Representations of Lie Algebras and PartialDifferential Equations

Xiaoping Xu

Representations of LieAlgebras and PartialDifferential Equations

123

Xiaoping XuInstitute of MathematicsAcademy of Mathematicsand Systems Sciences,Chinese Academy of Sciences

BeijingP.R. China

and

School of MathematicsUniversity of ChineseAcademy of Sciences

BeijingP.R. China

ISBN 978-981-10-6390-9 ISBN 978-981-10-6391-6 (eBook)DOI 10.1007/978-981-10-6391-6

Library of Congress Control Number: 2017950037

Mathematics Subject Classification (2010): 17B10, 35C05, 17B20, 33C67, 94B05

© Springer Nature Singapore Pte Ltd. 2017This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made. The publisher remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.

Printed on acid-free paper

This Springer imprint is published by Springer NatureThe registered company is Springer Nature Singapore Pte Ltd.The registered company address is: 152 Beach Road, #21-01/04GatewayEast, Singapore 189721, Singapore

To My Former Ph.D Thesis Advisors:Profs. James Lepowsky and Robert LeeWilson

Preface

Symmetry is an important phenomenon in the natural world. Lie algebra is notpurely an abstract mathematics but a fundamental tool of studying symmetry. In fact,Norwegian mathematician Sorphus Lie introduced Lie group and Lie algebra in1874 in order to study the symmetry of differential equations. Lie algebras are theinfinitesimal structures (bones) of Lie groups, which are symmetric manifolds. Lietheory has extensive and important applications in many other fields of mathematics,such as geometry, topology, number theory, control theory, integrable systems,operator theory, and stochastic process. Representations of finite-dimensionalsemisimple Lie algebras play fundamental roles in quantum mechanics. The con-trollability property of the unitary propagator of an N-level quantum mechanicalsystem subject to a single control field can be described in terms of the structuretheory of semisimple Lie algebras. Moreover, Lie algebras were used to explain thedegeneracies encountered in genetic codes as the result of a sequence of symmetrybreakings that have occurred during its evolution. The structures and representationsof simple Lie algebras are connected with solvable quantum many-body system inone-dimension. Our research also showed that the representation theory of Liealgebras is connected with algebraic coding theory.

The existing classical books on finite-dimensional Lie algebras, such as the onesby Jacobson and by Humphreys, purely focus on the algebraic structures of semisimple Lie algebras and their finite-dimensional representations. Explicit irreduciblerepresentations of simple Lie algebras had not been addressed extensively.Moreover, the relations of Lie algebras with the other subjects had not been narratedthat much. It seems to us that a book on Lie algebras with more extensive view isneeded in coupling with modern development of mathematics, sciences, andtechnology.

This book is mainly an exposition of the author’s works and his joint works withhis former students on explicit representations of finite-dimensional simple Liealgebras, related partial differential equations, linear orthogonal algebraic codes,combinatorics, and algebraic varieties. Various oscillator generalizations of theclassical representation theorem on harmonic polynomials are presented. Newfunctors from the representation category of a simple Lie algebra to that of another

vii

simple Lie algebra are given. Partial differential equations play key roles in solvingcertain representation problems. The weight matrices of the minimal and adjointrepresentations over the simple Lie algebras of types E and F are proved to generateternary orthogonal linear codes with large minimal distances. New multivariablehypergeometric functions related to the root systems of simple Lie algebras areintroduced in connection with quantum many-body system in one-dimension.Certain equivalent combinatorial properties on representation formulas are found.Irreducibility of representations is proved directly related to algebraic varieties.

This book is self-contained with the minimal prerequisite of calculus and linearalgebra. It is our wish that the results this book can also be easily understood bynonalgebraists and applied to the other mathematical fields and physics. It consistsof three parts. The first part is mainly the classical structure and finite-dimensionalrepresentation theory of finite-dimensional semisimple Lie algebras, written withHumphreys’ book “Introduction to Lie Algebras and Representation Theory” as themain reference, where we give more examples and direct constructions of simpleLie algebras of exceptional types, revise some arguments, and prove some newstatements. This part serves as the preparation for the main context. The second partis our explicit representation theory of finite-dimensional simple Lie algebras. Manyof the irreducible representations in this part are infinite-dimensional, and someof them are even not of highest-weight type. Certain important natural represen-tation problems are solved by means of solving partial differential equations. Inparticular, some of our irreducible presentations are completely characterized byinvariant partial differential equations. New representation functors are constructedfrom inhomogeneous oscillator representations of simple Lie algebras, which givefractional representations of the corresponding Lie groups, such as the projectiverepresentations of special linear Lie groups and the conformal representations ofcomplex orthogonal groups. The third part is an extension of the second part. Firstwe give supersymmetric generalizations of the classical representation theorem onharmonic polynomials. They can also be viewed as certain supersymmetric Howedualities. Then we present our theory of representation theoretic codes. Finally, wetalk about root-related integrable systems and our new multivariable hypergeo-metric functions, which are natural multivariable analogues of the classical Gausshypergeometric functions. The corresponding hypergeometric partial differentialequations are found.

Part of this book has been taught for many times at the University of ChineseAcademy of Sciences. The book can serve as a research reference book formathematicians and scientists. It can also be treated as a textbook for students aftera proper selection of materials.

Beijing, P.R. China Xiaoping Xu2017

viii Preface

Acknowledgements

The research in this book was partly supported by NSFC Grants 11671381,11321101 and Hua Loo-Keng Key Mathematical Laboratory, Chinese Academy ofSciences. We thank the reviewers and Dr. Ramond Peng for their helpfulcomments.

ix

Contents

Part I Fundament of Lie Algebras

1 Preliminary of Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Basic Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Lie Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5 Nilpotent and Solvable Lie Algebras . . . . . . . . . . . . . . . . . . . . 21References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2 Semisimple Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.1 Killing Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.3 Real and Complex Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 412.4 Weyl’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.5 Root Space Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.6 Properties of Roots and Root Subspaces . . . . . . . . . . . . . . . . . . 53References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3 Root Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.1 Definitions, Examples and Properties . . . . . . . . . . . . . . . . . . . . 613.2 Weyl Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.3 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.4 Automorphisms, Constructions and Weights . . . . . . . . . . . . . . . 86

4 Isomorphisms, Conjugacy and Exceptional Types . . . . . . . . . . . . . 954.1 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.2 Cartan Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.3 Conjugacy Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.4 Simple Lie Algebra of Exceptional Types . . . . . . . . . . . . . . . . 114

xi

5 Highest-Weight Representation Theory . . . . . . . . . . . . . . . . . . . . . . 1255.1 Universal Enveloping Algebras . . . . . . . . . . . . . . . . . . . . . . . . 1265.2 Highest-Weight Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.3 Formal Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.4 Weyl’s Character Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.5 Dimensional Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Part II Explicit Representations

6 Representations of Special Linear Algebras . . . . . . . . . . . . . . . . . . 1556.1 Fundamental Lemma on Polynomial Solutions . . . . . . . . . . . . . 1566.2 Canonical Oscillator Representations . . . . . . . . . . . . . . . . . . . . 1626.3 Noncanonical Representations I: General . . . . . . . . . . . . . . . . . 1696.4 Noncanonical Representations II: n1 þ 1\ n2 . . . . . . . . . . . . . . 1766.5 Noncanonical Representations III: n1 þ 1 ¼ n2 . . . . . . . . . . . . . 1846.6 Noncanonical Representations VI: n1 ¼ n2 . . . . . . . . . . . . . . . . 1936.7 Extensions of the Projective Representations . . . . . . . . . . . . . . 2006.8 Projective Oscillator Representations . . . . . . . . . . . . . . . . . . . . 211References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

7 Representations of Even Orthogonal Lie Algebras . . . . . . . . . . . . . 2177.1 Canonical Oscillator Representations . . . . . . . . . . . . . . . . . . . . 2187.2 Noncanonical Oscillator Representations . . . . . . . . . . . . . . . . . 2237.3 Extensions of the Conformal Representation . . . . . . . . . . . . . . . 2317.4 Conformal Oscillator Representations . . . . . . . . . . . . . . . . . . . . 242References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

8 Representations of Odd Orthogonal Lie Algebras . . . . . . . . . . . . . . 2538.1 Canonical Oscillator Representations . . . . . . . . . . . . . . . . . . . . 2548.2 Noncanonical Oscillator Representations . . . . . . . . . . . . . . . . . 2588.3 Extensions of the Conformal Representation . . . . . . . . . . . . . . . 2728.4 Conformal Oscillator Representations . . . . . . . . . . . . . . . . . . . . 283References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

