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Author: ARKADY L. ONISHCHIK.
Publisher:
ISBN: 9783037195024
Category : Lie algebras
Languages : en
Pages :
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Book Description
In 1914, E. Cartan posed the problem to find all irreducible real linear Lie algebras. An updated exposition of his work was given by Iwahori (1959). This theory reduces the classification of irreducible real representations of a real Lie algebra to a description of the so-called self-conjugate irreducible complex representations of this algebra and to the calculation of an invariant of such a representation (with values +1 or -1) which is called the index. Moreover, these two problems were reduced to the case when the Lie algebra is simple and the highest weight of its irreducible complex representation is fundamental. A complete case-by-case classification for all simple real Lie algebras was given (without proof) in the tables of Tits (1967). But actually a general solution of these problems is contained in a paper of Karpelevich (1955) (written in Russian and not widely known), where inclusions between real forms induced by a complex representation were studied. We begin with a simplified (and somewhat extended and corrected) exposition of the main part of this paper and relate it to the theory of Cartan-Iwahori. We conclude with some tables, where an involution of the Dynkin diagram which allows us to find self-conjugate representations is described and explicit formulas for the index are given. In a short addendum, written by J. v. Silhan, this involution is interpreted in terms of the Satake diagram. The book is aimed at students in Lie groups, Lie algebras and their representations, as well as researchers in any field where these theories are used. The reader is supposed to know the classical theory of complex semisimple Lie algebras and their finite dimensional representation; the main facts are presented without proofs in Section 1. In the remaining sections the exposition is made with detailed proofs, including the correspondence between real forms and involutive automorphisms, the Cartan decompositions and the con ...
Author: ARKADY L. ONISHCHIK.
Publisher:
ISBN: 9783037195024
Category : Lie algebras
Languages : en
Pages :
Get Book Here
Book Description
In 1914, E. Cartan posed the problem to find all irreducible real linear Lie algebras. An updated exposition of his work was given by Iwahori (1959). This theory reduces the classification of irreducible real representations of a real Lie algebra to a description of the so-called self-conjugate irreducible complex representations of this algebra and to the calculation of an invariant of such a representation (with values +1 or -1) which is called the index. Moreover, these two problems were reduced to the case when the Lie algebra is simple and the highest weight of its irreducible complex representation is fundamental. A complete case-by-case classification for all simple real Lie algebras was given (without proof) in the tables of Tits (1967). But actually a general solution of these problems is contained in a paper of Karpelevich (1955) (written in Russian and not widely known), where inclusions between real forms induced by a complex representation were studied. We begin with a simplified (and somewhat extended and corrected) exposition of the main part of this paper and relate it to the theory of Cartan-Iwahori. We conclude with some tables, where an involution of the Dynkin diagram which allows us to find self-conjugate representations is described and explicit formulas for the index are given. In a short addendum, written by J. v. Silhan, this involution is interpreted in terms of the Satake diagram. The book is aimed at students in Lie groups, Lie algebras and their representations, as well as researchers in any field where these theories are used. The reader is supposed to know the classical theory of complex semisimple Lie algebras and their finite dimensional representation; the main facts are presented without proofs in Section 1. In the remaining sections the exposition is made with detailed proofs, including the correspondence between real forms and involutive automorphisms, the Cartan decompositions and the con ...
Author: A. L. Onishchik
Publisher: European Mathematical Society
ISBN: 9783037190029
Category : Mathematics
Languages : en
Pages : 100
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Book Description
The book begins with a simplified (and somewhat extended and corrected) exposition of the main results of F. Karpelevich's 1955 paper and relates them to the theory of Cartan-Iwahori. It concludes with some tables, where an involution of the Dynkin diagram that allows for finding self-conjugate representations is described and explicit formulas for the index are given. In a short addendum, written by J. V. Silhan, this involution is interpreted in terms of the Satake diagram.
Author: Jean-Pierre Serre
Publisher: Springer
ISBN: 3540706348
Category : Mathematics
Languages : en
Pages : 180
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Book Description
The main general theorems on Lie Algebras are covered, roughly the content of Bourbaki's Chapter I.I have added some results on free Lie algebras, which are useful, both for Lie's theory itself (Campbell-Hausdorff formula) and for applications to pro-Jrgroups. of time prevented me from including the more precise theory of Lack semisimple Lie algebras (roots, weights, etc.); but, at least, I have given, as a last Chapter, the typical case ofal, . This part has been written with the help of F. Raggi and J. Tate. I want to thank them, and also Sue Golan, who did the typing for both parts. Jean-Pierre Serre Harvard, Fall 1964 Chapter I. Lie Algebras: Definition and Examples Let Ie be a commutativering with unit element, and let A be a k-module, then A is said to be a Ie-algebra if there is given a k-bilinear map A x A~ A (i.e., a k-homomorphism A0" A -+ A). As usual we may define left, right and two-sided ideals and therefore quo tients. Definition 1. A Lie algebra over Ie isan algebrawith the following properties: 1). The map A0i A -+ A admits a factorization A ®i A -+ A2A -+ A i.e., ifwe denote the imageof(x, y) under this map by [x, y) then the condition becomes for all x e k. [x, x)=0 2). (lx, II], z]+ny, z), x) + ([z, xl, til = 0 (Jacobi's identity) The condition 1) implies [x,1/]=-[1/, x).
Author: Alexander A. Kirillov
Publisher: Cambridge University Press
ISBN: 0521889693
Category : Mathematics
Languages : en
Pages : 237
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Book Description
This book is an introduction to semisimple Lie algebras. It is concise and informal, with numerous exercises and examples.
