be transformed to a diagonal form by some element of $F_4(K)$.

 

Chapter 3 Stiefel, Grassmann manifolds and generalizations

In this chapter we investigate and prove some properties of the

classical manifolds of Stiefel, Grassmann and flag manifolds.

All along this chapter $\mathcal$ denotes the field of

reals, $\mathbb$, the field of complex

numbers, $\mathbb$, the skew field of

quaternions, $\mathbb$ and, except if otherwise

said the octonion division algebra, $\mathbb$.

 

Chapter 4 More classical matrix varieties

In this chapter we generalize Stiefel, Grassmann and flag

manifolds, defined in Chapter 3, to what we call

here i-Stiefel, i-Grassmann and i-flag manifolds.

This “i” comes from idempotent. Those manifolds do not seem

to have being enough studied in the literature. In particular, they

do not have even a name. As in Chapter 2,$ \mathcal$ denotes the

field of reals, $\mathbb$, the field of complex numbers, $\mathbb$, theskew field of quaternions, $\mathbb$ and,

occasionally, the octonion division algebra $\mathbb$.

 

Chapter 5 Algebraic generalizations of matrix varieties

We use Chapters 1 and 2 to define and extend results of

Chapters 3 and 4 to matrix varieties over more general division

algebras.

That includes extending the classical definitions of Riemannian,

Hermitian and symplectic manifolds.

All along this chapter $K$ is a formally real

Pythagorean field and $\mathcal$ denotes either $K$, the

complex $K$-algebra $\mathbb(K)$, the quaternion $K$-algebra $\mathbb(K)$ or the

octonion $K$-algebra $\mathbb(K)$.

 

Chapter 6 Curvature, geodesics and distance on matrix varieties

In this chapter we study more closely the Riemannian structure

of classical matrix manifolds introduced in Chapters 3 and 4.

Here, $\mathcal = \mathbb,\, \mathbb,\, \mathbb$ and

occasionally $\mathbb$.

We also extend, whenever it is possible, definitions and

results to the general case treated in Chapter 5,

where $\mathcal = K,\, \mathbb(K),\, \mathbb(K),\, \mathbb(K)$

for $K$ a Pythagorean formally real field.

 

O autorze

Marek Golasiński is a Professor at the Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn (Poland) since 2012. He was previously Associate Professor at the Faculty of Mathematics and Computer Science, Nicolaus Copernicus University in Toruń (Poland) from 1971-2011. He was awarded the degrees of Ph.D. (1978) and Habilitation (2004), both in Algebraic Topology from the Faculty of Mathematics and Computer Science, Nicolaus Copernicus University in Toruń (Poland). He has written a previous book (with Juno Mukai) on Gottlieb and Whitehead Center Groups of Spheres, Projective and Moore Spaces.

Francisco Gómez Ruiz studied mathematics at the University of Barcelona. In 1978 received his doctorate at the University of Toronto (Stephen Halperin was his advisor). After 2 years at the department of mathematics of the Autonomous University of Barcelona and one year at the University of Cantabria, he has been 33 years professor at the department of algebra, geometry and topology of the University of Malaga. He has published over 30 research articles and 3 books.