| classical groups and geometric algebra: Classical Groups and Geometric Algebra Larry C. Grove, ''Classical groups'', named so by Hermann Weyl, are groups of matrices or quotients of matrix groups by small normal subgroups. Thus the story begins, as Weyl suggested, with ''Her All-embracing Majesty'', the general linear group $GL n(V)$ of all invertible linear transformations of a vector space $V$ over a field $F$. All further groups discussed are either subgroups of $GL n(V)$ or closely related quotient groups. Most of the classical groups consist of invertible linear transformations that respect a bilinear form having some geometric significance, e.g., a quadratic form, a symplectic form, etc. Accordingly, the author develops the required geometric notions, albeit from an algebraic point of view, as the end results should apply to vector spaces over more-or-less arbitrary fields, finite or infinite. The classical groups have proved to be important in a wide variety of venues, ranging from physics to geometry and far beyond. In recent years, they have played a prominent role in the classification of the finite simple groups. This text provides a single source for the basic facts about the classical groups and also includes the required geometrical background information from the first principles. It is intended for graduate students who have completed standard courses in linear algebra and abstract algebra. The author, L. C. Grove, is a well-known expert who has published extensively in the subject area. |
| classical groups and geometric algebra: Groups and Characters Larry C. Grove, 2011-09-26 An authoritative, full-year course on both group theory and ordinary character theory--essential tools for mathematics and the physical sciences One of the few treatments available combining both group theory and character theory, Groups and Characters is an effective general textbook on these two fundamentally connected subjects. Presuming only a basic knowledge of abstract algebra as in a first-year graduate course, the text opens with a review of background material and then guides readers carefully through several of the most important aspects of groups and characters, concentrating mainly on finite groups. Challenging yet accessible, Groups and Characters features: * An extensive collection of examples surveying many different types of groups, including Sylow subgroups of symmetric groups, affine groups of fields, the Mathieu groups, and symplectic groups * A thorough, easy-to-follow discussion of Polya-Redfield enumeration, with applications to combinatorics * Inclusive explorations of the transfer function and normal complements, induction and restriction of characters, Clifford theory, characters of symmetric and alternating groups, Frobenius groups, and the Schur index * Illuminating accounts of several computational aspects of group theory, such as the Schreier-Sims algorithm, Todd-Coxeter coset enumeration, and algorithms for generating character tables As valuable as Groups and Characters will prove as a textbook for mathematicians, it has broader applications. With chapters suitable for use as independent review units, along with a full bibliography and index, it will be a dependable general reference for chemists, physicists, and crystallographers. |
| classical groups and geometric algebra: Clifford Algebras and the Classical Groups Ian R. Porteous, 1995-10-05 The Clifford algebras of real quadratic forms and their complexifications are studied here in detail, and those parts which are immediately relevant to theoretical physics are seen in the proper broad context. Central to the work is the classification of the conjugation and reversion anti-involutions that arise naturally in the theory. It is of interest that all the classical groups play essential roles in this classification. Other features include detailed sections on conformal groups, the eight-dimensional non-associative Cayley algebra, its automorphism group, the exceptional Lie group G(subscript 2), and the triality automorphism of Spin 8. The book is designed to be suitable for the last year of an undergraduate course or the first year of a postgraduate course. |
| classical groups and geometric algebra: Classical Groups and Geometric Algebra Larry C. Grove, 2002 A graduate-level text on the classical groups: groups of matrices, or (more often) quotients of matrix groups by small normal subgroups. It pulls together into a single source the basic facts about classical groups defined over fields, together with the required geometrical background information, from first principles. The chief prerequisites are basic linear algebra and abstract algebra, including fundamentals of group theory and some Galois Theory. The author teaches at the U. of Arizona. c. Book News Inc. |
| classical groups and geometric algebra: Clifford Algebra to Geometric Calculus David Hestenes, Garret Sobczyk, 1984 Matrix algebra has been called the arithmetic of higher mathematics [Be]. We think the basis for a better arithmetic has long been available, but its versatility has hardly been appreciated, and it has not yet been integrated into the mainstream of mathematics. We refer to the system commonly called 'Clifford Algebra', though we prefer the name 'Geometric Algebra' suggested by Clifford himself. Many distinct algebraic systems have been adapted or developed to express geometric relations and describe geometric structures. Especially notable are those algebras which have been used for this purpose in physics, in particular, the system of complex numbers, the quaternions, matrix algebra, vector, tensor and spinor algebras and the algebra of differential forms. Each of these geometric algebras has some significant advantage over the others in certain applications, so no one of them provides an adequate algebraic structure for all purposes of geometry and physics. At the same time, the algebras overlap considerably, so they provide several different mathematical representations for individual geometrical or physical ideas. |
