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This book is a comprehensive treatment of the general (algebraic) theory of symmetric domains. Originally published in 1981. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
This book is a comprehensive treatment of the general (algebraic) theory of symmetric domains. Originally published in 1981. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Introduction. Holomorphic maps in banach spaces. Banach manifolds. Symmetric banach manifolds -- Jordan and Lie algebraic structures. Jordan algebras. Jordan triple systems. Lie algebras and Tits-Kantor-Koecher construction. Jordan and Lie structures in banach spaces. Cartan factors -- Bounded symmetric domains. Algebraic structures of symmetric manifolds. Realisation of bounded symmetric domains. Rank of a bounded symmetric domain. Boundary structures. Invariant metrics, Schwarz Lemma and dynamics. Siegel domains. Holomorphic homogeneous regular domains. Classification -- Function theory. The class S. Bloch constant and bloch maps. Banach spaces of bloch functions. Composition operators.
Explores the basic theory of quantum bounded symmetric domains. The area became active in the late 1990s at a junction of noncommutative complex analysis and extensively developing theory of quantum groups. In a surprising advance of the theory of quantum bounded symmetric domains, it turned out that many classical problems admit elegant quantum analogs. Some of those are expounded in the book.
A number of important topics in complex analysis and geometry are covered in this excellent introductory text. Written by experts in the subject, each chapter unfolds from the basics to the more complex. The exposition is rapid-paced and efficient, without compromising proofs and examples that enable the reader to grasp the essentials. The most basic type of domain examined is the bounded symmetric domain, originally described and classified by Cartan and Harish- Chandra. Two of the five parts of the text deal with these domains: one introduces the subject through the theory of semisimple Lie algebras (Koranyi), and the other through Jordan algebras and triple systems (Roos). Larger classes of domains and spaces are furnished by the pseudo-Hermitian symmetric spaces and related R-spaces. These classes are covered via a study of their geometry and a presentation and classification of their Lie algebraic theory (Kaneyuki). In the fourth part of the book, the heat kernels of the symmetric spaces belonging to the classical Lie groups are determined (Lu). Explicit computations are made for each case, giving precise results and complementing the more abstract and general methods presented. Also explored are recent developments in the field, in particular, the study of complex semigroups which generalize complex tube domains and function spaces on them (Faraut). This volume will be useful as a graduate text for students of Lie group theory with connections to complex analysis, or as a self-study resource for newcomers to the field. Readers will reach the frontiers of the subject in a considerably shorter time than with existing texts.
This monograph explicitly determines the "orbit structure" of all irreducible hermitian symmetric (IHS) spaces in a unified way by means of Lie algebra calculations, using J. Tits' models of the Lie algebras [script]e6 and [script]e7 in the two "exceptional" cases. An introduction to the theory of hermitian symmetric spaces is included, along with an elementary exposition of the facts from nonassociative algebra needed to understand and use Tits' constructions of all the complex exceptional simple Lie algebras and their real forms
Mumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book is the first comprehensive exploration of Mumford-Tate groups and domains. Containing basic theory and a wealth of new views and results, it will become an essential resource for graduate students and researchers. Although Mumford-Tate groups can be defined for general structures, their theory and use to date has mainly been in the classical case of abelian varieties. While the book does examine this area, it focuses on the nonclassical case. The general theory turns out to be very rich, such as in the unexpected connections of finite dimensional and infinite dimensional representation theory of real, semisimple Lie groups. The authors give the complete classification of Hodge representations, a topic that should become a standard in the finite-dimensional representation theory of noncompact, real, semisimple Lie groups. They also indicate that in the future, a connection seems ready to be made between Lie groups that admit discrete series representations and the study of automorphic cohomology on quotients of Mumford-Tate domains by arithmetic groups. Bringing together complex geometry, representation theory, and arithmetic, this book opens up a fresh perspective on an important subject.
This book is crafted to address the diverse and intricate topics of Abstract Algebra, tailored specifically for students preparing for the CSIR NET(JRF) Mathematical Sciences examination. Each chapter methodically explores essential algebraic structures and theories crucial for a deep understanding of modern algebra. Chapter 1 delves into Group Theory, beginning with foundational concepts of groups and subgroups, and advancing to normal subgroups, quotient groups, and homomorphisms. Key topics include Cayley’s Theorem, class equations, and Sylow theorems, providing a comprehensive overview of group structure and classification. Chapter 2 shifts focus to Ring Theory, exploring rings, subrings, and ideals. It discusses prime and maximal ideals, quotient rings, and various types of domains such as Unique Factorization Domains (UFD), Principal Ideal Domains (PID), and Euclidean Domains, highlighting their properties and interrelationships. Chapter 3 addresses Polynomial Rings, emphasizing their structure and irreducibility criteria. This chapter provides tools for understanding polynomial behavior and factorization. Chapter 4 introduces Field Theory, covering fields, their extensions, and the structures of finite fields. This foundational knowledge sets the stage for advanced topics. Chapter 5 presents Galois Theory, exploring field extensions, automorphisms, and the solvability of equations by radicals. The chapter connects these concepts to broader applications and theoretical implications. This book aims to provide a clear, structured approach to these topics, equipping students with the theoretical insights and problem-solving skills needed for success in the CSIR NET(JRF) examination.