Introduction to Finite and Infinite Dimensional Lie (Super)algebras

By Neelacanta Sthanumoorthy

Lie superalgebras are a natural generalization of Lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. Introduction to Finite and Infinite Dimensional Lie Algebras and Superalgebras introduces the theory of Lie superalgebras, their algebras, and their representations.

The material covered ranges from basic definitions of Lie groups to the classification of finite-dimensional representations of semi-simple Lie algebras. While discussing all classes of finite and infinite dimensional Lie algebras and Lie superalgebras in terms of their different classes of root systems, the book focuses on Kac-Moody algebras. With numerous exercises and worked examples, it is ideal for graduate courses on Lie groups and Lie algebras.

• Discusses the fundamental structure and all root relationships of Lie algebras and Lie superalgebras and their finite and infinite dimensional representation theory
• Closely describes BKM Lie superalgebras, their different classes of imaginary root systems, their complete classifications, root-supermultiplicities, and related combinatorial identities
• Includes numerous tables of the properties of individual Lie algebras and Lie superalgebras
• Focuses on Kac-Moody algebras

• Language: English
• Publisher: Academic Press
• Release date: Apr 26, 2016
• ISBN: 9780128046838

Neelacanta Sthanumoorthy

N. Sthanumoorthy has 45 years of teaching and research experience. Formerly Professor and Professor Emeritus, Dr Sthanumoorthy is presently the Principal Investigator of a Book-Writing Project funded by “Science and Engineering Research Board - Department of Science and Technology, Government of India” in RIASM, University of Madras, India. He has published several research papers on topics closely related to the title of the present book, guided many Ph.D. scholars and evaluated several Ph.D. theses. He was an editor of Kac-Moody Lie Algebras and Related topics, which published as volume 343 of ‘Contemporary Mathematics (AMS)’, and he is a reviewer for Mathematical Reviews. He delivered lectures in many institutions in the USA, Germany, Italy, China, and India. Many awards and honors were also conferred on the author.

Book preview

Introduction to Finite and Infinite Dimensional Lie (Super)algebras

First Edition

N. Sthanumoorthy

Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India

Table of Contents

Cover image

Title page

Copyright

Dedication

About the author

Acknowledgments

Preface

Acknowledgments

1: Finite-dimensional Lie algebras

Abstract

1.1 Basic definition of Lie algebras with examples and structure constants

1.3 Ideals, quotient Lie algebras, derived sub Lie algebras, and direct sum

1.4 Simple Lie algebras, semisimple Lie algebras, solvable and nilpotent Lie algebras

1.5 Isomorphism theorems, Killing form, and some basic theorems

1.6 Derivation of Lie algebras

1.8 Rootspace decomposition of semisimple Lie algebras

1.9 Root system in Euclidean spaces and root diagrams

1.10 Coxeter graphs and Dynkin diagrams

1.11 Cartan matrices, ranks, and dimensions of simple Lie algebras

1.12 Weyl groups and structure of Weyl groups of simple Lie algebras

1.13 Root systems of classical simple Lie algebras and highest long and short roots

1.14 Universal enveloping algebras of Lie algebras

1.15 Representation theory of semisimple Lie algebras

1.16 Construction of semisimple Lie algebras by generators and relations

1.17 Cartan-Weyl basis

1.18 Character of a finite-dimensional representation and Weyl dimension formula

1.19 Lie algebras of vector fields

2: Kac-Moody algebras

Abstract

2.1 Basic concepts in Kac-Moody algebras

2.2 Classification of finite, affine, hyperbolic, and extended-hyperbolic Kac-Moody algebras and their Dynkin diagrams

2.3 Invariant bilinear forms

2.4 Coxeter groups and Weyl groups

2.5 Real and imaginary roots of Kac-Moody algebras

2.6 Weyl groups of affine Lie algebras

2.7 Realization of affine Lie algebras

2.8 Different classes of imaginary roots (special imaginary roots, strictly imaginary roots, purely imaginary roots) in Kac-Moody algebras

2.9 Representations of Kac-Moody algebras, integrable highest weight modules, Verma modules, and character formulas

2.10 Graded Lie algebras and root multiplicities

3: Generalized Kac-Moody algebras

Abstract

3.1 Borcherds-Cartan matrix (BCM), Generalized Generalized Cartan matrix (GGCM), Borcherds Kac-Moody (BKM) algebras, and Generalized Kac-Moody (GKM) algebras