| classical groups and geometric algebra: The Subgroup Structure of the Finite Classical Groups Peter B. Kleidman, Martin W. Liebeck, 1990-04-26 With the classification of the finite simple groups complete, much work has gone into the study of maximal subgroups of almost simple groups. In this volume the authors investigate the maximal subgroups of the finite classical groups and present research into these groups as well as proving many new results. In particular, the authors develop a unified treatment of the theory of the 'geometric subgroups' of the classical groups, introduced by Aschbacher, and they answer the questions of maximality and conjugacy and obtain the precise shapes of these groups. Both authors are experts in the field and the book will be of considerable value not only to group theorists, but also to combinatorialists and geometers interested in these techniques and results. Graduate students will find it a very readable introduction to the topic and it will bring them to the very forefront of research in group theory. |
| classical groups and geometric algebra: Algebra , 1983-11-01 Algebra |
| classical groups and geometric algebra: Classical Topology and Combinatorial Group Theory John Stillwell, 2012-12-06 In recent years, many students have been introduced to topology in high school mathematics. Having met the Mobius band, the seven bridges of Konigsberg, Euler's polyhedron formula, and knots, the student is led to expect that these picturesque ideas will come to full flower in university topology courses. What a disappointment undergraduate topology proves to be! In most institutions it is either a service course for analysts, on abstract spaces, or else an introduction to homological algebra in which the only geometric activity is the completion of commutative diagrams. Pictures are kept to a minimum, and at the end the student still does nr~ understand the simplest topological facts, such as the rcason why knots exist. In my opinion, a well-balanced introduction to topology should stress its intuitive geometric aspect, while admitting the legitimate interest that analysts and algebraists have in the subject. At any rate, this is the aim of the present book. In support of this view, I have followed the historical development where practicable, since it clearly shows the influence of geometric thought at all stages. This is not to claim that topology received its main impetus from geometric recreations like the seven bridges; rather, it resulted from the l'isualization of problems from other parts of mathematics-complex analysis (Riemann), mechanics (Poincare), and group theory (Dehn). It is these connec tions to other parts of mathematics which make topology an important as well as a beautiful subject. |
| classical groups and geometric algebra: A Course in Group Theory J. F. Humphreys, 1996 Each chapter ends with a summary of the material covered and notes on the history and development of group theory. |
| classical groups and geometric algebra: Geometric Algebra for Physicists Chris Doran, Anthony Lasenby, 2007-11-22 Geometric algebra is a powerful mathematical language with applications across a range of subjects in physics and engineering. This book is a complete guide to the current state of the subject with early chapters providing a self-contained introduction to geometric algebra. Topics covered include new techniques for handling rotations in arbitrary dimensions, and the links between rotations, bivectors and the structure of the Lie groups. Following chapters extend the concept of a complex analytic function theory to arbitrary dimensions, with applications in quantum theory and electromagnetism. Later chapters cover advanced topics such as non-Euclidean geometry, quantum entanglement, and gauge theories. Applications such as black holes and cosmic strings are also explored. It can be used as a graduate text for courses on the physical applications of geometric algebra and is also suitable for researchers working in the fields of relativity and quantum theory. |
| classical groups and geometric algebra: Geometric Algebra Emil Artin, 2016-01-20 This concise classic presents advanced undergraduates and graduate students in mathematics with an overview of geometric algebra. The text originated with lecture notes from a New York University course taught by Emil Artin, one of the preeminent mathematicians of the twentieth century. The Bulletin of the American Mathematical Society praised Geometric Algebra upon its initial publication, noting that mathematicians will find on many pages ample evidence of the author's ability to penetrate a subject and to present material in a particularly elegant manner. Chapter 1 serves as reference, consisting of the proofs of certain isolated algebraic theorems. Subsequent chapters explore affine and projective geometry, symplectic and orthogonal geometry, the general linear group, and the structure of symplectic and orthogonal groups. The author offers suggestions for the use of this book, which concludes with a bibliography and index. |