9 Representations of Symplectic Lie Algebras . . . . . . . . . . . . . . . . . . 2939.1 Canonical Oscillator Representations . . . . . . . . . . . . . . . . . . . . 2939.2 Noncanonical Oscillator Representations . . . . . . . . . . . . . . . . . 297

9.2.1 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2979.2.2 Proof of the Theorem When n2 ¼ n . . . . . . . . . . . . . . 2999.2.3 Proof of the Theorem When n1\n2\n . . . . . . . . . . . . 3029.2.4 Proof of the Theorem When n1 ¼ n2\n . . . . . . . . . . . 308

9.3 Projective Oscillator Representations . . . . . . . . . . . . . . . . . . . . 317References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

xii Contents

10 Representations of G2 and F4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32910.1 Basic Oscillator Representations of G2 . . . . . . . . . . . . . . . . . . . 33010.2 Conformal Oscillator Representations of G2 . . . . . . . . . . . . . . . 33310.3 Basic Oscillator Representation of F4 . . . . . . . . . . . . . . . . . . . . 33810.4 Decomposition of the Representation of F4 . . . . . . . . . . . . . . . 342References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

11 Representations of E6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35111.1 Basic Oscillator Representation . . . . . . . . . . . . . . . . . . . . . . . . 35211.2 Decomposition of the Oscillator Representation . . . . . . . . . . . . 35511.3 Spin Oscillator Representation of D5 . . . . . . . . . . . . . . . . . . . . 365

11.3.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36511.3.2 Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36911.3.3 Symmetry and Equivalent Combinatorics . . . . . . . . . . . 372

11.4 Realization of E6 in 16-Dimensional Space . . . . . . . . . . . . . . . 37511.5 Functor from D5-Mod to E6-Mod . . . . . . . . . . . . . . . . . . . . . . 38111.6 Irreducibility of the Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . 38511.7 Representations on Exponential-Polynomial Functions . . . . . . . 395References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

12 Representations of E7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40112.1 Basic Oscillator Representation of E7 . . . . . . . . . . . . . . . . . . . . 40312.2 Constructions of Singular Vectors . . . . . . . . . . . . . . . . . . . . . . 41112.3 Decomposition of the Oscillator Representation . . . . . . . . . . . . 41912.4 Realization of E7 in 27-Dimensional Space . . . . . . . . . . . . . . . 43112.5 Functor from E6-Mod to E7-Mod . . . . . . . . . . . . . . . . . . . . . . . 44012.6 Irreducibility of the Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . 44712.7 Combinatorics of the Representation of E6 . . . . . . . . . . . . . . . . 46312.8 Representations on Exponential-Polynomial Functions . . . . . . . 467References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

Part III Related Topics

13 Oscillator Representations of glðnjmÞ and ospðnj2mÞ . . . . . . . . . . . 48113.1 Canonical Oscillator Representation of glðnjmÞ . . . . . . . . . . . . . 482

13.1.1 Representations of the Even Subalgebra . . . . . . . . . . . . 48313.1.2 Singular Vectors of glðnjmÞ . . . . . . . . . . . . . . . . . . . . 48713.1.3 Structure of the Representation for glðnjmÞ . . . . . . . . . 490

13.2 Noncanonical Oscillator Representations of glðnjmÞ . . . . . . . . . 49413.3 Oscillator Representations of ospð2nj2mÞ . . . . . . . . . . . . . . . . . 50313.4 Oscillator Representations of ospð2nþ 1j2mÞ . . . . . . . . . . . . . . 512References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

Contents xiii

14 Representation Theoretic Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 52314.1 Basics Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52414.2 Codes Related to Representations of slðn;CÞ . . . . . . . . . . . . . . 52614.3 Codes Related to Representations of oð2m;CÞ . . . . . . . . . . . . . 53614.4 Exceptional Lie Algebras and Ternary Codes . . . . . . . . . . . . . . 546References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554

15 Path Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 55515.1 Integrable Systems and Weyl Functions . . . . . . . . . . . . . . . . . . 55715.2 Etingof Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56715.3 Path Hypergeometric Functions of Type A . . . . . . . . . . . . . . . . 57615.4 Path Hypergeometric Functions of Type C . . . . . . . . . . . . . . . . 58615.5 Properties of Path Hypergeometric Functions . . . . . . . . . . . . . . 603References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

xiv Contents

Notational Conventions

N f0; 1; 2; 3; . . .g, the set of nonnegative integersZ The ring of integersQ The field of rational numbersR The field of real numbersC The field of complex numbersF A field with characteristic 0, such as, Q;R;C

i; iþ j fi; i þ 1; i þ 2; . . .; i þ jg, an index set. di;j ¼ 1 if i ¼ j, 0 ifi 6¼ j

A An associative algebraG A Lie algebraP The set of positive simple rootsU The set of rootsX Casimir operatorx Casimir elementW The Weyl groupF½x1; x2; . . .; xn� The algebra of polynomials in x1; x2; . . .; xn.Mð‚Þ The Verma module of highest weight ‚.Vð‚Þ The irreducible module with highest weight ‚.Der U The derivation Lie algebra of the algebra U.Wn The Witt algebra, that is, Der F½x1; x2; . . .; xn�ri The ith simple reflection.End V The algebra of linear transformations on the vector space V .

xv

Introduction

Algebraic study of partial differential equations traces back to Norwegian mathe-matician Sophus Lie [Lie], who invented the powerful tool of continuous groups(known as Lie groups) in 1874 in order to study symmetry of differential equations.Lie’s idea has been carried on mainly by the mathematicians in the former states ofSoviet Union, East Europe and some mathematicians in North America. Now it hasbecome an important mathematical field known as “group analysis of differentialequations,” whose main objective is to find symmetry group of differential equa-tions, related conservation laws, and similarity solutions. One of the classicallong-standing problems in the area was to determine all invariant partial differentialequation under the natural representations of classical groups (it was also mentionedby Olver [Op]). The problem was eventually settled down theoretically by theauthor [X5], where we found the complete set of functional generators for thedifferential invariants of classical groups.

Gel’fand, Dikii, and Dorfman [GDi1, GDi2, GDo1-GDo3] introduced in 1970s atheory of Hamiltonian operators in order to study the integrability of nonlinearevolution partial differential equations. Our first experience with partial differentialequation was in the works [X3, X4, X9, X10, X11] on the structure of Hamiltonianoperators and their supersymmetric generalizations. In particular, we [X10] provedthat Lie conformal algebras in the sense of Kac [Kv2] are equivalent to linearHamiltonian operators as mathematical structures. Since Borcherds’ vertex algebras(cf. [Bo]) are determined by their positive parts (modes in physics terminology) ofvertex operators that form Lie conformal algebras (e.g., cf. [Kv2, X6]), our resultessentially established an equivalence between vertex algebras and Hamiltonianoperators. The result was later generalized by Barakat, De Sole, and Kac [BDK] tovertex Poisson algebras. Moreover, we used the techniques and thoughts from Liealgebras to solve various physical partial differential equations such as the equationof transonic gas flows, the equation of geopotential forecast, the nonlinearSchrödinger equations, the Navier–Stokes equations, and the classical boundarylayer equations (cf. [X21] and the references therein). In this book, we present anexplicit representation theory of finite-dimensional simple Lie algebras and show

xvii

that partial differential equations can be used to solve representation problems ofLie algebras.