3.2 Dynkin diagrams of GKM algebras

3.3 Root systems and Weyl groups of GKM algebras

3.4 Special imaginary roots in GKM algebras and their complete classifications

3.5 Strictly imaginary roots in GKM algebras and their complete classifications

3.6 Purely imaginary roots in GKM algebras and their complete classifications

3.7 Representations of GKM algebras

3.8 Homology modules and root multiplicities in GKM algebras

4: Lie superalgebras

Abstract

4.1 Basic concepts in Lie superalgebras with examples

4.2 Coloring matrices, θ-colored Lie superalgebras, and examples

4.3 Subsuperalgebras, ideals of Lie superalgebras, abelian Lie superalgebras, solvable, and nilpotent Lie superalgebras

4.4 General linear Lie superalgebras

4.5 Simple and semisimple Lie superalgebras and bilinear forms

4.6 Representations of Lie superalgebras

4.7 Different classes of classical Lie superalgebras

4.8 Universal enveloping algebras of Lie superalgebras and θ-colored Lie superalgebras

4.9 Cartan subalgebras and root systems of Lie superalgebras

4.10 Killing forms on Lie superalgebras

4.11 Dynkin diagrams of Lie superalgebras [34, 129]

4.12 Lie superalgebras over an algebraically closed field of characteristic zero

4.13 Classification of non-classical Lie superalgebras

4.14 Lie superalgebras of vector fields

5: Borcherds Kac-Moody Lie superalgebras

Abstract

5.1 BKM supermatrices and BKM Lie superalgebras

5.2 Dynkin diagrams of BKM Lie superalgebras and in particular Kac Moody Lie superalgebras and some examples

5.3 Domestic type and Alien type imaginary roots in BKM Lie superalgebras

5.4 Special imaginary roots in BKM Lie superalgebras and their complete classifications

5.5 Strictly and Purely imaginary roots in BKM Lie superalgebras and complete classification of BKM Lie superalgebras possessing purely imaginary roots

5.6 BKM Lie superalgebras possessing purely imaginary property but not strictly imaginary property

5.7 Complete classification of BKM Lie superalgebras possessing strictly imaginary property (SIM property)

5.8 Borcherds superalgebras and root supermultiplicities

5.9 Root supermultiplicities of Borcherds superalgebras which are extensions of Kac-Moody algebras and some combinatorial identities

5.10 Description of finite and infinite dimensional Lie algebras and Lie superalgebras and their different classes of root systems

6: Lie algebras of Lie groups, Kac-Moody groups, supergroups, and some specialized topics in finite- and infinite-dimensional Lie algebras

Abstract

6.1 Lie groups and Lie algebras of Lie groups

6.2 Kac-Moody groups, supergroups, and some applications

6.3 Homogeneous spaces, corresponding Lie algebras, and spectra of some differential operators on homogeneous spaces

6.4 Spectral invariants of zeta function of the Laplace-Beltrami operator

6.5 Generalization of Macdonald’s identities for some Kac-Moody algebras

6.6 Some special infinite dimensional Lie algebras

6.7 Hirota bilinear differential operators and soliton solutions for KdV equation

6.8 Principal vertex operator construction of basic representation and homogeneous vertex operator construction of the basic representation

6.10 Fermionic Fock space, Clifford algebra, Bosonic Fock space, and Boson-Fermion correspondence

6.11 Remark on Quantum groups, String theory and Mathematical Physics with some references

Appendix

Bibliography

Index

Copyright

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Dedication

In memory of my beloved parents R. Neelakanda Pillai and L. Anantham Neelakanda Pillai.

