Several Complex Variables With Connections To Algebraic Geometry And Lie Groups

Author : Joseph L. Taylor
Publisher : American Mathematical Soc.
Page : 530 pages
File Size : 45,8 Mb
Release : 2002
Category : Functions of several complex variables
ISBN : 9780821831786

Get Book

Several Complex Variables with Connections to Algebraic Geometry and Lie Groups by Joseph L. Taylor Pdf

This text presents an integrated development of core material from several complex variables and complex algebraic geometry, leading to proofs of Serre's celebrated GAGA theorems relating the two subjects, and including applications to the representation theory of complex semisimple Lie groups. It includes a thorough treatment of the local theory using the tools of commutative algebra, an extensive development of sheaf theory and the theory of coherent analytic and algebraicsheaves, proofs of the main vanishing theorems for these categories of sheaves, and a complete proof of the finite dimensionality of the cohomology of coherent sheaves on compact varieties. The vanishing theorems have a wide variety of applications and these are covered in detail. Of particular interest arethe last three chapters, which are devoted to applications of the preceding material to the study of the structure theory and representation theory of complex semisimple Lie groups. Included are introductions to harmonic analysis, the Peter-Weyl theorem, Lie theory and the structure of Lie algebras, semisimple Lie algebras and their representations, algebraic groups and the structure of complex semisimple Lie groups. All of this culminates in Milicic's proof of the Borel-Weil-Bott theorem,which makes extensive use of the material developed earlier in the text. There are numerous examples and exercises in each chapter. This modern treatment of a classic point of view would be an excellent text for a graduate course on several complex variables, as well as a useful reference for theexpert.

Author : Wolfgang Ebeling
Publisher : American Mathematical Soc.
Page : 334 pages
File Size : 46,9 Mb
Release : 2007
Category : Functions of several complex variables
ISBN : 9780821833193

Get Book

Functions of Several Complex Variables and Their Singularities by Wolfgang Ebeling Pdf

The book provides an introduction to the theory of functions of several complex variables and their singularities, with special emphasis on topological aspects. The topics include Riemann surfaces, holomorphic functions of several variables, classification and deformation of singularities, fundamentals of differential topology, and the topology of singularities. The aim of the book is to guide the reader from the fundamentals to more advanced topics of recent research. All the necessary prerequisites are specified and carefully explained. The general theory is illustrated by various examples and applications.

Author : Alfonso Di Bartolo
Publisher : Springer Science & Business Media
Page : 223 pages
File Size : 45,7 Mb
Release : 2008-04-17
Category : Mathematics
ISBN : 9783540785835

Get Book

Algebraic Groups and Lie Groups with Few Factors by Alfonso Di Bartolo Pdf

This volume treats algebraic groups from a group theoretical point of view and compares the results with the analogous issues in the theory of Lie groups. It examines a classification of algebraic groups and Lie groups having only few subgroups.

Author : Eric Bedford
Publisher : American Mathematical Soc.
Page : 368 pages
File Size : 43,5 Mb
Release : 1991
Category : Mathematics
ISBN : 9780821814918

Get Book

Several Complex Variables and Complex Geometry, Part III by Eric Bedford Pdf

Author : Robert C. Gunning,Hugo Rossi
Publisher : American Mathematical Society
Page : 334 pages
File Size : 46,6 Mb
Release : 2022-08-25
Category : Mathematics
ISBN : 9781470470661

Get Book

Analytic Functions of Several Complex Variables by Robert C. Gunning,Hugo Rossi Pdf