| classical groups and geometric algebra: Classical Algebraic Geometry Igor V. Dolgachev, 2012-08-16 Algebraic geometry has benefited enormously from the powerful general machinery developed in the latter half of the twentieth century. The cost has been that much of the research of previous generations is in a language unintelligible to modern workers, in particular, the rich legacy of classical algebraic geometry, such as plane algebraic curves of low degree, special algebraic surfaces, theta functions, Cremona transformations, the theory of apolarity and the geometry of lines in projective spaces. The author's contemporary approach makes this legacy accessible to modern algebraic geometers and to others who are interested in applying classical results. The vast bibliography of over 600 references is complemented by an array of exercises that extend or exemplify results given in the book. |
| classical groups and geometric algebra: The Four Pillars of Geometry John Stillwell, 2005-08-09 This book is unique in that it looks at geometry from 4 different viewpoints - Euclid-style axioms, linear algebra, projective geometry, and groups and their invariants Approach makes the subject accessible to readers of all mathematical tastes, from the visual to the algebraic Abundantly supplemented with figures and exercises |
| classical groups and geometric algebra: An Introduction to Algebraic Geometry and Algebraic Groups Meinolf Geck, 2013-03-14 An accessible text introducing algebraic groups at advanced undergraduate and early graduate level, this book covers the conjugacy of Borel subgroups and maximal tori, the theory of algebraic groups with a BN-pair, Frobenius maps on affine varieties and algebraic groups, zeta functions and Lefschetz numbers for varieties over finite fields. |
| classical groups and geometric algebra: Buildings and Classical Groups Paul B. Garrett, 1997-04-01 Buildings are highly structured, geometric objects, primarily used in the finer study of the groups that act upon them. In Buildings and Classical Groups, the author develops the basic theory of buildings and BN-pairs, with a focus on the results needed to apply it to the representation theory of p-adic groups. In particular, he addresses spherical and affine buildings, and the spherical building at infinity attached to an affine building. He also covers in detail many otherwise apocryphal results. Classical matrix groups play a prominent role in this study, not only as vehicles to illustrate general results but as primary objects of interest. The author introduces and completely develops terminology and results relevant to classical groups. He also emphasizes the importance of the reflection, or Coxeter groups and develops from scratch everything about reflection groups needed for this study of buildings. In addressing the more elementary spherical constructions, the background pertaining to classical groups includes basic results about quadratic forms, alternating forms, and hermitian forms on vector spaces, plus a description of parabolic subgroups as stabilizers of flags of subspaces. The text then moves on to a detailed study of the subtler, less commonly treated affine case, where the background concerns p-adic numbers, more general discrete valuation rings, and lattices in vector spaces over ultrametric fields. Buildings and Classical Groups provides essential background material for specialists in several fields, particularly mathematicians interested in automorphic forms, representation theory, p-adic groups, number theory, algebraic groups, and Lie theory. No other available source provides such a complete and detailed treatment. |
| classical groups and geometric algebra: Geometric Group Theory Clara Löh, 2017-12-19 Inspired by classical geometry, geometric group theory has in turn provided a variety of applications to geometry, topology, group theory, number theory and graph theory. This carefully written textbook provides a rigorous introduction to this rapidly evolving field whose methods have proven to be powerful tools in neighbouring fields such as geometric topology. Geometric group theory is the study of finitely generated groups via the geometry of their associated Cayley graphs. It turns out that the essence of the geometry of such groups is captured in the key notion of quasi-isometry, a large-scale version of isometry whose invariants include growth types, curvature conditions, boundary constructions, and amenability. This book covers the foundations of quasi-geometry of groups at an advanced undergraduate level. The subject is illustrated by many elementary examples, outlooks on applications, as well as an extensive collection of exercises. |
| classical groups and geometric algebra: Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics D.H. Sattinger, O.L. Weaver, 2013-11-11 This book is intended as an introductory text on the subject of Lie groups and algebras and their role in various fields of mathematics and physics. It is written by and for researchers who are primarily analysts or physicists, not algebraists or geometers. Not that we have eschewed the algebraic and geo metric developments. But we wanted to present them in a concrete way and to show how the subject interacted with physics, geometry, and mechanics. These interactions are, of course, manifold; we have discussed many of them here-in particular, Riemannian geometry, elementary particle physics, sym metries of differential equations, completely integrable Hamiltonian systems, and spontaneous symmetry breaking. Much ofthe material we have treated is standard and widely available; but we have tried to steer a course between the descriptive approach such as found in Gilmore and Wybourne, and the abstract mathematical approach of Helgason or Jacobson. Gilmore and Wybourne address themselves to the physics community whereas Helgason and Jacobson address themselves to the mathematical community. This book is an attempt to synthesize the two points of view and address both audiences simultaneously. We wanted to present the subject in a way which is at once intuitive, geometric, applications oriented, mathematically rigorous, and accessible to students and researchers without an extensive background in physics, algebra, or geometry. |