About the author

N. Sthanumoorthy has more than 45 years of teaching and research experience. He obtained Ph.D. degree in Mathematics, with a highly commended thesis, from the University of Madras. He is presently working as the Principal Investigator of a Book-Writing Project funded by Science and Engineering Research Board (SERB)—Department of Science and Technology (DST), Government of India and was formerly working as Professor and Professor Emeritus (granted by University Grants Commission (UGC), Government of India) in the Ramanujan Institute for Advanced Study in Mathematics (RIASM), University of Madras, Chennai, India. UGC awarded Visiting Associate-ship to him for two years during 1994–1996. He also served as a Visiting Professor offered by NBHM during 2004 and a Visiting Fellow in the Tata Institute of Fundamental Research, Mumbai during 1986. Among his 36 research publications on topics closely related to the title of the present book, two articles were published in Science Encyclopedia and four articles were published in Source Books brought out in Algebra, Applied Algebra, and Real Analysis (Part 1 and Part 2). He was the convener and coordinator of Ramanujan International Symposium on Kac-Moody Lie Algebras and Applications held in RIASM, during 2002 and was an editor of Contemporary Mathematics (AMS), Vol. 343 – Kac-Moody Algebras and Related Topics which was the proceedings of this symposium. As the coordinator of a refresher course on Differential Geometry, organized by UGC-Academic Staff College, University of Madras, he brought out a reading material on Differential Geometry in 2005. He supervised and guided six research fellows for the Ph.D. degree and 35 research fellows for the M.Phil. degree. He successfully completed four Major Research Projects, two of them funded by DST, Government of India, rest funded by UGC, Government of India and one minor research project, also funded by the UGC. He delivered lectures in many international institutions located in countries like United States, Germany, Italy, China, and India. Awards and Honors conferred on the author: (i) he was a recipient of Performance Linked Cash Incentive award with highest credit in his department in the University of Madras for 2 years (2002–2004); (ii) he was a subject of biographical record for outstanding achievement published by Marquis ‘Who’s Who’ in the World during 2009; (iii) he was a recipient of Vijay Shree Award with Certificate of Excellence from the India International Friendship Society during 2006 for outstanding services, series of achievements and contributions and also selected for Rajiv Gandhi Excellence Award; and (iv) he received Best Citizen of India Award from International Publishing House, during 2005. He refereed papers for many journals, evaluated several Ph.D. theses, reviewed research papers for Mathematical Reviews like AMS (USA) and Zentralblatt fur Mathematik (Germany), evaluated many research projects and served as an expert committee member for UGC. He was a member of many Professional societies. Now he is an Associate Editor of International Journal of Theoretical and Computational Mathematics (ISSN: 2395–6607).

Acknowledgments

Catalysed and Supported by the Science & Engineering Research Board, Department of Science & Technology, Government of India Under its Utilisation of Scientific Expertise of Retired Scientists Scheme

Preface

The aim of writing this book is to bring the essence in finite and infinite-dimensional Kac-Moody Lie algebras, Generalized Kac-Moody (GKM) algebras, Lie superalgebras, and Borcherds Kac-Moody superalgebras and some applications of Lie groups and Kac-Moody algebras to Number theory, Differential geometry, and Differential equations in one book. The present book may not contain the proofs of some important theorems proved in other standard books, instead will give many examples to illustrate the standard results. This makes the point that we can cover so much material by not giving all the detailed proofs of general theorems (which can already be found in other textbooks). The idea is to present enough examples so that the readers can really understand the main concepts behind the theory. Expected readers may be graduate students, research scholars, and teachers. It is hoped that the graduate students can learn the fundamentals from this book, whereas the researchers will have many options to choose and will get much working knowledge in the concerned topics after reading the book and teachers may find it interesting to teach the topics.

First I briefly explain the historical development of the topics mentioned above.