The theory of analytic functions of several complex variables enjoyed a period of remarkable development in the middle part of the twentieth century. After initial successes by Poincaré and others in the late 19th and early 20th centuries, the theory encountered obstacles that prevented it from growing quickly into an analogue of the theory for functions of one complex variable. Beginning in the 1930s, initially through the work of Oka, then H. Cartan, and continuing with the work of Grauert, Remmert, and others, new tools were introduced into the theory of several complex variables that resolved many of the open problems and fundamentally changed the landscape of the subject. These tools included a central role for sheaf theory and increased uses of topology and algebra. The book by Gunning and Rossi was the first of the modern era of the theory of several complex variables, which is distinguished by the use of these methods. The intention of Gunning and Rossi's book is to provide an extensive introduction to the Oka-Cartan theory and some of its applications, and to the general theory of analytic spaces. Fundamental concepts and techniques are discussed as early as possible. The first chapter covers material suitable for a one-semester graduate course, presenting many of the central problems and techniques, often in special cases. The later chapters give more detailed expositions of sheaf theory for analytic functions and the theory of complex analytic spaces. Since its original publication, this book has become a classic resource for the modern approach to functions of several complex variables and the theory of analytic spaces. Further information about this book, including updates, can be found at the following URL: www.ams.org/publications/authors/books/postpub/chel-368.

Author : Ngaiming Mok
Publisher : Unknown
Page : 434 pages
File Size : 51,8 Mb
Release : 2001
Category : Algebra
ISBN : CORNELL:31924098353497

Get Book

Aspects of Mathematics by Ngaiming Mok Pdf

Author : Anthony W. Knapp
Publisher : Princeton University Press
Page : 522 pages
File Size : 51,5 Mb
Release : 1988-05-21
Category : Mathematics
ISBN : 9780691084985

Get Book

Lie Groups, Lie Algebras, and Cohomology by Anthony W. Knapp Pdf

This book starts with the elementary theory of Lie groups of matrices and arrives at the definition, elementary properties, and first applications of cohomological induction, which is a recently discovered algebraic construction of group representations. Along the way it develops the computational techniques that are so important in handling Lie groups. The book is based on a one-semester course given at the State University of New York, Stony Brook in fall, 1986 to an audience having little or no background in Lie groups but interested in seeing connections among algebra, geometry, and Lie theory. These notes develop what is needed beyond a first graduate course in algebra in order to appreciate cohomological induction and to see its first consequences. Along the way one is able to study homological algebra with a significant application in mind; consequently one sees just what results in that subject are fundamental and what results are minor.

Author : L. Pukanszky
Publisher : American Mathematical Soc.
Page : 149 pages
File Size : 42,6 Mb
Release : 1999
Category : Characters of groups
ISBN : 9780821810880

Get Book

Characters of Connected Lie Groups by L. Pukanszky Pdf

This book adds to the great body of research that extends back to A. Weil and E. P. Wigner on the unitary representations of locally compact groups and their characters, i.e. the interplay between classical group theory and modern analysis. The groups studied here are the connected Lie groups of general type (not necessarily nilpotent or semisimple). Final results reflect Kirillov's orbit method; in the case of groups that may be non-algebraic or non-type I, the method requires considerable sophistication. Methods used range from deep functional analysis (the theory of $C*$-algebras, factors from F. J. Murray and J. von Neumann, and measure theory) to differential geometry (Lie groups and Hamiltonian actions). This book presents for the first time a systematic and concise compilation of proofs previously dispersed throughout the literature. The result is an impressive example of the deepness of Pukanszky's work.

Author : Robert Everist Greene,Steven George Krantz
Publisher : American Mathematical Soc.
Page : 536 pages
File Size : 48,8 Mb
Release : 2006
Category : Mathematics
ISBN : 0821839624

Get Book

Function Theory of One Complex Variable by Robert Everist Greene,Steven George Krantz Pdf

Complex analysis is one of the most central subjects in mathematics. It is compelling and rich in its own right, but it is also remarkably useful in a wide variety of other mathematical subjects, both pure and applied. This book is different from others in that it treats complex variables as a direct development from multivariable real calculus. As each new idea is introduced, it is related to the corresponding idea from real analysis and calculus. The text is rich with examples andexercises that illustrate this point. The authors have systematically separated the analysis from the topology, as can be seen in their proof of the Cauchy theorem. The book concludes with several chapters on special topics, including full treatments of special functions, the prime number theorem,and the Bergman kernel. The authors also treat $Hp$ spaces and Painleve's theorem on smoothness to the boundary for conformal maps. This book is a text for a first-year graduate course in complex analysis. It is an engaging and modern introduction to the subject, reflecting the authors' expertise both as mathematicians and as expositors.