Abstractly, a Lie algebraG is a vector space with a bilinear map ½�; �� : G� G ! Gsuch that

½u; v� ¼ �½v; u�; ½u; ½v;w�� ¼ ½½u; v�;x� þ ½v; ½u;w��; u; v;w 2 G ð1Þ

A Lie algebra G is called simple if it does not contain any nonzero “invariant”proper subspace I ; that is,

½u; I� � I ; u 2 G ð2Þ

Finite-dimensional complex simple Lie algebras were classified by Killing andCartan in later nineteenth century. For a vector space V , we denote by V� the spaceof linear functions on V . It turns out that such a Lie algebra G must contain asubspace H, called toral Cartan subalgebra, such that

G ¼ �a2H�

Ga; Ga ¼ fu 2 Gj½h; u� ¼ aðhÞu for h 2 Hg; ð3Þ

and G0 ¼ H, where

U ¼ fa 2 H�nf0gjGa 6¼ f0gg ð4Þ

is called the root system of G. Moreover,

dimGa ¼ 1 for a 2 U ð5Þ

and there exists a subset Uþ of U containing a basis P = a1; . . .; anf g of H� suchthat

U ¼ �Uþ[

Uþ ; �Uþ\

Uþ ¼ ; ð6Þ

and for any b 2 Uþ ,

b ¼Xni¼1

kiai with 0 ki 2 Z; ð7Þ

where Z is the ring of integers. Furthermore, there exists hi 2 ½Gai ;G�ai � such that

aiðhiÞ ¼ 2 for i ¼ 1; 2; . . .; n: ð8Þ

It turns out that the structure of G is completely determined by the matrixðaiðhjÞÞn�n, which is called the Cartan matrix of G. Consequently, there are onlynine classes of finite-dimensional simple Lie algebras, which are called of types

xviii Introduction

An;Bn;Cn;Dn;G2;F4;E6;E7;E8. The first four are infinite series, called “classicalLie algebras.” The last five are fixed simple Lie algebras, called “exceptionaltypes.” Using the root lattices Kr ¼

Pa2P Za of type A and the Coxeter auto-

morphism of Kr, we constructed in [X1] two families of self-dual complex latticesthat are also real self-dual (unimodular) lattices. These lattices can be used instudying geometry of numbers and conformal field theory.

Since every finite-dimensional complex simple Lie algebra must contain a toralCartan subalgebra, it is natural to ask whether there exists a complex simple Liealgebra that does not contain any toral Cartan subalgebra. In [X7, X8], we con-structed six families of infinite-dimensional complex simple Lie algebras withoutany toral Cartan subalgebras.

For a vector space M, we denote by End M the space of all linear transforma-tions on M. A representation m of a Lie algebra G on M is a linear map from G toEnd M such that

mð½u; v�Þ ¼ mðuÞmðvÞ � mðvÞmðuÞ for u; v 2 G: ð9Þ

The space M is called a G- module. In this book, we sometimes use the notions:

mðnÞ ¼ njM ; mðnÞðwÞ ¼ nðwÞ for n 2 G; w 2 M: ð10Þ

A subspace N of a G-module M is called a submodule of M if

nðwÞ 2 N forn 2 G; w 2 N: ð11Þ

If M dose not contain any proper nonzero submodule, we say that M is an irre-ducible G-module. Let G be a Lie algebra with a decomposition (3) satisfying (4),(6), and (7). A G-module M is called a weight module if

M ¼ �l2H�

Ml; Ml ¼ fw 2 M j hðwÞ ¼ lðhÞwg; ð12Þ

where H� is the space of linear functions on H. The set

KðMÞ ¼ fl 2 H�jMl 6¼ f0gg ð13Þ

is called the weight set of M and the elements in KðMÞ are called the weights of M.A vector inMl with l 2 KðMÞ is called a weight vector. A nonzero weight vector xis called a singular vector if

nðwÞ ¼ 0 for n 2[

b2Uþ

Gb: ð14Þ

If M is generated by a singular vector � with weight ‚, we call M a highest-weightG-module, � a highest-weight vector and ‚ the highest weight of M.

Introduction xix

Finite-dimensional representations of a complex finite-dimensional simple Liealgebra were essentially determined by Cartan in early twentieth century. Theapproaches were simplified and further developed by Weyl in 1920s. Let G be afinite-dimensional complex simple Lie algebra and take the settings in (3–8). Set

Kþ ¼ ‚ 2 H�j0 ‚ðhiÞ 2 Z for i ¼ 1; . . .; nf g ð15Þ

It turns out that any finite-dimensional G-module is a direct sum of its irreduciblesubmodules, and any finite-dimensional irreducible G-module is a highest-weightmodule with its highest weight in Kþ . Conversely, for any element ‚ 2 Kþ , thereexists a unique finite-dimensional irreducible G-module with highest weight ‚. Theabove conclusion is now called Weyl’s theorem on completely reducibility.

Coding theory began with Shannon’s famous paper “A mathematical theory ofcommunication” published in 1948 (cf. [Sc]). Naturally, people had speculated theexistence of some kind of relations between the representation theory and codingtheory. However, it had not been found until our work [X20] published in 2012. Forany finite-dimensional G-module M,

l hið Þ 2 Z fori 2 1; . . .; nf g; l 2 K Mð Þ: ð16Þ

Write K Mð Þn 0f g = l1; l2; . . .; lmf g. We found in [X20] that the set

l1 hið Þ; l2 hið Þ; . . .; lm hið Þð Þ j i ¼ 1; . . .; nf g ð17Þ

spans a binary or ternary orthogonal code for certain G and M. In particular, weshowed that when G is of type F4; E6; E7 and E8, and M is of minimal dimensionor M ¼ G with left multiplication as the representation, the set (17) spans a ternaryorthogonal code with large minimal distance d, which can be used to correct d=2½ �½ �errors in information technology.

A module of a finite-dimensional simple Lie algebra is called cuspidal if it is notinduced from its proper “parabolic subalgebras.” Infinite-dimensional irreducibleweight modules of finite-dimensional simple Lie algebras with finite-dimensionalweight subspaces had been intensively studied by the authors in [BBL, BFL, BHL,BL1, BL2, Fs, Fv, Mo]. In particular, Fernando [Fs] proved that such modules mustbe cuspidal or parabolically induced. Moreover, such cuspidal modules exist onlyfor special linear Lie algebras and symplectic Lie algebras. A similar result wasindependently obtained by Futorny [Fv]. Mathieu [Mo] proved that these cuspidalmodules are irreducible components in the tensor modules of their multiplicity-freemodules with finite-dimensional modules. Thus, the structures of irreducible weightmodules of finite-dimensional simple Lie algebras with finite-dimensional weightsubspaces were essentially determined by Fernando’s result in [Fs] and Methieu’sresult in [Mo]. However, explicit presentations of such modules need furtherexploration.

An important feature of this book is the connection between representations ofsimple Lie algebras and partial differential equations. Let Er;s be the square matrix

xx Introduction

with 1 as its ðr; sÞ-entry and 0 as the others. Denote by R the field of real numbers.Let n 3 be an integer. The compact orthogonal Lie algebra oðn;RÞ ¼P

1 r\s n RðEr;s � Es;rÞ; whose representation on the polynomial algebra A ¼R½x1; . . .; xn� is given by ðEr;s � Es;rÞjA ¼ xr@xs � xs@xr . Denote by Ak the sub-space of homogeneous polynomials in A with degree k. Recall that the Laplaceoperator D ¼ @2

x1 þ @2x2 þ � � � þ @2

xn and its corresponding invariantg ¼ x21 þ x22 þ � � � þ x2n. It is well known that the subspaces

Hk ¼ ff 2 AkjDðf Þ ¼ 0g ð18Þ

of harmonic polynomials form irreducible oðn;RÞ-submodules and

A ¼ �1i;k¼0

giHk ð19Þ

is a direct sum of irreducible submodules. In other words, the irreducible sub-modules are characterized by the Laplace operator D and its dual invariant g givesthe complete reducibility of the polynomial algebra A. The above conclusion iscalled the classical theorem on harmonic polynomials.

The Navier equations

i1Dð~uÞþ ði1 þ i2Þðrt � rÞð~uÞ ¼ 0 ð20Þ

are used to describe the deformation of a homogeneous, isotropic, and linear elasticmedium in the absence of body forces, where~u is an n-dimensional vector-valuedfunction, r ¼ ð@x1 ; @x2 ; . . .; @xnÞ is the gradient operator, i1 and i2 are Lamé con-stants with i1 [ 0, 2i1 þ i2 [ 0 and i1 þ i2 6¼ 0. In fact, rt � r is the well-knownHessian operator. Mathematically, the above system is a natural vector generaliza-tion of the Laplace equation. In [X16], we found methods of solving linear flagpartial differential equations for polynomial solutions. Cao [Cb] used a method of usto prove that the subspaces of homogeneous polynomial-vector solutions are exactlydirect sums of three explicitly given irreducible oðn;RÞ-submodules if n 6¼ 4 and offour explicitly given irreducible oðn;RÞ-submodules if n ¼ 4. This gave a vectorgeneralization of the classical theorem on harmonic polynomials.