The creators of the Lie theory considered Lie group as a group of symmetries of an algebraic or geometric object and Lie algebra as the set of infinitesimal transformations. Lie [5–7] considered the problem of classification of GLn and infinite-dimensional group of transformations. Starting from the works of Sophus Lie, Wilhelm Killing, and Elie Cartan, the theory of finite-dimensional Lie groups and Lie algebras developed in depth and scope. As stated in Kac [8], the following are four classes of infinite-dimensional Lie groups and Lie algebras that underwent intensive study: (i) Lie algebras of vector fields and corresponding groups of diffeomorphisms on manifolds, (ii) Lie groups and Lie algebras of smooth mappings on a given manifold into finite-dimensional Lie groups or Lie algebras, respectively, (iii) classical Lie groups and algebras of operators in a Hilbert’s space or Banach space, and (iv) infinite-dimensional Lie algebras, namely, Kac-Moody algebras. Using the researches of Sophus Lie and Wilhelm Killing, Cartan [9] in his 1894 thesis, completed the classification of finite-dimensional simple Lie algebras over C. The nine types of this classification (consisting of the four classes of classical simple Lie algebras and five exceptional simple Lie algebras) correspond to the nine types of finite Cartan matrices and to the nine types of Dynkin Diagrams [10, 11]. Wilhelm Killing and Elie Cartan had developed a process from finite-dimensional simple Lie algebras to finite Cartan matrices, whereas, Chevalley [12] and Harish-Chandra [13] constructed a scheme that began with a finite Cartan matrix and produced finite-dimensional simple Lie algebra. During 1976, Serre [14–16] proved the defining relations on the generators and Cartan integers (elements of the Cartan matrix) of the finite-dimensional complex semi-simple Lie algebras. On the other hand, Moody [17, 18] in Canada and Kac [19–21] in Russia worked simultaneously and independently to extend the construction that Jacobson [22] had presented in Chapter 7 of his book to infinite-dimensional setting. Both of them gave the classification of the generalized Cartan matrices of affine type and gave standard realizations of them. Kac-Moody Lie algebras, in general, appear as infinite-dimensional generalizations of semi-simple Lie algebras over generalized Cartan matrices. For a finite-dimensional complex semisimple Lie algebra, all the corresponding principal minors of the Cartan matrix are positive whereas Kac and Moody suggested to drop this condition and enlarge the corresponding abelian Lie algebra (Cartan subalgebra), so that the linear functions on the linear span of the abelian subalgebra will be linearly independent. Hence for a Kac-Moody algebra, all the principal minors of the corresponding generalized Cartan matrix need not be positive but may be also singular and the corresponding Kac-Moody algebra is in general infinite dimensional. It was in their paper on Lie algebra Homology and Macdonald-Kac Formulas by Garland and Lepowsky [23], the term Kac-Moody Lie algebras was first used in the literature. Within a decade, this subfield of mathematics had found many surprising physical applications. In the theory of the finite-dimensional complex semi-simple Lie algebras, there are four main tools: the root string (or more generally, the weight string), the Weyl group, the Killing form, and the Casimir operator. All of them were generalized to Kac-Moody algebras by Kac and Moody, namely, the root string and the Weyl group were generalized to arbitrary Kac-Moody algebras whereas the Killing form and the Casimir operator were generalized to symmetrizable Kac-Moody Lie algebras. The algebraic work of Moody and Kac led to deep results in Physics and won for them, the prestigious Wigner Medal in 1994. To classify the Kac-Moody algebras, it is sufficient to classify the generalized Cartan matrices. The generalized Cartan matrices of finite type are simply the Cartan matrices, and their classification is well-known. The classification of the generalized Cartan matrices of affine type were achieved by Kac and Moody independently. In the class of indefinite type, there are hyperbolic types [8], extended hyperbolic types [24], and other indefinite types. For details regarding the history of the early development of Kac-Moody algebras, one can refer to Berman and Parshall [25] and Macdonald [128]. For details regarding Flag varieties and conjugacy theorems, one can refer Kac and Peterson [26].

GKM algebras, also known as Borcherds algebras, are a natural generalization of Kac-Moody algebras allowing imaginary simple roots. They were constructed by Borcherds [27] in his study of vertex algebras and Conway and Norton’s moonshine conjectures [28] for the Monster sporadic simple group. GKM algebras contain as a subclass, the symmetrizable Kac-Moody algebras, and in particular, the affine and finite-dimensional simple Lie algebras. Additionally here, the generalized Cartan matrix may be infinite. The structure and the representation theory of GKM algebras are very similar to those of Kac-Moody algebras, and many basic facts about Kac-Moody algebras can be extended to GKM algebras. But there are some differences, too. For example, GKM algebras may have imaginary simple roots with norms ≤0 and multiplicities can be >1. GKM algebras can also be obtained as quotients of vertex algebras [29–33]. GKM algebras were originally described in terms of generators and relations which are certain generalizations of the defining conditions for Kac-Moody algebras.

Lie superalgebras appeared as Lie algebras of certain generalized groups, nowadays called Lie supergroups, whose function algebras are algebras with commuting and anticommuting variables. Recently, a satisfying theory, similar to Lie’s theory, has been developed on the connection between Lie supergroups and Lie superalgebras. If G is a finite-dimensional Lie superalgebra, then G contains a unique maximal solvable ideal R (the solvable radical). The Lie superalgebra G/R is semisimple (that is, it has no solvable ideals). Therefore, the theory of finite-dimensional Lie superalgebras is reduced, in a certain sense, to the theories of semisimple and solvable Lie superalgebras. The main fact in the theory of solvable Lie algebras is Lie’s theorem, which asserts that every finite-dimensional irreducible representation of a solvable Lie algebra over C is one-dimensional. For Lie superalgebras this is not true, in general. Next, it is well known that a semisimple Lie algebra is a direct sum of simple ones. This is by no means true for Lie superalgebras. However, there is a construction that allows us to describe finite-dimensional semisimple Lie superalgebras in terms of simple ones. So we come to the fundamental problem of classifying the finite-dimensional simple Lie superalgebras. The principal difficulty lies in the fact that the Killing form may be degenerate, which cannot happen in the case of simple Lie algebras. Therefore, the classical technique of Killing-Cartan is not applicable here. A detailed exposition of the theory of Lie superalgebras can also be seen in Kac [34] and Scheunert [35]. Kac and Wakimoto [36] further developed the theory of Lie superalgebras and presented some interesting applications of affine Kac-Moody superalgebras to number theory.