| classical groups and geometric algebra: An Introduction to Invariants and Moduli Shigeru Mukai, 2003-09-08 Sample Text |
| classical groups and geometric algebra: The Random Matrix Theory of the Classical Compact Groups Elizabeth S. Meckes, 2019-08-01 This is the first book to provide a comprehensive overview of foundational results and recent progress in the study of random matrices from the classical compact groups, drawing on the subject's deep connections to geometry, analysis, algebra, physics, and statistics. The book sets a foundation with an introduction to the groups themselves and six different constructions of Haar measure. Classical and recent results are then presented in a digested, accessible form, including the following: results on the joint distributions of the entries; an extensive treatment of eigenvalue distributions, including the Weyl integration formula, moment formulae, and limit theorems and large deviations for the spectral measures; concentration of measure with applications both within random matrix theory and in high dimensional geometry; and results on characteristic polynomials with connections to the Riemann zeta function. This book will be a useful reference for researchers and an accessible introduction for students in related fields. |
| classical groups and geometric algebra: Topics in Groups and Geometry Tullio Ceccherini-Silberstein, Michele D'Adderio, 2022-01-01 This book provides a detailed exposition of a wide range of topics in geometric group theory, inspired by Gromov’s pivotal work in the 1980s. It includes classical theorems on nilpotent groups and solvable groups, a fundamental study of the growth of groups, a detailed look at asymptotic cones, and a discussion of related subjects including filters and ultrafilters, dimension theory, hyperbolic geometry, amenability, the Burnside problem, and random walks on groups. The results are unified under the common theme of Gromov’s theorem, namely that finitely generated groups of polynomial growth are virtually nilpotent. This beautiful result gave birth to a fascinating new area of research which is still active today. The purpose of the book is to collect these naturally related results together in one place, most of which are scattered throughout the literature, some of them appearing here in book form for the first time. In this way, the connections between these topics are revealed, providing a pleasant introduction to geometric group theory based on ideas surrounding Gromov's theorem. The book will be of interest to mature undergraduate and graduate students in mathematics who are familiar with basic group theory and topology, and who wish to learn more about geometric, analytic, and probabilistic aspects of infinite groups. |
| classical groups and geometric algebra: Geometry of Crystallographic Groups Andrzej Szczepański, 2012 Crystallographic groups are groups which act in a nice way and via isometries on some n-dimensional Euclidean space. This book gives an example of the torsion free crystallographic group with a trivial center and a trivial outer automorphism group. |
| classical groups and geometric algebra: Representations of Algebraic Groups Jens Carsten Jantzen, 2003 Gives an introduction to the general theory of representations of algebraic group schemes. This title deals with representation theory of reductive algebraic groups and includes topics such as the description of simple modules, vanishing theorems, Borel-Bott-Weil theorem and Weyl's character formula, and Schubert schemes and lne bundles on them. |
| classical groups and geometric algebra: The Classical Groups Hermann Weyl, 1946 The author discusses symmetric, full linear, orthogonal, and symplectic groups and determines their different invariants and representations. Using basic concepts from algebra, he examines the various properties of the groups. The book also covers topics such as matrix algebras, semigroups, commutators, and spinors, which are important in understanding the group-theoretic structure of quantum mechanics. |
| classical groups and geometric algebra: Representations of Finite and Compact Groups Barry Simon, 1996 This text is a comprehensive pedagogical presentation of the theory of representation of finite and compact Lie groups. It considers both the general theory and representation of specific groups. Representation theory is discussed on the following types of groups: finite groups of rotations, permutation groups, and classical compact semisimple Lie groups. Along the way, the structure theory of the compact semisimple Lie groups is exposed. This is aimed at research mathematicians and graduate students studying group theory. |