Denote by B ¼ C½x1; . . .; xn; y1; . . .; yn� the polynomial algebra in x1; . . .; xn andy1; . . .; yn over the field C of complex numbers. Fix n1; n2 2 f1; . . .; ng withn1 n2. We redefine

deg xi ¼ �1; deg xj ¼ 1; deg yr ¼ 1; deg ys ¼ �1 ð21Þ

for i 2 f1; . . .; n1g; j 2 fn1 þ 1; ::; ng; r 2 f1; . . .; n2g and s 2 fn2 þ 1; . . .; ng.Denote by Bh‘1;‘2i the subspace of homogeneous polynomials with degree ‘1 in

Introduction xxi

fx1; . . .; xng and degree ‘2 in fy1; . . .; yng. We deform the Laplace operatorPni¼1 @xi@yi to the operator

~D ¼ �Xn1i¼1

xi@yi þXn2

r¼n1 þ 1

@xr@yr �Xn

s¼n2 þ 1

ys@xs ð22Þ

and its dualPn

i¼1 xiyi to the operator

g ¼Xn1i¼1

yi@xi þXn2

r¼n1 þ 1

xryr þXn

s¼n2 þ 1

xs@ys : ð23Þ

Define

Hh‘1;‘2i ¼ ff 2 Bh‘1;‘2ij~Dðf Þ ¼ 0g: ð24Þ

Luo and the author [LX1] constructed a new representation on B for the simple Liealgebra G of type An�1 such that the operators in (22) and (23) commute with theelements in GjB and used a method in [X16] to prove that Hh‘1;‘2i with ‘1; ‘2 2 Z

such that ‘1 þ ‘2 n1 � n2 þ 1� dn1;n2 are irreducible G-modules. Moreover,Bh‘1;‘2i ¼ �1

m¼0 gmðHh‘1�m;‘2�miÞ is a decomposition of irreducible G-submodules.

This establishes a Z2-graded analogue of the classical theorem on harmonicpolynomials for the simple Lie algebra of type An�1.

Set

Bhki ¼X‘2Z

Bh‘;k�‘i; Hhki ¼X‘2Z

Hh‘;k�‘i for k 2 Z: ð25Þ

Luo and the author [LX2] extended the above representation of G on B to arepresentation of the simple Lie algebra G1 of type Dn on B and proved that Hhkiwith n1 � n2 þ 1� dn1;n2 k 2 Z are irreducible G1-modules, and Bhki ¼�1

i¼0 giðHhk�2iiÞ is a decomposition of irreducible G1-submodules. This gives a Z-

graded analogue of the classical theorem on harmonic polynomials for the simpleLie algebra of type Dn.

Let B0 ¼ C½x0; x1; . . .; xn; y1; . . .; yn� be the polynomial algebra in x0; x1; . . .; xnand y1; . . .; yn. We define deg x0 ¼ 1 and take (21). Write the deformed Laplaceoperator

~D0 ¼ @2x0 � 2

Xn1i¼1

xi@yi þ 2Xn2

r¼n1 þ 1

@xr@yr � 2Xn

s¼n2 þ 1

ys@xs ð26Þ

xxii Introduction

and its dual operator

g0 ¼ x20 þ 2Xn1i¼1

yi@xi þ 2Xn2

r¼n1 þ 1

xryr þ 2Xn

s¼n2 þ 1

xs@ys : ð27Þ

Set

B0hki ¼

X1i¼0

Bhk�iixi0; H0hki ¼ f f 2 B0

hkij~D0ð f Þ ¼ 0g ð28Þ

Luo and the author [LX2] extend the above representation of G1 on B to a rep-resentation of the simple Lie algebra G2 of type Bn onB0 and proved thatH0

hki withk 2 Z are irreducible G2-modules, andB0 ¼ �k2Z�1

i¼0g0iðH0

hkiÞ is a decompositionof irreducible G2-submodules. This is a Z-graded analogue of the classical theoremon harmonic polynomials for the simple Lie algebra of type Bn.

When n1\n2, the bases of the subspaces Hh‘1;‘2i; Hhki and H0hki had been

obtained. Using Fourier transformation, we can identify the subspacesHh‘1;‘2i; Hhki and H0

hki with the subspaces of homogeneous solutions for thecorresponding usual Laplace equations in certain spaces of generalized functions. In[LX3], Luo and the author extended the above representation of G on B to arepresentation of the simple Lie algebra G3 of type Cn on B and proved that ifn1\n2 or k 6¼ 0, the subspace Bhki with k 2 Z are irreducible G3-modules, andwhen n1 ¼ n2, the subspace Bh0i is a direct sum of two explicitly given irreducibleG3-submodules.

The above irreducible modules except n1 ¼ n2 ¼ n in the case of type An�1 areexplicit infinite-dimensional weight modules of finite-dimensional weight sub-spaces. They are not unitary. The irreducible modules for G1;G2 and G3 aregenerically neither of highest-weight type nor of unitary type. Bai [B1, B2] provedthat some of irreducible representations in [LX1–LX3] have distinguished Gelfand–Kirillov dimensions.

A supersymmetry relating mesons and baryons was first proposed by Miyazawa[Mh] in 1966. It was largely ignored because it did not involve space-time. In early1970s, physicists rediscovered it in the context of quantum field theory, a radicallynew type of symmetry of space-time and fundamental fields, which establishes arelationship among elementary particles of quantum nature, bosons and fermions,and unifies space-time and internal symmetries of microscopic phenomena (e.g., cf.[GS]). Supersymmetry with a consistent Lie-algebraic-graded structure on whichthe Gervais–Sakita rediscovery [GS] based directly first arose during 1971 in thecontext of a early version of string theory (cf. [R1]). It was later widely applied tonuclear physics, critical phenomena, quantum mechanics, and statistical physics. In[LX4], Luo and the author generalized the above results for G;G1 and G2 to thosefor the corresponding Lie superalgebras.

Introduction xxiii

Dickson [D] (1901) first realized that there exists an E6-invariant trilinear formon its 27-dimensional basic irreducible module, which corresponds to the uniquefundamental cubic polynomial invariant and constant-coefficient differential oper-ator. We proved in [X17] that the space of homogeneous polynomial solutions withdegree m for the cubic Dickson invariant differential equation is exactly a directsum of m=2þ 1½ �½ � explicitly determined irreducible E6-submodules, and the wholepolynomial algebra is a free module over the polynomial algebra in the Dicksoninvariant generated by these solutions. Thus, we obtained a cubic E6-generalizationof the classical theorem on harmonic polynomials.

Next we want to describe our new multivariable hypergeometric functions oftype A and their connections with the representations of type-A simple Lie algebrasand the Calogero–Sutherland model. For a 2 C and 0\n 2 Z, we denote

ðaÞn ¼ aðaþ 1Þ � � � ðaþ n� 1Þ; ðaÞ0 ¼ 1 ð29Þ

The well-known Gauss hypergeometric function is

2F1ða; b; c; zÞ ¼X1n¼0

ðaÞnðbÞnn!ðcÞn

zn ð30Þ

Many well-known elementary functions and orthogonal polynomials are specialcases of 2F1ða; b; c; zÞ (e.g., cf. [WG, AAR, X21]). It satisfies the classicalhypergeometric equation

zð1� zÞy00 þ ½c� ðaþ bþ 1Þz�y0 � aby ¼ 0: ð31Þ

Denoting D ¼ z ddz, we can rewrite the above equation as

ðcþDÞ ddz

ðyÞ ¼ ðaþDÞðbþDÞðyÞ ð32Þ

Moreover, it has the differential property:

ddz 2

F1ða; b; c; zÞ ¼ abc 2 F1ðaþ 1; bþ 1; cþ 1; zÞ ð33Þ

Furthermore, the Euler integral representation

2F1ða; b; c; zÞ ¼ CðcÞCðbÞCðc� bÞ

Z 1

0tb�1ð1� tÞc�b�1ð1� ztÞ�adt ð34Þ

holds in the z plane cut along the real axis from 1 to 1.

xxiv Introduction

Let N be the additive semigroup of nonnegative integers and let

CA ¼X

1 p\q n

N�q;p ð35Þ

be the additive semigroup of rank nðn� 1Þ=2 with �q;p as base elements. Fora ¼ P

1 p\q n aq;p�q;p 2 CA, we denote

a1� ¼ a�n ¼ 0; ak� ¼Xk�1

r¼1

ak;r; a�l ¼Xns¼lþ 1

as;l ð36Þ

Given # 2 Cnf�kj0\k 2 Zg and sr 2 C with r 2 f1; . . .; ng, we definedðnðn� 1Þ=2Þ-variable hypergeometric function of type A in [X14] by

XAðs1; ::; sn;#Þfzj;kg ¼Xb2CA

Qn�1s¼1 ðss � bs�Þb�s

h iðsnÞbn�

b!ð#Þbn�

zb; ð37Þ

where

b! ¼Y

1 k\j n

bj;k!; zb ¼Y

1 k\j n

zbj;kj;k ð38Þ

The variables fzj;kg correspond to the negative root vectors of the simple Liealgebra of type An�1 and the above definition has essentially used the structure of itsroot system.