Author : Arkadij L. Onishchik,Ernest B. Vinberg
Publisher : Springer Science & Business Media
Page : 347 pages
File Size : 40,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642743344

Get Book

Lie Groups and Algebraic Groups by Arkadij L. Onishchik,Ernest B. Vinberg Pdf

This book is based on the notes of the authors' seminar on algebraic and Lie groups held at the Department of Mechanics and Mathematics of Moscow University in 1967/68. Our guiding idea was to present in the most economic way the theory of semisimple Lie groups on the basis of the theory of algebraic groups. Our main sources were A. Borel's paper [34], C. ChevalIey's seminar [14], seminar "Sophus Lie" [15] and monographs by C. Chevalley [4], N. Jacobson [9] and J-P. Serre [16, 17]. In preparing this book we have completely rearranged these notes and added two new chapters: "Lie groups" and "Real semisimple Lie groups". Several traditional topics of Lie algebra theory, however, are left entirely disregarded, e.g. universal enveloping algebras, characters of linear representations and (co)homology of Lie algebras. A distinctive feature of this book is that almost all the material is presented as a sequence of problems, as it had been in the first draft of the seminar's notes. We believe that solving these problems may help the reader to feel the seminar's atmosphere and master the theory. Nevertheless, all the non-trivial ideas, and sometimes solutions, are contained in hints given at the end of each section. The proofs of certain theorems, which we consider more difficult, are given directly in the main text. The book also contains exercises, the majority of which are an essential complement to the main contents.

Author : Anthony W. Knapp
Publisher : Princeton University Press
Page : 526 pages
File Size : 55,7 Mb
Release : 2021-01-12
Category : Mathematics
ISBN : 9780691223803

Get Book

Lie Groups, Lie Algebras, and Cohomology. (MN-34), Volume 34 by Anthony W. Knapp Pdf

This book starts with the elementary theory of Lie groups of matrices and arrives at the definition, elementary properties, and first applications of cohomological induction, which is a recently discovered algebraic construction of group representations. Along the way it develops the computational techniques that are so important in handling Lie groups. The book is based on a one-semester course given at the State University of New York, Stony Brook in fall, 1986 to an audience having little or no background in Lie groups but interested in seeing connections among algebra, geometry, and Lie theory. These notes develop what is needed beyond a first graduate course in algebra in order to appreciate cohomological induction and to see its first consequences. Along the way one is able to study homological algebra with a significant application in mind; consequently one sees just what results in that subject are fundamental and what results are minor.

Author : James E. Humphreys
Publisher : American Mathematical Soc.
Page : 320 pages
File Size : 51,7 Mb
Release : 2024-05-20
Category : Categories (Mathematics)
ISBN : 0821872524

Get Book

Representations of Semisimple Lie Algebras in the BGG Category O by James E. Humphreys Pdf

This is the first textbook treatment of work leading to the landmark 1979 Kazhdan-Lusztig Conjecture on characters of simple highest weight modules for a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb {C}$. The setting is the module category $\mathscr {O}$ introduced by Bernstein-Gelfand-Gelfand, which includes all highest weight modules for $\mathfrak{g}$ such as Verma modules and finite dimensional simple modules. Analogues of this category have become influential in many areas of representation theory. Part I can be used as a text for independent study or for a mid-level one semester graduate course; it includes exercises and examples. The main prerequisite is familiarity with the structure theory of $\mathfrak{g}$. Basic techniques in category $\mathscr {O}$ such as BGG Reciprocity and Jantzen's translation functors are developed, culminating in an overview of the proof of the Kazhdan-Lusztig Conjecture (due to Beilinson-Bernstein and Brylinski-Kashiwara). The full proof however is beyond the scope of this book, requiring deep geometric methods: $D$-modules and perverse sheaves on the flag variety. Part II introduces closely related topics important in current research: parabolic category $\mathscr {O}$, projective functors, tilting modules, twisting and completion functors, and Koszul duality theorem of Beilinson-Ginzburg-Soergel.