| classical groups and geometric algebra: Vertex Algebras and Algebraic Curves Edward Frenkel, David Ben-Zvi, 2004-08-25 Vertex algebras are algebraic objects that encapsulate the concept of operator product expansion from two-dimensional conformal field theory. Vertex algebras are fast becoming ubiquitous in many areas of modern mathematics, with applications to representation theory, algebraic geometry, the theory of finite groups, modular functions, topology, integrable systems, and combinatorics. This book is an introduction to the theory of vertex algebras with a particular emphasis on the relationship with the geometry of algebraic curves. The notion of a vertex algebra is introduced in a coordinate-independent way, so that vertex operators become well defined on arbitrary smooth algebraic curves, possibly equipped with additional data, such as a vector bundle. Vertex algebras then appear as the algebraic objects encoding the geometric structure of various moduli spaces associated with algebraic curves. Therefore they may be used to give a geometric interpretation of various questions of representation theory. The book contains many original results, introduces important new concepts, and brings new insights into the theory of vertex algebras. The authors have made a great effort to make the book self-contained and accessible to readers of all backgrounds. Reviewers of the first edition anticipated that it would have a long-lasting influence on this exciting field of mathematics and would be very useful for graduate students and researchers interested in the subject. This second edition, substantially improved and expanded, includes several new topics, in particular an introduction to the Beilinson-Drinfeld theory of factorization algebras and the geometric Langlands correspondence. |
| classical groups and geometric algebra: Introduction to Representation Theory Pavel I. Etingof, Oleg Golberg, Sebastian Hensel , Tiankai Liu , Alex Schwendner , Dmitry Vaintrob , Elena Yudovina , 2011 Very roughly speaking, representation theory studies symmetry in linear spaces. It is a beautiful mathematical subject which has many applications, ranging from number theory and combinatorics to geometry, probability theory, quantum mechanics, and quantum field theory. The goal of this book is to give a ``holistic'' introduction to representation theory, presenting it as a unified subject which studies representations of associative algebras and treating the representation theories of groups, Lie algebras, and quivers as special cases. Using this approach, the book covers a number of standard topics in the representation theories of these structures. Theoretical material in the book is supplemented by many problems and exercises which touch upon a lot of additional topics; the more difficult exercises are provided with hints. The book is designed as a textbook for advanced undergraduate and beginning graduate students. It should be accessible to students with a strong background in linear algebra and a basic knowledge of abstract algebra. |
| classical groups and geometric algebra: Geometric Differentiation I. R. Porteous, 2001-12-13 This is a revised version of the popular Geometric Differentiation, first edition. |
| classical groups and geometric algebra: A Course in Algebra Ėrnest Borisovich Vinberg, 2003-04-10 Presents modern algebra. This book includes such topics as affine and projective spaces, tensor algebra, Galois theory, Lie groups, and associative algebras and their representations. It is suitable for independent study for advanced undergraduates and graduate students. |
| classical groups and geometric algebra: Theory of Finite Simple Groups Gerhard Michler, 2006-09-21 The first representation theoretic and algorithmic approach to the theory of abstract finite simple groups. |
| classical groups and geometric algebra: Group Cohomology and Algebraic Cycles Burt Totaro, 2014-06-26 This book presents a coherent suite of computational tools for the study of group cohomology algebraic cycles. |
| classical groups and geometric algebra: New Foundations in Mathematics Garret Sobczyk, 2012-10-26 The first book of its kind, New Foundations in Mathematics: The Geometric Concept of Number uses geometric algebra to present an innovative approach to elementary and advanced mathematics. Geometric algebra offers a simple and robust means of expressing a wide range of ideas in mathematics, physics, and engineering. In particular, geometric algebra extends the real number system to include the concept of direction, which underpins much of modern mathematics and physics. Much of the material presented has been developed from undergraduate courses taught by the author over the years in linear algebra, theory of numbers, advanced calculus and vector calculus, numerical analysis, modern abstract algebra, and differential geometry. The principal aim of this book is to present these ideas in a freshly coherent and accessible manner. New Foundations in Mathematics will be of interest to undergraduate and graduate students of mathematics and physics who are looking for a unified treatment of many important geometric ideas arising in these subjects at all levels. The material can also serve as a supplemental textbook in some or all of the areas mentioned above and as a reference book for professionals who apply mathematics to engineering and computational areas of mathematics and physics. |
| classical groups and geometric algebra: Groups, Combinatorics and Geometry Martin W. Liebeck, 1992-09-10 This volume contains a collection of papers on the subject of the classification of finite simple groups. |
| classical groups and geometric algebra: Algebras of Functions on Quantum Groups: Part I Leonid I. Korogodski, Yan S. Soibelman, 1998 The text is devoted to the study of algebras of functions on quantum groups. The book includes the theory of Poisson-Lie algebras (quasi-classical version of algebras of functions on quantum groups), a description of representations of algebras of functions and the theory of quantum Weyl groups. It can serve as a text for an introduction to the theory of quantum groups and is intended for graduate students and research mathematicians working in algebra, representation theory and mathematical physics. |