Our functions XAðs1; ::; sn;#Þfzj;kg are indeed natural generalizations of theGauss hypergeometric function 2F1ða; b; c; zÞ. Denote

Dp� ¼Xp�1

r¼1

zp;r@zp;r ; D�q ¼

Xns¼qþ 1

zs;q@zs;q for p 2 f2; . . .; ng; q 2 f1; . . .; n� 1g

ð39Þ

We have the following analogous system of partial differential equations of (32):

ðsr2 � 1�Dr2� þD�r2Þ@zr2 ;r1 ðXAÞ ¼ ðsr2 � 1�Dr2�Þðsr1 �Dr1� þD�

r1ÞðXAÞ ð40Þ

for 1 r1\r2 n� 1, and

ð#þDn�Þ@zn;r ðXAÞ ¼ ðsn þDn�Þðsr �Dr� þD�r ÞðXAÞ ð41Þ

for r 2 f1; . . .; n� 1g. Define

Introduction xxv

1 0 0 � � � 0P½1;2� 1 0 � � � 0

P½1;3� P½2;3� 1 . .. ..

.

..

. ... . .

. . ..

0P½1;n� P½2;n� � � � P½n�1;n� 1

0BBBBBB@

1CCCCCCA ¼

1 0 0 � � � 0z2;1 1 0 � � � 0

z3;1 z3;2 1 . .. ..

.

..

. ... . .

. . ..

0zn;1 zn;2 � � � zn;n�1 1

0BBBBBB@

1CCCCCCA

�1

ð42Þ

and treat P½i;i� ¼ 1. Then P½i;j� are polynomials in fzr;sj1 s\r ng. Moreover, for1 r1\r2 n� 1 and r 2 f1; . . .; n� 1g, we have the differential property:

@zr2 ;r1 ðXAÞ ¼Xr1s¼1

ssP½s;r1�XA½s; r2�; ð43Þ

@zn;rðXAÞ ¼ sn#

Xr

s¼1

ssP½s;r�XA½s; n�; ð44Þ

where XA½i; j� is obtained from XA by changing si to si þ 1 and sj to sj � 1 for1 i\j n� 1 and XA½k; n� is obtained from XA by changing si to si þ 1, sn tosn þ 1 and # to #þ 1 for k 2 1; n� 1. The system of (43) and (44) is an analogue of(33). Furthermore, if Re sn [ 0 and Re ð#� snÞ[ 0, then we have the Eulerintegral representation

XA ¼ Cð#ÞCð#� snÞCðsnÞ

Z 1

0

Yn�1

r¼1

ðXn�1

s¼r

P½r;s� þ tP½r;n�Þ�sr

" #tsn�1ð1� tÞ#�sn�1dt ð45Þ

on the region P½r;n�. Pn�1

s¼r P½r;s�� �

62 ð�1;�1Þh i

for r 2 f1; . . .; n� 1g.Expression (45) is exactly an analogue of (34).

The Calogero–Sutherland model is an exactly solvable quantum many-bodysystem in one-dimension (cf. [Cf, Sb]), whose Hamiltonian is given by

HCS ¼Xni¼1

@2xi þK

X1 p\q n

1

sinh2ðxp � xqÞ; ð46Þ

where K is a constant. Set

nAr2;r1 ¼Yr2�1

s¼r1

e2xr2

e2xr2 � e2xsfor 1 r1\r2 n: ð47Þ

Take ð‚1; . . .;‚nÞ 2 Cn such that

xxvi Introduction

‚1 � ‚2 ¼ � � � ¼ ‚n�2 � ‚n�1 ¼ l and ‚n�1 � ‚n ¼ r 62 N; ð48Þ

for some constants l and r. Based on Etingof’s work [Ep] related to representationsof simple Lie algebra of type An�1, we proved in [X14] that

Ynr¼1

e2ð‚r þðnþ 1Þ=2�rÞxrXAðlþ 1; ::; lþ 1;�l;�rÞfnAr2;r1g ð49Þ

is a solution of the Calogero–Sutherland model.We also introduced in [X14] new multivariable hypergeometric functions related

to the simple Lie algebras of types Bn;Cn;Dn, where the functions of type C giverise to the solutions of the corresponding Olshanestsky–Perelomov model(cf. [OP]).

Partial differential equations are also used by us [X12, X17, X18, X19, X22] toexplicitly determine singular vectors in certain weight modules offinite-dimensional simple Lie algebras and the structures of the modules (someof the results were abstractly determined by Brion [B] before us). In particular, weobtained combinatorial identities on the dimensions of an infinite family of irre-ducible modules of F4; E6 and E7, respectively. Differential-operator representa-tions of Lie algebras are called oscillator representations in physics terminology(e.g., cf. [FC, FSS, G]). In the oscillator representation of the simple Lie algebra oftype F4 lifted from its basic irreducible module, we showed in [X18] that thenumber of irreducible submodules contained in the space of homogeneous har-monic polynomials with degree k 2 is k=3½ �½ � þ ðk � 2Þ=3½ �½ � þ 2. Moreover, inthe polynomial algebra over the 56-dimensional basic irreducible module of thesimple Lie algebra E7, we found in [X22] two three-parameter families of irre-ducible submodules in the solution space of the Cartan’s fourth-order E7-invariantpartial differential equation.

For any ‚ 2 H�, there exists a unique highest-weight G-module Mð‚Þ withhighest weight ‚ such that any other highest-weight G-module with highest weight‚ must be a quotient module of Mð‚Þ. The module Mð‚Þ is called a Verma module(cf. [V1, V2]). Suppose that G is of type An�1. The Weyl group of G in this case isexactly the group Sn of permutations on f1; 2; . . .; ng. In [X19], we derived asystem of variable-coefficient second-order linear partial differential equations thatdetermine the singular vectors in Mð‚Þ, and a differential-operator representation ofSn on the related space of truncated power series. We proved that the solution spaceof the system of partial differential equations is exactly spanned by frð1Þjr 2 Sng.Moreover, the singular vectors in Mð‚Þ are given by those rð1Þ that are polyno-mials. Xiao [XW1] found the explicit formula for rð1Þ when r is the reflectionassociated with a positive root and determined the condition for it to be a poly-nomial. A similar result as that in [X19] for the simple Lie algebra of type C2 wasproved by us [X12]. Xiao [XW2] generalized our result of An�1 in [X19] and that ofC2 in [X12] to all finite-dimensional simple Lie algebras.

Introduction xxvii

The second important feature of this book is the construction of irreduciblerepresentations of simple Lie algebras from their natural inhomogeneous oscillatorrepresentations, which give fractional representations of the corresponding Liegroups.

Suppose that n 2 is an integer. Let G be the simple Lie algebra of typesAn; Bn; Dn;E6 and E7. It is known that it has the following subalgebradecomposition

G ¼ G� � G0 � Gþ ð50Þ

with

½Gþ ;G�� � G0; ½G0;G�� � G�; ½G�;G�� ¼ f0g; ð51Þ

where G0 is a direct sum of a one-dimensional trivial Lie algebra with the simpleLie algebra G0

0 of types An�1; Bn�1; Dn�1; D5 and E6, respectively. Moreover, G�forms the minimal natural irreducible G0

0-module with respect to theleft-multiplication representation of G if G is not of type E6, and G� gives the “spinrepresentation” of G0

0 when G is of type E6. Moreover, Gþ is its dual G00 -module ofG�. We lift the representation of G0

0 on G� to an oscillator representation of G0 onthe algebra A ¼ C½x1; . . .; xm� of polynomial functions on G�, where m ¼ dimG�.