Author : Steven H. Weintraub
Publisher : American Mathematical Soc.
Page : 226 pages
File Size : 44,6 Mb
Release : 2003
Category : Finite groups
ISBN : 9780821832226

Get Book

Representation Theory of Finite Groups: Algebra and Arithmetic by Steven H. Weintraub Pdf

``We explore widely in the valley of ordinary representations, and we take the reader over the mountain pass leading to the valley of modular representations, to a point from which (s)he can survey this valley, but we do not attempt to widely explore it. We hope the reader will be sufficiently fascinated by the scenery to further explore both valleys on his/her own.'' --from the Preface Representation theory plays important roles in geometry, algebra, analysis, and mathematical physics. In particular, representation theory has been one of the great tools in the study and classification of finite groups. There are some beautiful results that come from representation theory: Frobenius's Theorem, Burnside's Theorem, Artin's Theorem, Brauer's Theorem--all of which are covered in this textbook. Some seem uninspiring at first, but prove to be quite useful. Others are clearly deep from the outset. And when a group (finite or otherwise) acts on something else (as a set of symmetries, for example), one ends up with a natural representation of the group. This book is an introduction to the representation theory of finite groups from an algebraic point of view, regarding representations as modules over the group algebra. The approach is to develop the requisite algebra in reasonable generality and then to specialize it to the case of group representations. Methods and results particular to group representations, such as characters and induced representations, are developed in depth. Arithmetic comes into play when considering the field of definition of a representation, especially for subfields of the complex numbers. The book has an extensive development of the semisimple case, where the characteristic of the field is zero or is prime to the order of the group, and builds the foundations of the modular case, where the characteristic of the field divides the order of the group. The book assumes only the material of a standard graduate course in algebra. It is suitable as a text for a year-long graduate course. The subject is of interest to students of algebra, number theory and algebraic geometry. The systematic treatment presented here makes the book also valuable as a reference.

Author : Louis Halle Rowen
Publisher : American Mathematical Soc.
Page : 680 pages
File Size : 46,8 Mb
Release : 2008
Category : Mathematics
ISBN : 0821884085

Get Book

Graduate Algebra by Louis Halle Rowen Pdf

"This book is an expanded text for a graduate course in commutative algebra, focusing on the algebraic underpinnings of algebraic geometry and of number theory. Accordingly, the theory of affine algebras is featured, treated both directly and via the theory of Noetherian and Artinian modules, and the theory of graded algebras is included to provide the foundation for projective varieties." --Book Jacket.

Author : Donu Arapura
Publisher : Springer Science & Business Media
Page : 329 pages
File Size : 41,5 Mb
Release : 2012-02-15
Category : Mathematics
ISBN : 9781461418092

Get Book

Algebraic Geometry over the Complex Numbers by Donu Arapura Pdf

This is a relatively fast paced graduate level introduction to complex algebraic geometry, from the basics to the frontier of the subject. It covers sheaf theory, cohomology, some Hodge theory, as well as some of the more algebraic aspects of algebraic geometry. The author frequently refers the reader if the treatment of a certain topic is readily available elsewhere but goes into considerable detail on topics for which his treatment puts a twist or a more transparent viewpoint. His cases of exploration and are chosen very carefully and deliberately. The textbook achieves its purpose of taking new students of complex algebraic geometry through this a deep yet broad introduction to a vast subject, eventually bringing them to the forefront of the topic via a non-intimidating style.