The theories of Lie superalgebras, Kac-Moody superalgebras, and GKM algebras were combined to give rise to the theory of Borcherds Kac-Moody Lie superalgebras. Wakimoto [3, 4] introduced the definitions of Borcherds Kac-Moody supermatrix (BKM supermatrix) and Borcherds Kac-Moody Lie superalgebra (BKM Lie superalgebra) by imposing an additional structure in the index set in the definition of Borcherds Cartan matrix. Kang and Kim [37] introduced the definition of Borcherds superalgebras by introducing a coloring matrix to the Borcherds Cartan matrix. Using this coloring matrix, one can define a bimultiplicative form θ [38] which in turn will give a θ-colored Lie superalgebra to make it finally as the Borcherds superalgebra. On the other hand, the additional structure due to the introduction of a subset (of the index set in the definition of Borcherds Cartan matrix) corresponding to odd roots of the BKM Lie superalgebra coincides with the introduction of the coloring matrix in the definition of Borcherds superalgebra.

During the last more than four decades, the theory of finite and infinite-dimensional Lie algebras and Lie superalgebras attracted researchers from different areas of Mathematics and Physics because of its close connections with combinatorics, differential equations, group theory, modular forms, singularities, knot theory, statistical mechanics, quantum field theory, and string theory to name a few.

Below I give briefly the chapter-wise description of this book. To understand the fundamentals about all types of finite- and infinite-dimensional Lie algebras and Lie superalgebras, it is better that one knows the fundamentals of finite-dimensional Lie algebras. So in this volume, I start with the finite-dimensional Lie algebras.

. I also explain the structure of Weyl groups of simple Lie algebras, root systems of classical Lie algebras, Cartan-Weyl basis, universal enveloping algebras, and character of finite-dimensional representations. Moreover, Lie algebras of vector fields are also explained in this chapter.

Chapter 2 deals with the infinite-dimensional Kac-Moody Lie algebras: Starting from the basic concepts, various classes of Kac-Moody algebras and different classes of imaginary roots are explained with examples. Weyl groups of affine Lie algebras, realization of affine Lie algebras, representation theory of Kac-Moody algebras, character formulae, graded algebras, and root multiplicities are given in detail with examples. In particular, recently obtained results on different classes of root systems and root multiplicities are also described in this chapter.

In Chapter 3, I discuss about GKM algebras: Starting from the definition of Borcherds Cartan Matrices and Dynkin Diagrams, different classes of root systems of GKM algebras are given with examples. Homology modules, root multiplicities, and representations of GKM algebras are discussed in this chapter. Newly obtained results on complete classification theorems on special, strictly and purely imaginary roots, and dimension formulae are also given in this chapter. Monstrous Lie algebras are also explained here.

Chapter 4 describes about Lie superalgebras: Lie superalgebras are defined here basically in two different ways with examples. Then the basic concepts in Lie superalgebras, coloring matrices, θ-colored Lie superalgebras, the representations of Lie superalgebras, universal enveloping algebras of Lie superalgebras, and θ-colored Lie superalgebras are explained. Moreover, in this chapter, the Killing forms, Dynkin diagrams, classifications of Lie superalgebras and nonclassical Lie superalgebras, and Lie superalgebras of vector fields are also given.

Chapter 5 is about Borcherds Kac-Moody Lie superalgebras: All basic concepts in BKM Lie superalgebras, their Dynkin diagrams along with all types of imaginary roots, namely, domestic, alien, purely, strictly, and special imaginary roots and their complete classifications are explained. Root supermultiplicities of Borcherds superalgebras along with recently found out superdimension-formulae and corresponding combinatorial identities are also given. Diagrammatic description of finite and infinite dimensional Lie algebras, Lie superalgebras and their different classes of root systems is also given.