Denote by Ak the subspace of homogeneous polynomials with degree k. Weextend the oscillator representation of G0 to an inhomogeneous oscillator repre-sentation of G by fixing Gþ jA ¼ Pm

i¼1 C@xi and solving G�jA inPm

i¼1 A2@xi .When G is of type An, the extended representation is exactly the one induced fromthe corresponding projective transformations (cf. [ZX]). If G is of types Bn and Dn,the extended representation is exactly the one induced from the correspondingconformal transformations (cf. [XZ]). In the cases of E6 and E7, we found thecorresponding fractional representations of the corresponding Lie groups (cf. [X23,X26]).

We define a Lie algebra

GA ¼ GjA �AG0 ð52Þ

with the Lie bracket

½d1 þ f1u; d2 þ f2v� ¼ ðd1d2 � d2d1Þþ ðd1ð f2Þv� d2ð f1Þuþ f1 f2½u; v�Þ ð53Þ

for d1; d2 2 GjA, f1; f2 2 A and u; v 2 G0. Using Shen’s idea of mixed product (cf.[Sg]), we found a Lie algebra monomorphism i : G ! GA such that i Gð Þ 6� GjA.Let M be any G00 -module. We extend it to a G0-module by letting the centralelement take a constant map. Define a GA-module structure on bM ¼ A�C M by

xxviii Introduction

ðdþ fuÞðg� wÞ ¼ dðgÞ � wþ f g� uðwÞ ð54Þ

for d 2 GjA; f ; g 2 A; u 2 G0 and w 2 M . Then bM becomes a G-module withthe action

nð-Þ ¼ iðnÞð-Þ for n 2 G; - 2 bM : ð55Þ

The map M 7! bM defines a new functor from the category of G00 -modules to thecategory of G-modules. When M is an irreducible G00 -module, C½G��ð1�MÞ isnaturally an irreducible G-module due to the fact Gþ jA ¼ Pm

i¼1 C@xi , where C½G��is the polynomial algebra generated by G�. Thus, the less explicit functorM 7!C½G��ð1�MÞ gives a polynomial extension from irreducible G0

0-modules toirreducible G-modules. In particular, it can be used to construct Gel’fand–Zetlin-type bases for E6 from those for D5 and then construct Gel’fand–Zetlin-typebases for E7 from those for E6 (cf. [GZ1, GZ2, Ma]). When M is afinite-dimensional irreducible G0

0-module, Zhao and the author [ZX, XZ] deter-

mined the condition for bM ¼ C½G��ð1�MÞ if G is of types An; Bn and Dn. Thecondition for E6 and E7 was found by the author in [X23, X26]. In these works, theidea of Kostant’s characteristic identities in [Kb] played a key role. Our approachesheavily depend on the explicit decompositions of A and the tensor module of G�with any finite-dimensional G0

0-module into direct sums of irreducible G00-sub-

modules. This is a reason why we adopted the case-by-case approach.The above works also yield a one-parameter (c) family of inhomogeneous

first-order differential-representations of G. Letting these operators act on the spaceof exponential–polynomial functions that depend on a parametric vector ~a 2 Cm,we proved that the space forms an irreducible G-module for any c 2 C if~a is not onan explicitly given algebraic variety when G is not of type An (cf. [X25, X27]). Theequivalent combinatorial properties of the representations played key roles. In thecase of An, we [X24] got new nonweight irreducible modules of the simple Liealgebras of types An and C‘ if n ¼ 2‘ is an even integer. By partially swappingdifferential operators and multiplication operators, we obtain more general oscillatorrepresentations of G. If c 62 Z, we obtained in [X24] explicit infinite-dimensionalweight modules with finite-dimensional weight subspaces for the simple Liealgebras of types An and C‘ when n ¼ 2‘ is an even integer. When c 62 Z=2; , we[X25] found explicit infinite-dimensional weight modules with finite-dimensionalweight subspaces for the simple Lie algebras of types Bn and Dn. These weightmodules are not of highest type.

Below we give a chapter-by-chapter introduction. Throughout this book, F isalways a field with characteristic 0 such as the field Q of rational numbers, the fieldR of real numbers, and the field C of complex numbers. All the vector spaces areassumed over F unless they are specified.

The first part of this book is the classical theory of finite-dimensional Liealgebras and their representations, which serves as the preparation of our later maincontents.

Introduction xxix

In Chap. 1, we give basic concepts and examples of Lie algebras. Moreover,Engel’s theorem on nilpotent Lie algebras and Lie’s theorem on solvable Liealgebras are proved. Furthermore, we derive the Jordan–Chevalley decompositionof a linear transformation and use it to show Cartan’s criterion on the solvability.

In Chap. 2, we first introduce the Killing form and prove that the semisimplicityof a finite-dimensional Lie algebra over C is equivalent to the nondegeneracy of itsKilling form. Then we use the Killing form to derive the decomposition of afinite-dimensional semisimple Lie algebra over C into a direct sum of simple ideals.Moreover, it is showed that a derivation of G must be a left-multiplication operatorof G for such a Lie algebra. Furthermore, we study the completely reduciblemodules of a Lie algebra and prove the Weyl’s theorem of complete reducibility.The equivalence of the complete reducibility of real and complex modules is alsogiven. Cartan’s root-space decomposition of a finite-dimensional semisimple Liealgebra over C is derived. In particular, we prove that such a Lie algebra is gen-erated by two elements. The complete reducibility of finite-dimensional modulesof the simple Lie algebra of type A1 plays an important role in proving the prop-erties of the corresponding root systems.

In Chap. 3, we start with the axiom of root system and give the root systems ofspecial linear algebras, orthogonal Lie algebras, and symplectic Lie algebras. Thenwe derive some basic properties of root systems; in particular, the existence of thebases of root systems. As finite symmetries of root systems, the Weyl groups areintroduced and studied in detail. The classification and explicit constructions of rootsystems are presented. The automorphism groups of root systems are determined.As a preparation for later representation theory of Lie algebras, the correspondingweight lattices and their saturated subsets are investigated.

In Chap. 4, we show that the structure of a finite-dimensional semisimple Liealgebra over C is completely determined by its root system. Moreover, we provethat any two Cartan subalgebras of such a Lie algebra G are conjugated under thegroup of inner automorphisms of G. In particular, the automorphism group of G isdetermined when it is simple. Furthermore, we give explicit constructions of thesimple Lie algebras of exceptional types.

By Weyl’s Theorem, any finite-dimensional representation of a finite-dimensional semisimple Lie algebra over C is completely reducible. The maingoal in Chap. 5 is to study finite-dimensional irreducible representations of afinite-dimensional semisimple Lie algebra over C. First we introduce the universalenveloping algebra of a Lie algebra and prove the Poincaré–Birkhoff–Witt(PBW) Theorem on its basis. Then we use the universal enveloping algebra of afinite-dimensional semisimple Lie algebra G to construct the Verma modules of G.Moreover, we prove that any finite-dimensional G-module is the quotient of aVerma module modulo its maximal proper submodule, whose generators areexplicitly given. Furthermore, the Weyl’s character formula of a finite-dimensionalirreducible G-module is derived, and the dimensional formula of the module isdetermined. Finally, we decompose the tensor module of two finite-dimensionalirreducible G-modules into a direct sum of irreducible G-submodules in terms ofcharacters.

xxx Introduction

Part II is about the explicit representations of finite-dimensional simple Liealgebra over C, which is the main content of this book.

In Chap. 6, we give various explicit representations of the simple Lie algebras oftype A. First we present a fundamental lemma of solving flag partial differentialequations for polynomial solutions, which was due to our work [X16]. Then wepresent the canonical bosonic and fermionic oscillator representations over theirminimal natural modules and minimal orthogonal modules. Moreover, we deter-mine the structure of the noncanonical oscillator representations obtained from thecanonical bosonic oscillator representations over their minimal natural modules andminimal orthogonal modules by partially swapping differential operators andmultiplication operators. The case over their minimal natural modules was essen-tially due to Howe [Hr4]. The results in the case over their minimal orthogonalmodules are generalizations of the classical theorem on harmonic polynomials. Theresults were due to Luo and the author [LX1]. We construct a functor from thecategory of An�1-modules to the category of An-modules, which is related to n-dimensional projective transformations. This work was due to Zhao and the author[ZX]. Finally, we present multiparameter families of irreducible projective oscil-lator representations of the algebras given in [X24].