| classical groups and geometric algebra: Algebraic Geometry Joe Harris, 2013-11-11 This book succeeds brilliantly by concentrating on a number of core topics...and by treating them in a hugely rich and varied way. The author ensures that the reader will learn a large amount of classical material and perhaps more importantly, will also learn that there is no one approach to the subject. The essence lies in the range and interplay of possible approaches. The author is to be congratulated on a work of deep and enthusiastic scholarship. --MATHEMATICAL REVIEWS |
| classical groups and geometric algebra: The $K$-book Charles A. Weibel, 2013-06-13 Informally, $K$-theory is a tool for probing the structure of a mathematical object such as a ring or a topological space in terms of suitably parameterized vector spaces and producing important intrinsic invariants which are useful in the study of algebr |
| classical groups and geometric algebra: Symmetry, Representations, and Invariants Roe Goodman, Nolan R. Wallach, 2009-07-30 Symmetry is a key ingredient in many mathematical, physical, and biological theories. Using representation theory and invariant theory to analyze the symmetries that arise from group actions, and with strong emphasis on the geometry and basic theory of Lie groups and Lie algebras, Symmetry, Representations, and Invariants is a significant reworking of an earlier highly-acclaimed work by the authors. The result is a comprehensive introduction to Lie theory, representation theory, invariant theory, and algebraic groups, in a new presentation that is more accessible to students and includes a broader range of applications. The philosophy of the earlier book is retained, i.e., presenting the principal theorems of representation theory for the classical matrix groups as motivation for the general theory of reductive groups. The wealth of examples and discussion prepares the reader for the complete arguments now given in the general case. Key Features of Symmetry, Representations, and Invariants: (1) Early chapters suitable for honors undergraduate or beginning graduate courses, requiring only linear algebra, basic abstract algebra, and advanced calculus; (2) Applications to geometry (curvature tensors), topology (Jones polynomial via symmetry), and combinatorics (symmetric group and Young tableaux); (3) Self-contained chapters, appendices, comprehensive bibliography; (4) More than 350 exercises (most with detailed hints for solutions) further explore main concepts; (5) Serves as an excellent main text for a one-year course in Lie group theory; (6) Benefits physicists as well as mathematicians as a reference work. |
| classical groups and geometric algebra: Groups and Geometric Analysis Sigurdur Helgason, 2022-03-17 Group-theoretic methods have taken an increasingly prominent role in analysis. Some of this change has been due to the writings of Sigurdur Helgason. This book is an introduction to such methods on spaces with symmetry given by the action of a Lie group. The introductory chapter is a self-contained account of the analysis on surfaces of constant curvature. Later chapters cover general cases of the Radon transform, spherical functions, invariant operators, compact symmetric spaces and other topics. This book, together with its companion volume, Geometric Analysis on Symmetric Spaces (AMS Mathematical Surveys and Monographs series, vol. 39, 1994), has become the standard text for this approach to geometric analysis. Sigurdur Helgason was awarded the Steele Prize for outstanding mathematical exposition for Groups and Geometric Analysis and Differential Geometry, Lie Groups and Symmetric Spaces. |
| classical groups and geometric algebra: Representations and Invariants of the Classical Groups Roe Goodman, Nolan R. Wallach, 2000-01-13 More than half a century has passed since Weyl's 'The Classical Groups' gave a unified picture of invariant theory. This book presents an updated version of this theory together with many of the important recent developments. As a text for those new to the area, this book provides an introduction to the structure and finite-dimensional representation theory of the complex classical groups that requires only an abstract algebra course as a prerequisite. The more advanced reader will find an introduction to the structure and representations of complex reductive algebraic groups and their compact real forms. This book will also serve as a reference for the main results on tensor and polynomial invariants and the finite-dimensional representation theory of the classical groups. It will appeal to researchers in mathematics, statistics, physics and chemistry whose work involves symmetry groups, representation theory, invariant theory and algebraic group theory. |
| classical groups and geometric algebra: Linear Algebraic Groups T.A. Springer, 2010-10-12 The first edition of this book presented the theory of linear algebraic groups over an algebraically closed field. The second edition, thoroughly revised and expanded, extends the theory over arbitrary fields, which are not necessarily algebraically closed. It thus represents a higher aim. As in the first edition, the book includes a self-contained treatment of the prerequisites from algebraic geometry and commutative algebra, as well as basic results on reductive groups. As a result, the first part of the book can well serve as a text for an introductory graduate course on linear algebraic groups. |