Chapter 6 is about Lie groups and Lie algebras, Kac-Moody groups, supergroups, some specialized topics, and some applications of finite- and infinite-dimensional Lie algebras to Differential Geometry, Number theory, and Differential Equations. First I discuss about Lie algebras of Lie groups, Kac-Moody groups, supergroups, and some applications. Then spectra of some differential operators on some homogeneous spaces and the spectral invariants of the zeta function of the Laplacian on 1-forms on the above spaces and 2-forms on (4r − 1)-dimensional sphere and eta function and spectral asymmetry of the operator B = ±(*d − d*) acting on 2-forms on (4r , and corresponding nonlinear differential equations are discussed in this chapter. Some preliminaries on Fermionic Fock space, Clifford algebra, Bosonic Fock space, and Boson-Fermion correspondence and a remark on Quantum groups, String theory and Mathematical Physics with some references are also given.

functions on differentiable manifold, product manifold, Lie group SO(3) and Lie algebra so(3), pseudo Riemannian manifold, Riemannian manifold, Riemannian symmetric space with examples, Riemann surface, symmetries of heat equation, Lie algebra realized as operators, Maya diagram, Young tableau, presheaf and sheaf of sets, symmetric group with an example are given. Finally definitions and Dynkin diagrams of quasi finite, quasi affine and quasi hyperbolic Kac-Moody algebras are also given here.

N. Sthanumoorthy, Ramanujan Institute for Advanced Study in Mathematics

September 18, 2015

References

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Acknowledgments

I am grateful to my research guide and supervisor T.S. Bhanu Murthy for his research guidance during 1978–1982 to me in Ramanujan Institute for Advanced Study in Mathematics, University of Madras for my research leading to Ph.D. degree. I am thankful to the authorities who were responsible for my academic accomplishments at the International level: (i) visits to Abdus Salam International Center for Theoretical Physics, Trieste, Italy to attend international conferences during April 17–28, 1989; March 15 to April 2, 1993; August 3 to September 1, 1995; and September 4–15, 1995; (ii) invited talk delivered in the International Congress of Mathematicians (ICM 2002) held in Beijing, China, during August 20–28, 2002; (iii) lectures given in the Max Plank Institute Fur Gravitational Physics, Albert Einstein Institute, Germany during July 4 to August 2, 2004; (iv) visit to School of Mathematics, University of Minnesota, Minneapolis, USA on September 29, 2006; (v) lectures given in the Department of Mathematics, University of Texas at Arlington, Texas, USA on December 28, 2006; (vi) lectures given in the Department of Mathematics, University of Nebraska, Lincoln, USA on May 6, 2010; and (vii) participation and lectures given in many conferences in India. My visits to Tata Institute of Fundamental Research (TIFR), Mumbai, India, several times, in particular, as a Visiting Fellow (offered by TIFR) and Visiting Professor offered by National Board for Higher Mathematics (NBHM) were also useful to me. I also acknowledge American Mathematical Society (AMS) for my work as an editor of Contemporary Mathematics (Vol. 343), published by AMS.

I thank the authorities of the University of Madras for their approval to do this book project, sanctioned and financially supported by SERB, Department of Science and Technology (DST), Government of India and also thank the Director and Head, entire teaching and non-teaching staff of Ramanujan Institute for Advanced Study in Mathematics for the necessary facilities offered to me to complete this book project.

I am thankful to Alex Feingold, State University of New York, Binghamton, New York, for his encouragement and best wishes for the successful completion of this book.

I thank the authorities of Elsevier Academic Press, in particular, Cathleen Sether (Publishing Director), Nikki Levy (Publisher), Graham Nisbet (Senior Acquisitions Editor), Steven Mathews (Editorial Project Manager), Susan Ikeda (Editorial Project Manager), Poulouse Joseph (Senior Project Manager), and Matthew Limbert (Designer) for their kind cooperation in bringing out this volume on time.

I thank A. Uma Maheswari for her valuable assistance in the proofreading and also thank K. Priyadharsini for her skillful technical assistance and sincere role in typesetting the manuscript of the book.

I am thankful to my wife, Anantha Radha Sthanumoorthy, for her support and encouragement throughout my career in teaching and research. I thank my sons, Neelakantan Sthanumoorthy and Raja Sthanumoorthy, for their constant encouragement during the preparation of this book. I also thank all other members of my family for their best wishes. Last but not least, I would like to mention that the warmth and affection of my grand children, Riya, Shrihan and Ishan were great source of inspiration to me.