Author: William Fulton
Publisher: Springer Science & Business Media
ISBN: 9780387974958
Category : Mathematics
Languages : en
Pages : 616
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Book Description
Introducing finite-dimensional representations of Lie groups and Lie algebras, this example-oriented book works from representation theory of finite groups, through Lie groups and Lie algrbras to the finite dimensional representations of the classical groups.
Author: Wu-yi Hsiang
Publisher: World Scientific
ISBN: 981474073X
Category : Mathematics
Languages : en
Pages : 161
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Book Description
This volume consists of nine lectures on selected topics of Lie group theory. We provide the readers a concise introduction as well as a comprehensive 'tour of revisiting' the remarkable achievements of S Lie, W Killing, É Cartan and H Weyl on structural and classification theory of semi-simple Lie groups, Lie algebras and their representations; and also the wonderful duet of Cartan's theory on Lie groups and symmetric spaces.With the benefit of retrospective hindsight, mainly inspired by the outstanding contribution of H Weyl in the special case of compact connected Lie groups, we develop the above theory via a route quite different from the original methods engaged by most other books.We begin our revisiting with the compact theory which is much simpler than that of the general semi-simple Lie theory; mainly due to the well fittings between the Frobenius-Schur character theory and the maximal tori theorem of É Cartan together with Weyl's reduction (cf. Lectures 1-4). It is a wonderful reality of the Lie theory that the clear-cut orbital geometry of the adjoint action of compact Lie groups on themselves (i.e. the geometry of conjugacy classes) is not only the key to understand the compact theory, but it actually already constitutes the central core of the entire semi-simple theory, as well as that of the symmetric spaces (cf. Lectures 5-9). This is the main reason that makes the succeeding generalizations to the semi-simple Lie theory, and then further to the Cartan theory on Lie groups and symmetric spaces, conceptually quite natural, and technically rather straightforward.
Author: J.E. Humphreys
Publisher: Springer Science & Business Media
ISBN: 1461263980
Category : Mathematics
Languages : en
Pages : 189
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Book Description
This book is designed to introduce the reader to the theory of semisimple Lie algebras over an algebraically closed field of characteristic 0, with emphasis on representations. A good knowledge of linear algebra (including eigenvalues, bilinear forms, euclidean spaces, and tensor products of vector spaces) is presupposed, as well as some acquaintance with the methods of abstract algebra. The first four chapters might well be read by a bright undergraduate; however, the remaining three chapters are admittedly a little more demanding. Besides being useful in many parts of mathematics and physics, the theory of semisimple Lie algebras is inherently attractive, combining as it does a certain amount of depth and a satisfying degree of completeness in its basic results. Since Jacobson's book appeared a decade ago, improvements have been made even in the classical parts of the theory. I have tried to incor porate some of them here and to provide easier access to the subject for non-specialists. For the specialist, the following features should be noted: (I) The Jordan-Chevalley decomposition of linear transformations is emphasized, with "toral" subalgebras replacing the more traditional Cartan subalgebras in the semisimple case. (2) The conjugacy theorem for Cartan subalgebras is proved (following D. J. Winter and G. D. Mostow) by elementary Lie algebra methods, avoiding the use of algebraic geometry.
Author: J.A. Wolf
Publisher: Springer Science & Business Media
ISBN: 940098961X
Category : Science
Languages : en
Pages : 498
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Book Description
This book presents the text of the lectures which were given at the NATO Advanced Study Institute on Representations of Lie groups and Harmonic Analysis which was held in Liege from September 5 to September 17, 1977. The general aim of this Summer School was to give a coordinated intro duction to the theory of representations of semisimple Lie groups and to non-commutative harmonic analysis on these groups, together with some glance at physical applications and at the related subject of random walks. As will appear to the reader, the order of the papers - which follows relatively closely the order of the lectures which were actually give- follows a logical pattern. The two first papers are introductory: the one by R. Blattner describes in a very progressive way a path going from standard Fourier analysis on IR" to non-commutative harmonic analysis on a locally compact group; the paper by J. Wolf describes the structure of semisimple Lie groups, the finite-dimensional representations of these groups and introduces basic facts about infinite-dimensional unitary representations. Two of the editors want to thank particularly these two lecturers who were very careful to pave the way for the later lectures. Both these chapters give also very useful guidelines to the relevant literature.
Author: James E. Humphreys
Publisher: American Mathematical Soc.
ISBN: 1470463261
Category : Education
Languages : en
Pages : 289
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Book Description
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 g g over C C. The setting is the module category O O introduced by Bernstein–Gelfand–Gelfand, which includes all highest weight modules for g 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 g g. Basic techniques in category O 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 D-modules and perverse sheaves on the flag variety. Part II introduces closely related topics important in current research: parabolic category O O, projective functors, tilting modules, twisting and completion functors, and Koszul duality theorem of Beilinson–Ginzburg–Soergel.
Author: Marina Avitabile
Publisher: American Mathematical Soc.
ISBN: 1470410230
Category : Mathematics
Languages : en
Pages : 258
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Book Description
This volume contains the proceedings of the Workshop on Lie Algebras, in honor of Helmut Strade's 70th Birthday, held from May 22-24, 2013, at the Università degli Studi di Milano-Bicocca, Milano, Italy. Lie algebras are at the core of several areas of mathematics, such as, Lie groups, algebraic groups, quantum groups, representation theory, homogeneous spaces, integrable systems, and algebraic topology. The first part of this volume combines research papers with survey papers by the invited speakers. The second part consists of several collections of problems on modular Lie algebras, their representations, and the conjugacy of their nilpotent elements as well as the Koszulity of (restricted) Lie algebras and Lie properties of group algebras or restricted universal enveloping algebras.