| classical groups and geometric algebra: An Introduction to Symplectic Geometry Rolf Berndt, 2001 Symplectic geometry is a central topic of current research in mathematics. Indeed, symplectic methods are key ingredients in the study of dynamical systems, differential equations, algebraic geometry, topology, mathematical physics and representations of Lie groups. This book is a true introduction to symplectic geometry, assuming only a general background in analysis and familiarity with linear algebra. It starts with the basics of the geometry of symplectic vector spaces. Then, symplectic manifolds are defined and explored. In addition to the essential classic results, such as Darboux's theorem, more recent results and ideas are also included here, such as symplectic capacity and pseudoholomorphic curves. These ideas have revolutionized the subject. The main examples of symplectic manifolds are given, including the cotangent bundle, Kähler manifolds, and coadjoint orbits. Further principal ideas are carefully examined, such as Hamiltonian vector fields, the Poisson bracket, and connections with contact manifolds. Berndt describes some of the close connections between symplectic geometry and mathematical physics in the last two chapters of the book. In particular, the moment map is defined and explored, both mathematically and in its relation to physics. He also introduces symplectic reduction, which is an important tool for reducing the number of variables in a physical system and for constructing new symplectic manifolds from old. The final chapter is on quantization, which uses symplectic methods to take classical mechanics to quantum mechanics. This section includes a discussion of the Heisenberg group and the Weil (or metaplectic) representation of the symplectic group. Several appendices provide background material on vector bundles, on cohomology, and on Lie groups and Lie algebras and their representations. Berndt's presentation of symplectic geometry is a clear and concise introduction to the major methods and applications of the subject, and requires only a minimum of prerequisites. This book would be an excellent text for a graduate course or as a source for anyone who wishes to learn about symplectic geometry. |
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algebra, which we began in the first unit. The axioms split naturally into several groups which are discussed separately; namely, incidence, betweenness, separation, linear measurement, …
LANGLANDS CORRESPONDENCE L G
theory of p-adic groups and the (categorical) representation theory of loop groups. This result had been conjectured by V. Drinfeld, and it plays an important role in his and A. Beilinson’s approach …
Geometric Langlands duality and representations of alg…
Jan 24, 2004 · representations of algebraic groups over commutative rings By I. Mirkovic´ and K. Vilonen* 1. Introduction In this paper we give a geometric version of the Satake isomorphism [Sat]. As such, …
The Geometric Satake Equivalence - University of T…
The Geometric Satake Equivalence German Stefanich The classical Satake correspondence identi es the spherical Hecke algebra of a reductive group over a local ring with the representation ring of …
arXiv:1404.4000v4 [math.RT] 18 Oct 2017
The algebra K and its identification as a coideal algebra 23 5. A second geometric construction in type B 31 6. Convolution algebras from geometry of type C 35 ... being the quantum groups of classical …
GEOMETRIC SCHUR DUALITY OF CLASSICAL T…
feld–Jimbo quantum groups [Dr86] would provide the answer. However, the expec-tation for an answer being the quantum groups of classical type was somewhat diminished after Nakajima’s quiver …
Primer on Geometric Algebra - davidhestenes.net
Primer on Geometric Algebra OUTLINE I. Prolog: On optimizing the design of introductory mathematics. II. Standard algebraic tools for linear geometry PART I. Introduction to Geometric Algebra and …
Quantum Theory, Groups and Representations: An In…
Quantum Theory, Groups and Representations: An Introduction Revised and expanded version, under construction Peter Woit Department of Mathematics, Columbia University
1. New Algebraic Tools for Classical Geometry†
framework. In this and subsequent chapters we show how classical geometry fits neatly into the broader mathematical system of Geometric Algebra (GA) and its extension to a complete Geometric …
Classical Solutions of the Quantum Yang-Baxter Equat…
310 A. Weinstein and P. Xu These conditions imply that R satisfies the quantum Yang-Baxter equation, ^13^23^12 = ^12^23^13 (3) In the equations above, A: si-» si ® si is the …
Contents
7 Geometric algebra 101 7.1 GL(n) (a prototype) 102 7.2 Bilinear and hermitian forms 106 7.3 Extending isometries 111 7.4 Parabolics 114 ... This book describes the structure of the classical groups, …
Geometric Algebra: An Introduction with Application…
Cli ord to the study of classical physics. Hestenes’ success with applying Cli ord’s geometric product and his continued publications have inspired a new generation of physicists and …
Introduction to Abstract Algebra (Math 113) - Univers…
• If P and Q are two statements, then P ⇒ Q means that if P is true then Q is true. For example: x odd ⇒ x ∕= 2. We say that P implies Q. • If P ⇒ Q and Q ⇒ P then we write P ⇐⇒ Q, which should be read as …
Applied Geometric Algebra - MIT OpenCourseWare
geometry, algebra and infinitesimal analysis. The interplay of these elements has undergone a considerable change since the turn of the century. In classical physics, analysis, in particular …
Chapter 5 Classical Invariant Theory - Springer