This book is dedicated to the memory of my beloved parents, R. Neelakanda Pillai (father) and L. Anantham Neelakanda Pillai (mother) and to the memory of my sister, A. Visalakshi. My late father-in-law, M. Subramania Pillai, would be immensely happy to know the publication of this book.

Above all, I thank the Almighty, for his grace and kindness to complete this work.

N. Sthanumoorthy

September 18, 2015

1

Finite-dimensional Lie algebras

Abstract

History of the development of finite-dimensional Lie algebras is described in the preface itself. Lie theory has its name from the work of Sophus Lie [6], who studied certain transformation groups, that is, the groups of symmetries of algebraic or geometric objects that are now called Lie groups. Using the researches of Sophus Lie and Wilhelm Killing, Cartan [9] in his 1894 thesis, completed the classification of finite-dimensional simple Lie algebras over C. The nine types of this classification (consisting of the four classes of classical simple Lie algebras and five exceptional simple Lie algebras) correspond to the nine types of finite Cartan matrices and to the nine types of Dynkin Diagrams [10, 11]. Chevalley [12] and Harish-Chandra [13] constructed a scheme that began with a finite Cartan matrix and produced finite-dimensional simple Lie algebra. During 1976, Serre [14–16] proved the defining relations on the generators and Cartan integers (elements of the Cartan matrix) of the finite-dimensional complex semi-simple Lie algebras.

are also explained. Moreover, the rootspace decomposition of semisimple Lie algebras, Dynkin diagrams, Cartan matrices, rank and dimensions of simple Lie algebras, Weyl groups, universal enveloping algebras, construction of semisimple Lie algebras by generators and relations, Cartan-Weyl basis, and character of finite-dimensional representations are also explained. Cartan matrices of all classical simple algebras, structures of Weyl groups of simple algebras, Weyl groups, root systems highest short and long roots of all classical simple algebras are given. Basic properties of Lie algebras of vector fields are also given.

Throughout this chapter, all Lie algebras are finite dimensional unless otherwise stated.

Keywords

Lie algebras; structure constants; Classes of subalgebras of gl (n, C); Ideals; quotient Lie algebras; derived sub Lie algebras; direct sum; Simple; semisimple; Solvable; Nilpotent Lie algebras; Isomorphism theorems; Killing form; Derivations of Lie algebras; Representations of sl(2, C); Rootspace decomposition; Root systems in Euclidean spaces; Root diagrams; Coxeter graphs; Dynkin diagrams; Cartan matrices; Ranks and dimensions of simple Lie algebras; Weyl groups; Root systems; Long, short roots of classical simple Lie algebras; Universal enveloping algebras; Representation theory; Construction of semisimple Lie algebras by generators; Cartan-Weyl basis; Character; Weyl dimension formula; Lie algebras of vector fields

Chapter Outline

1.1 Basic definition of Lie algebras with examples and structure constants   2

1.2   8

1.3 Ideals, quotient Lie algebras, derived sub Lie algebras, and direct sum   10

1.4 Simple Lie algebras, semisimple Lie algebras, solvable and nilpotent Lie algebras   12

1.5 Isomorphism theorems, Killing form, and some basic theorems   14

1.6 Derivation of Lie algebras   19

1.7   21

1.8 Rootspace decomposition of semisimple Lie algebras   28

1.9 Root system in Euclidean spaces and root diagrams   30

1.10 Coxeter graphs and Dynkin diagrams   34

1.11 Cartan matrices, ranks, and dimensions of simple Lie algebras   36

1.12 Weyl groups and structure of Weyl groups of simple Lie algebras   39

1.13 Root systems of classical simple Lie algebras and highest long and short roots   40

1.14 Universal enveloping algebras of Lie algebras   44

1.15 Representation theory of semisimple Lie algebras   49

1.16 Construction of semisimple Lie algebras by generators and relations   50

1.17 Cartan-Weyl basis   51

1.18 Character of a finite-dimensional representation and Weyl dimension formula   52

1.19 Lie algebras of vector fields   53

1.1 Basic definition of Lie algebras with examples and structure constants

Definition 1

, called the Lie bracket of x and yis called a Lie algebra if the following axioms are satisfied:

(1) The bracket operation [ , ] is bilinear.

(2) [x,x(skew symmetry).