Chapter 7 is devoted to natural explicit representations of the simple Lie algebraof type D. First we present the canonical bosonic and fermionic oscillator repre-sentations over their minimal natural modules. Then we determine the structureof the noncanonical oscillator representations obtained from the above bosonicrepresentations by partially swapping differential operators and multiplicationoperators, which are generalizations of the classical theorem on harmonic poly-nomials (cf. [LX2]). Furthermore, we speak about a functor from the category ofDn-modules to the category of Dnþ 1-modules, which is related to 2n-dimensionalconformal transformations (cf. [XZ]). In addition, we present multiparameterfamilies of irreducible conformal oscillator representations of the algebras given in[X25], and some of them are related to an explicitly given algebraic variety.

Chapter 8 is about natural explicit representations of the simple Lie algebra oftype B. We give the canonical bosonic and fermionic oscillator representations overtheir minimal natural modules. Moreover, we determine the structure of the non-canonical oscillator representations obtained from the above bosonic representa-tions by partially swapping differential operators and multiplication operators,which are generalizations of the classical theorem on harmonic polynomials (cf.[LX2]). Furthermore, we present a functor from the category of Bn-modules to thecategory of Bnþ 1-modules, which is related to ð2nþ 1Þ-dimensional conformaltransformations (cf. [XZ]). Besides, we present multiparameter families of irre-ducible conformal oscillator representations of the algebras given in [X25], andsome of them are related to an explicitly given algebraic variety.

Chapter 9 determines the structure of the canonical bosonic and fermionicoscillator representations of the simple Lie algebras of type C over their minimalnatural modules. Moreover, we study the noncanonical oscillator representationsobtained from the above bosonic representations by partially swapping differentialoperators and multiplication operators, and obtain a two-parameter family of new

Introduction xxxi

infinite-dimensional irreducible representations (cf. [LX3]). Finally, we presentmultiparameter families of irreducible projective oscillator representations of thealgebras given in [X24].

In Chap. 10, we determine the structure of the canonical bosonic and fermionicoscillator representations of the simple Lie algebra of type G2 over itsseven-dimensional module. Moreover, we present a one-parameter family of con-formal oscillator representations of G2 derived from those of the simple Lie algebraof type B3 and determine their irreducibility. The result is newly obtained by theauthor. Furthermore, we use partial differential equations to find the explicit irre-ducible decomposition of the space of polynomial functions on 26-dimensionalbasic irreducible module of the simple Lie algebra of type F4 (cf. [X18]).

Chapter 11 studies explicit representations of the simple Lie algebra of type E6.First we prove the cubic E6-generalization of the classical theorem on harmonicpolynomials given [X17]. Then we study the functor from the module category ofD5 to the module category of E6 developed in [X26]. Finally, we give a family ofinhomogeneous oscillator representations of the simple Lie algebra of type E6 on aspace of exponential–polynomial functions and prove that their irreducibility isrelated to an explicit given algebraic variety. The work was due to an unpublishedpaper of the author.

Explicit representations of the simple Lie algebra of type E7 are given in Chap. 12.By solving certain partial differential equations, we find the explicit decompositionof the polynomial algebra over the 56-dimensional basic irreducible module ofthe simple Lie algebra E7 into a sum of irreducible submodules (cf. [X22]). Then westudy the functor from the module category of E6 to the module category of E7

developed in [X23]. Moreover, we construct a family of irreducible inhomogeneousoscillator representations of the simple Lie algebra of type E7 on a space of expo-nential–polynomial functions, related to an explicitly given algebraic variety(cf. [X27]).

How to construct explicit irreducible representations of E8 is still a challengingproblem. Part III is an extension of Part II.

In Chap. 13, we first establish two-parameter Z2-graded supersymmetric oscil-lator generalizations of the classical theorem on harmonic polynomials for thegeneral linear Lie superalgebra. Then we extend the result to two-parameter Z-graded supersymmetric oscillator generalizations of the classical theorem on har-monic polynomials for the orthosymplectic Lie superalgebras. This chapter is areformulation of Luo and the author’s work [LX4].

Linear codes with large minimal distances are important error correcting codes ininformation theory. Orthogonal codes have more applications in the other fields ofmathematics. In Chap. 14, we study the binary and ternary orthogonal codesgenerated by the weight matrices of finite-dimensional modules of simple Liealgebras. The Weyl groups of the Lie algebras act on these codes isometrically. Itturns out that certain weight matrices of the simple Lie algebras of types A and Dgenerate doubly-even binary orthogonal codes and ternary orthogonal codes withlarge minimal distances. Moreover, we prove that the weight matrices of F4, E6, E7

xxxii Introduction

and E8 on their minimal irreducible modules and adjoint modules all generateternary orthogonal codes with large minimal distances. In determining the minimaldistances, we have used the Weyl groups and branch rules of the irreducible rep-resentations of the related simple Lie algebras. The above results are taken from[X20].

In Chap. 15, we prove that certain variations of the classical Weyl functions aresolutions of the Calogero–Sutherland model and its generalizations—theOlshanestsky–Perelomov model in various cases. New multivariable hypergeo-metric functions related to the root systems of classical simple Lie algebras areintroduced. In particular, those of type A give rise to solutions of the Calogero–Sutherland model based on Etingof’s work [Ep] and those of type C yield solutionsof the Olshanestsky–Perelomov model of type C based on Etingof and Styrkas’work [ES]. The differential properties and multivariable hypergeometric equationsfor these multivariable hypergeometric functions are given. The Euler integralrepresentations of the type-A functions are found. These results come from theauthor’s work [X14].

References

[AAR] G. Andrews, R. Askey, R. Roy, Special Functions (Cambridge University Press, 1999)[B] M. Brion, Invariants d’un sous-groupe unipotent maxaimal d’un groupe semi-simple. Ann.

Inst. Fourier (Grenoble) 33(1), 1–27 (1983)[B1] Z. Bai, Gelfand-Kirillov dimensionas of the Z2-graded oscillator representations of sl(n),

Acta Math. Sinica (Engl. Ser.) 31(6), 921–937 (2015)[B2] Z. Bai, Gelfand-Kirillov dimensionas of the Z-graded oscillator representations of oðn;CÞ

and spð2n;CÞ, Commun. Algebra (to appear)[BBL] G. Benkart, D. Britten, F.W. Lemire, Modules with bounded multiplicities for simple Lie

algebras, Math. Z. 225, 333–353 (1997)[BDK] A. Barakat, A. De Sole, V.G. Kac, Poisson vertex algebras in the theory of Hamiltonian

equations, Jpn. J. Math. 4(2), 141–252 (2009)[BFL] D. Britten, V. Futorny, F.W. Lemire, Simple A2-modules with a finite dimensional weight

space, Commun. Algebra 23, 467–510 (1995)[BHL] D. Britten, J. Hooper, F.W. Lemire, Simple Cn-modules with multiplicities 1 and

applications, Canad. J. Phys. 72, 326–335 (1994)[BL1] D. Britten, F.W. Lemire, A classification of simple Lie modules having 1-dimensional

weight space. Trans. Amer. Math. Soc. 299, 683–697 (1987)[BL2] D. Britten, F.W. Lemire, On modules of bounded multiplicities for symplectic algebras.

Trans. Amer. Math. Soc. 351, 3413–3431 (1999)[Bo] R. Beerends, E. Opdam, Certain hypergeometric series related to the root system BC. Trans.

Amer. Math. Soc. 119, 581–609 (1993)[Cb] B. Cao, Solutions of Navier equations and their representation structure. Adv. Appl. Math.