As a special case, we obtain the classical theory of spherical harmonics. The chapter concludes with several more applications of the FFT. 5.1 Polynomial Invariants for Reductive Groups For an algebraic …
Abstract Algebra Theory and Applications - MIT Mathemat…
abstract algebra. Traditionally, these courses have covered the theoretical aspects of groups, rings, and elds. However, with the development of computing in the last several decades, …
GEOMETRIC QUANTIZATION AND REPRESENTATION TH…
7. Nilpotent groups 55 8. Background on symplectic geometry 61 9. Duistermaat-Heckman localization 66 10. The Kirillov character formula for Un 78 11. Classical mechanics 88 12. Holomorphic …
Tutorial on Geometric Calculus - davidhestenes.net
• Lie groups as Spin groups Origins of the Geometric Calculus book Though Geometric Calculus (GC) had important precursors [6], its systematic development as a unified language for mathematics …
CLASSICAL INVARIANT THEORY - IIT Bombay
1.2 Invariants 3 (g,f) → gf, gf(w):=f(g−1w) for g ∈ G, f ∈ K[W],w∈ W.This is usually called the regular representation of G on the coordinate ring. (The inverse g−1 in this definition is necessary in order to get a …
Abstract Algebra Teaching and Learning - hal.science
As a mathematical discipline, Abstract Algebra, also called modern algebra or structural algebra, denotes, by opposition to classical algebra which focuses on formal manipulation of abstract …
Lie Algebras, Algebraic Groups, and Lie Groups - J…
DRAMATIS PERSONÆ JACOBI (1804–1851). In his work on partial differential equations, he discovered the Jacobi identity. Jacobi’s work helped Lie to develop an analytic framework for his …
The Weyl Groups - 中国科学技术大学
LECTURE 27: THE WEYL GROUPS AND WEYL INTEGRATION FORMULA 1. The Weyl Groups Let Gbe a compact connected Lie group, and T ˆGa maximal torus. The normalizer of Tis N(T) = …
Classical Algebra - download.e-bookshelf.de
1.1. “Classical” and modern algebra The “environment” of algebra: Number systems Important concepts and principles in this lesson Lesson 2. Equations and Their Solutions 1. 2. The classification of …
Geometric Group Theory - uni-regensburg.de
This course provides an introduction to geometric group theory. Groups are an abstract concept from algebra, formalising the study of symmetries of various mathematical objects. What is …
Symplectic structures associated to Lie-Poisson gr…
The method of geometric quantization [9] provides a set of Poisson manifolds associated to each Lie group G. ... The theory of Lie-Poisson groups is a quasiclassical version of the theory of …
Math 249B. Root systems for split classical groups
The exceptional Lie groups were discovered by searching for groups which would realize the 5 exceptional reduced irreducible root systems. But the 4 in nite families of \classical" reduced …
arXiv:2009.03007v2 [math.RT] 12 Jan 2021
to an associative algebra H(W;q). Secondly, a ne Hecke algebras occur in the representation theory of reductive groups Gover p-adic elds. They can be isomorphic to the algebra of G-endo …
John C. Baez - Department of Mathematics
Feb 14, 2022 · symmetries and conservation laws. To compare classical and quantum versions of this theorem, we take an algebraic approach. In both classical and quantum mechanics, …
Geometric Modular Forms - mat.uab.cat
3 Reinterpreting Classical Modular Forms (on C) 7 4 Geometric Modular Forms “`a la Katz” 10 5 Higher Levels (level-N Structures) 10 ... 9 Explicit Description of the Sheaf Cohomology Groups 21 ...
A First Encounter with Classical Groups - webuser…
Week 1: The General and Special Linear Groups Lecture 1 (Introduction; the General Linear Group) 1.1Thegeneralandspeciallineargroups 1.1.1Thegenerallineargroup
The Algebraic and Geometric Theory of Quadratic Forms
K-homology groups 238 53. Euler classes and projective bundle theorem 243 54. Chern classes 247 55. Gysin and pull-back homomorphisms 250 56. K-cohomology ring of smooth schemes 257 Chapter X. …
Lectures on the Geometry of Quantization - University of …
work, one of the first to treat the connections between classical and quantum mechanics from a geometric viewpoint, is [41]. The book [29] treats further topics in symplectic geometry …
THE UNIVERSITY OF CHICAGO SYMMETRY AND …
Mar 4, 2010 · classical and geometric complexity theory. The Geometric Complexity Theory program is aimed at resolving central questions in complexity such as P versus NP using techniques …
GEOMETRIC BANACH PROPERTY (T) FOR METRI…
May 6, 2025 · geometric property (T)achieves a relavantformal parallel with the classical property(T) for groups. Later, J. Winkel generalized geometric property (T) to non-discrete spaces in [Win21], …
INTERACTIONS BETWEEN LIE THEORY AND ALGEBRAIC …
Lie algebra over a eld kof characteristic zero. Then we have the Poincar e-Birkho -Witt (PBW) isomorphism I PBW: Sg !Ug (2.1) of g-modules, where Sg is the symmetric algebra of g and Ug is the …
Tutorial on Geometric Calculus - ICDST
• Lie groups as Spin groups Origins of the Geometric Calculus book Though Geometric Calculus (GC) had important precursors [6], its systematic development as a unified language for mathematics …
SEMISIMPLE LIE ALGEBRAS AND THE CHEVALLEY GR…
for a semisimple Lie algebra and prove some properties. Section 12 discusses root systems, which arise naturally from considering root space decompositions, but are an interesting geometric …
Quantum Groups and WZNW Models - Project Euclid
In Sect. 3 we quantize the classical exchange algebra using the free field parametrization of the chiral WZNW model of [7, 8]. Instead of the r-matrix Poisson brackets we will have a quantum …