(3) [x,[y,z]] + [y,[z,x]] + [z,[x,y(Jacobi identity).

Remark 1

Note that (1) and (2) applied to [x + y,x + y− [y,x], which is anticommutativity.

, then apply y = x in

.

A Lie algebra can also be defined starting from the definition of an algebra

Definition 2

endowed with a multiplication ab, which is bilinear, that is,

Definition 3

with product [a,b], called the bracket of a and b, subject to the following two axioms:

• anticommutativity: [a,a] = 0 or skew symmetry

• Jacobi identity: [a,[b,c]] + [b,[c,a]] + [c,[a,b]] = 0.

Definition 4

is abelian if [a,b

Definition 5

is associative if (ab)c = a(bc. In particular, in an associative algebra with bilinear multiplication ab, define the Lie bracket [,] to be the commutator [a,b] = ab − baSo a commutative associative algebra is an abelian Lie algebra.

Definition 6

A homomorphism, say φis a linear map that preserves bracket: φ([X,Y ]) = [φ(X),φ(Y we speak of an endomorphism.)

The Kernel of a homomorphism for it. A homomorphism ϕ is called an isomorphism

We give below the formal definition of isomorphism.

Definition 7

are isomorphic satisfying ϕ([x,y]) = [ϕ(x),ϕ(y)] for x,y .

An isomorphism of a Lie algebra with itself is an automorphism.

Example 1 (General linear Lie algebras)

Let V Lie algebras arise in nature as vector spaces of linear transformations on V endowed with a new operation [ , ], called the commutator [f,g] = f ° g − g ° f (which is neither commutative nor associative) in terms of the associative composition operation ° in

. Actually this End(V , End(V ) has dimension n². It can be directly verified that this bracket [f,g] for all f,g ∈ End(V ) satisfies the axioms required for a Lie algebra. With this operation, End(V

In order to distinguish this new algebra structure from the old associative one, we write gl(V ) for End(V ) viewed as a Lie algebra and call it, the general linear algebra, gl(V ).

Any subalgebra of gl(V ) is called a linear Lie algebra.

There is an isomorphism of End(V ) or gl(V of n × n after a choice of basis is made for V. The standard basis of gl(V ) consists of the matrices (eij) (having 1 in the (i,j)-th position and 0 elsewhere). Since eijekl = δjkeil

then gl(V

Remark 2

In particular, if V and we get the following homomorphism "adby (adx)(y) := [x,yUsing Jacobi identity, we can easily verify that

Theorem 1

Proof

Let Vand {e1,e2,e3} be a basis for V3. For any two elements x = x1e1 + x2e2 + x3e3 and y = y1e1 + y2e2 + y3e3 in V3, define

Then V3 becomes a Lie algebra with this bracket. The bracket here is nothing but the cross product of vectors x and y in V3. The required axioms for Lie algebras can be directly verified.

the set of all 3 × 3skew-symmetric matrices over

Any element of o(3) can be taken as

with a basis {Rx,Ry,Rz}, where

with [Rx,Ry] = Rz,[Ry,Rz] = Rx, and [Rz,Rx] = Ry. Actually, Rx,Ry, and Rz will form the infinitesimal rotations around x-, y-, and z-axis, respectively.

On the other hand, if we take

Now one can establish an isomorphism between V3 and o

Remark 3

(1) One can define a Lie algebra structure on a three-dimensional vector space with a basis {x,y,z} and Lie brackets satisfying the following relations, [x,y] = z,[y,z] = x,[x,z] = y. All the three axioms for Lie algebra can be easily verified. This algebra is isomorphic to the algebras V3 and o(3). Moreover, the Lie algebra satisfying the above relationships can also be considered as a Real Lie algebra with three elements in the basis. This is also angular momentum algebra in three dimensions, equivalent to so(3).

Below we give some more examples to Lie algebras.

(2) Every finite-dimensional Lie algebra is isomorphic to a linear Lie algebra.

as a Lie algebra by setting [x,y. Such an algebra having trivial Lie multiplication is Abelian.

(ii) Lorentz algebra: The Lie algebra satisfying

is a real Lie algebra with three elements in the basis, called the Lorentz algebra in 2 + 1 dimension, denoted as so(2,1).

be two Lie algebras over the same field Kbecomes a Lie algebra over K if we define the bracket as

is the direct sum of these ideals.

the set of all complex n × n matrices X satisfying XM + MXt are isomorphic whenever M1 and M2 are congruent. In