43, 331–374 (2009)[Cf] F. Calogero, Solution of the one-dimensional n-body problem with quadratic and /or inversely

quadratic pair potentials, J. Math. Phys. 12, 419–432 (1971)[D] L. Dickson, A class of groups in an arbitrary realm connected with the configuration of the 27

lines on a cubic surface. J. Math. 33, 145–123 (1901)[Ep] P. Etingof, Quantum integrable systems and representations of Lie algebras. J. Math. Phys. 36

(6), 2637–2651 (1995)

Introduction xxxiii

[ES] P. Etingof, K. Styrkas, Algebraic integrability of Schrӧdinger operators and representations ofLie algebras. Compositio Math. 98(1), 91–112 (1995)

[FC] F.M. Fernández, E.A. Castro, Algebraic Methods in Quantum Chemistry and Physics (CRCPress, Inc., 1996)

[Fs] S.L. Fernando, Lie algebra modules with finite-dimensional weight spaces, I. Trans. Amer.Math. Soc. 322, 757–781 (1990)

[Fv] V. Futorny, The weight representations of semisimple finite-dimensional Lie algebras. Ph.D.Thesis, Kiev University, 1987

[G] H. Georgi, Lie Algebras in Particle Physics, 2nd edn. (Perseus Books Group, 1999)[GDi1] I.M. Gel’fand, L.A. Dikii, Asymptotic behaviour of the resolvent of Sturm-Liouville

equations and the algebra of the Korteweg-de Vries equations. Rus. Math. Surv. 30(5), 77–113(1975)

[GDi2] I.M. Gel’fand, L.A. Dikii, A Lie algebra structure in a formal variational calculation. Func.Anal. Appl. 10, 16–22 (1976)

[GDo1] I. M. Gel’fand, I. Ya. Dorfman, Hamiltonian operators and algebraic structures related tothem, Funkts. Anal. Prilozhen 13, 13–30 (1979)

[GDo3] I. M. Gel’fand, I. Ya, Dorfman, Hamiltonian operators and infinite- dimensional Liealgebras, Funkts. Anal. Prilozhen 14, 23–40 (1981)

[GS] J. Gervais, B. Sakita, Field theory interpretation of supergauges in dual models. Nucl. Phys.B34(2), 632–639 (1971)

[Hr4] R. Howe, in Perspectives on invariant theory: Schur duality, multiplicity-free actions andbeyond. Schur Lect. (1992) (Tel Aviv), 1–182. Israel Mathematics Conference Proceedings, 8,Bar-Ilan University, Ramat Gan (1995)

[Kb] B. Kostant, On the tensor product of a finite and an infinite dimensional representation.J. Fund. Anal. 20, 257–285 (1975)

[Kv2] V.G. Kac, Vertex Algebras for Beginners. University Lecture Series, vol. 10 (AMS,Providendence RI, 1996)

[Lie] S. Lie, Begrüdung einer Invariantentheorie der Berührungstransformationen. Math. Ann. 8,215–288 (1874)

[LX1] C. Luo, X. Xu, Z2-graded oscillator generalizations of slðnÞ, Commun. Algebra 41, 3147–3173 (2013)

[LX2] C. Luo, X. Xu, Z-Graded oscillator generalizations of the classical theorem on harmonicpolynomials. J. Lie Theory 23, 979–1003 (2013)

[LX3] C. Luo, X. Xu, Z-graded oscillator representations of Symplectic Lie algebras. J. Algebra403, 401–425 (2014)

[LX4] C. Luo, X. Xu, Supersymmetric analogues of the classical theorem on harmonicpolynomials. J. Algebra Appl. 13(6),1450011 42pp. (2014)

[Ma] A. Molev, A basis for representations of symplectic Lie algebras. Comm. Math. Phys. 201,591–618 (1999)

[Mo] O. Mathieu, Classification of irreducible weight modules. Ann. Inst. Fourier (Grenoble) 50,537–592 (2000)

[Mh] H. Miyazawa, Baryon number changing currents. Prog. Theor. Phys. 36(6), 1266–1276(1966)

[Op] P.J. Olver, Equivalence, Invariants and Symmetry (Cambridge University Press, Cambridge,1995)

[PHB] V. Pless, W.C. Huffman, R.A. Brualdi, An introduction to algebraic codes, in Handbook ofCoding Theory, ed. by V. Pless, W. Huffman (Chap. 1) (Elsevier Science B.V., 1998)

[R1] P. Ramond, Dual theory of free fermions. Phys. Review D 3(10), 2415–2418 (1971)[Sb] B. Sutherland, Exact results for a quantum many-body problem in one-dimension. Phys. Rev.

A 5, 1372–1376 (1972)[Sc] C. Shannon, A mathematical theory of communication. Bell System Tech. J. 27, 379–423,

623–656 (1948)

xxxiv Introduction

[Sg] G. Shen, Graded modules of graded Lie algebras of Cartan type (I)—mixed product ofmodules. Sci. Chin. A 29, 570–581 (1986)

[V1] D.-N. Verma, Structure of certain induced representations of complex semisimple Liealgebras. Thesis, Yale University, 1966

[V2] D.-N. Verma, Structure of certain induced representations of complex semisimple Liealgebras. Bull. Amer. Math. Soc. 74, 160–166 (1968)

[WG] Z. Wang, D. Guo, Special Functions. (World Scientific, Singapore, 1998)[X3] X. Xu, Hamiltonian superoperators. J. Phys. A 28, 1681–1698 (1995)[X4] X. Xu, Hamiltonian operators and associative algebras with a derivation. Lett. Math.Phys. 33,

1–6 (1995)[X5] X. Xu, Differential invariants of classical groups. Duke Math. J. 94, 543–572 (1998)[X6] X. Xu, Introduction to Vertex Operator Superalgebras and their Modules. (Kluwer

Academic Publishers, Dordrecht/Boston/London, 1998)[X7] X. Xu, Generalizations of the Block algebras. Manuscripta Math. 100, 489–518 (1999)[X8] X. Xu, New generalized simple Lie algebras of Cartan type over a field with characteristic 0.

J. Algebra 224, 23–58 (2000)[X9] X. Xu, Variational calculus of supervariables and related algebraic structures. J. Algebra 223,

386-437 (2000)[X10] X. Xu, Equivalence of conformal superalgebras to Hamiltonian superoperators. Algebra

Colloq. 9, 63–92 (2001)[X11] X. Xu, Poisson and Hamiltonian superpairs over polarized associative algebras. J. Phys. A

34, 4241–4265 (2001)[X12] X. Xu, Differential equations for singular vectors of sp(2n). Commun. Algebra 33, 4177–

4196 (2005)[X14] X. Xu, Path hypergeometric functions. J. Algebra Appl. 6, 595–653 (2007)[X16] X.Xu, Flag partial differential equations and representations of Lie algebras. Acta Appl.

Math. 102, 249–280 (2008)[X17] X. Xu, A cubic E6-generalization of the classical theorem on harmonic polynomials. J. Lie

Theory 21, 145–164 (2011)[X18] X. Xu, Partial differential equation approach to F4. Front. Math. China 6, 759–774 (2011)[X19] X. Xu, Differential-operator representation of Sn and singular vectors in Verma modules.

Algebr. Represent. Theor. 15, 211–231 (2012)[X20] X. Xu, Representations of Lie algebras and coding theory. J. Lie Theory 22(3), 647–682

(2012)[X21] X. Xu, Algebraic Approaches to Partial Differential Equations, (Springer, Heidelberg/New

York/Dordrecht/ London, 2013)[X22] X. Xu, Partial differential equation approach to E7. Acta Math Sin. (Engl. Ser.) 31, 177–200

(2015)[X23] X. Xu, A new functor from E6-Mod to E7-Mod. Commun. Algebra 43(9), 3589–3636

(2015)[X24] X. Xu, Projective oscillator representations of sl(n + 1) and sp(2m + 2). J. Lie Theory 26,

96–114 (2016)[X25] X. Xu, Conformal oscillator representations of orthogonal Lie algebras. Sci. China: Math.

(Engl Ser.) 59(1), 37–48 (2016)[X26] X. Xu, A new functor from D5-Mod to E6-Mod, Acta Math Sin. (Engl. Ser.) 32(8), 867–

892 (2016)[X27] X. Xu, Representations of E7, combinatorics and algebraic varieties. J. Algebra Appl. (to

appear)[XW1] W. Xiao, Differential equations and singular vectors in Verma modules over slðn;CÞsl(n;

C). Acta Math. Sinica, (Engl. Ser.) 31(7), 1057–1066 (2015)[XW2] W. Xiao, Differential-operator representations of Weyl groups and singular vectors in

Verma modules. Sci. China: Math. (Engl. Ser.) (to appear)

Introduction xxxv

[XZ] X. Xu, Y. Zhao, Extensions of the conformal representations for orthogonal Lie algebras.J. Algebra 377, 97–124 (2013)

[ZX] Y. Zhao, X. Xu, Generalized projective representations for sl(n+1). J. Algebra 328, 132–154(2011)

xxxvi Introduction