Representations of Lie Algebras and Partial Differential Equations

642 Pages·2017·7.755 MB·English

by  Xiaoping Xu

Xiaoping Xu
Representations
of Lie Algebras
and Partial
Differential
Equations
Representations of Lie Algebras and Partial
Differential Equations
Xiaoping Xu
Representations of Lie
Algebras and Partial
Differential Equations
123
XiaopingXu
Institute of Mathematics
Academy of Mathematics
andSystemsSciences,
ChineseAcademy of Sciences
Beijing
P.R.China
and
Schoolof Mathematics
University of Chinese
Academyof Sciences
Beijing
P.R.China
ISBN978-981-10-6390-9 ISBN978-981-10-6391-6 (eBook)
DOI 10.1007/978-981-10-6391-6
LibraryofCongressControlNumber:2017950037
MathematicsSubjectClassification(2010):17B10,35C05,17B20,33C67,94B05
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To My Former Ph.D Thesis Advisors:
Profs. James Lepowsky and Robert Lee
Wilson
Preface
Symmetry is an important phenomenon in the natural world. Lie algebra is not
purelyanabstractmathematicsbutafundamentaltoolofstudyingsymmetry.Infact,
Norwegian mathematician Sorphus Lie introduced Lie group and Lie algebra in
1874 in order to study the symmetry of differential equations. Lie algebras are the
infinitesimal structures (bones) of Lie groups, which are symmetric manifolds. Lie
theoryhasextensiveandimportantapplicationsinmanyotherfieldsofmathematics,
such as geometry, topology, number theory, control theory, integrable systems,
operator theory, and stochastic process. Representations of finite-dimensional
semisimple Lie algebras play fundamental roles in quantum mechanics. The con-
trollability property of the unitary propagator of an N-level quantum mechanical
system subject to a single control field can be described in terms of the structure
theoryofsemisimpleLiealgebras.Moreover,Liealgebraswereusedtoexplainthe
degeneraciesencounteredingeneticcodesastheresultofasequenceofsymmetry
breakingsthathaveoccurredduringitsevolution.Thestructuresandrepresentations
of simple Lie algebras are connected with solvable quantum many-body system in
one-dimension. Our research also showed that the representation theory of Lie
algebrasis connected with algebraic coding theory.
Theexistingclassicalbooksonfinite-dimensionalLiealgebras,suchastheones
by Jacobson and by Humphreys, purely focus on the algebraic structures of semi
simpleLiealgebrasandtheirfinite-dimensionalrepresentations.Explicitirreducible
representations of simple Lie algebras had not been addressed extensively.
Moreover,therelationsofLiealgebraswiththeothersubjectshadnotbeennarrated
that much. It seems to us that a book on Lie algebras with more extensive view is
needed in coupling with modern development of mathematics, sciences, and
technology.
Thisbookismainlyanexpositionoftheauthor’sworksandhisjointworkswith
his former students on explicit representations of finite-dimensional simple Lie
algebras, related partial differential equations, linear orthogonal algebraic codes,
combinatorics, and algebraic varieties. Various oscillator generalizations of the
classical representation theorem on harmonic polynomials are presented. New
functorsfromtherepresentationcategoryofasimpleLiealgebratothatofanother
vii
viii Preface
simpleLiealgebraaregiven.Partialdifferentialequationsplaykeyrolesinsolving
certain representation problems. The weight matrices of the minimal and adjoint
representationsoverthesimpleLiealgebrasoftypesEandFareprovedtogenerate
ternary orthogonal linear codes with large minimal distances. New multivariable
hypergeometric functions related to the root systems of simple Lie algebras are
introduced 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 linear
algebra. It is our wish that the results this book can also be easily understood by
nonalgebraists andappliedto theother mathematical fields and physics. It consists
of three parts. The first part is mainly the classical structure and finite-dimensional
representation theory of finite-dimensional semisimple Lie algebras, written with
Humphreys’book“IntroductiontoLieAlgebrasandRepresentationTheory”asthe
main reference, where we give more examples and direct constructions of simple
Lie algebras of exceptional types, revise some arguments, and prove some new
statements.Thispartservesasthepreparationforthemaincontext.Thesecondpart
isourexplicitrepresentationtheoryoffinite-dimensionalsimpleLiealgebras.Many
of the irreducible representations in this part are infinite-dimensional, and some
of them are even not of highest-weight type. Certain important natural represen-
tation problems are solved by means of solving partial differential equations. In
particular, some of our irreducible presentations are completely characterized by
invariant partial differential equations. Newrepresentation functors are constructed
from inhomogeneous oscillator representations of simple Lie algebras, which give
fractional representations of the corresponding Lie groups, such as the projective
representations of special linear Lie groups and the conformal representations of
complexorthogonal groups.Thethirdpartisanextensionofthesecond part. First
we give supersymmetric generalizations of the classical representation theorem on
harmonic polynomials. They can also be viewed as certain supersymmetric Howe
dualities.Thenwepresentourtheoryofrepresentationtheoreticcodes.Finally,we
talk about root-related integrable systems and our new multivariable hypergeo-
metric functions, which are natural multivariable analogues of the classical Gauss
hypergeometric functions. The corresponding hypergeometric partial differential
equations are found.
Part of this book has been taught for many times at the University of Chinese
Academy of Sciences. The book can serve as a research reference book for
mathematiciansandscientists.Itcanalsobetreatedasatextbookforstudentsafter
a proper selection of materials.
Beijing, P.R. China Xiaoping Xu
2017
Acknowledgements
The research in this book was partly supported by NSFC Grants 11671381,
11321101andHuaLoo-KengKeyMathematicalLaboratory,ChineseAcademyof
Sciences. We thank the reviewers and Dr. Ramond Peng for their helpful
comments.
ix
Contents
Part I Fundament of Lie Algebras
1 Preliminary of Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Basic Definitions and Examples. . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Lie Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Ideals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Homomorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5 Nilpotent and Solvable Lie Algebras . . . . . . . . . . . . . . . . . . . . 21
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2 Semisimple Lie Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1 Killing Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3 Real and Complex Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.4 Weyl’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.5 Root Space Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.6 Properties of Roots and Root Subspaces. . . . . . . . . . . . . . . . . . 53
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3 Root Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.1 Definitions, Examples and Properties . . . . . . . . . . . . . . . . . . . . 61
3.2 Weyl Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.3 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.4 Automorphisms, Constructions and Weights. . . . . . . . . . . . . . . 86
4 Isomorphisms, Conjugacy and Exceptional Types . . . . . . . . . . . . . 95
4.1 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2 Cartan Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.3 Conjugacy Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.4 Simple Lie Algebra of Exceptional Types . . . . . . . . . . . . . . . . 114
xi
xii Contents
5 Highest-Weight Representation Theory. . . . . . . . . . . . . . . . . . . . . . 125
5.1 Universal Enveloping Algebras . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2 Highest-Weight Modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.3 Formal Characters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.4 Weyl’s Character Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.5 Dimensional Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Part II Explicit Representations
6 Representations of Special Linear Algebras . . . . . . . . . . . . . . . . . . 155
6.1 Fundamental Lemma on Polynomial Solutions . . . . . . . . . . . . . 156
6.2 Canonical Oscillator Representations . . . . . . . . . . . . . . . . . . . . 162
6.3 Noncanonical Representations I: General . . . . . . . . . . . . . . . . . 169
6.4 Noncanonical Representations II: n þ1\n . . . . . . . . . . . . . . 176
1 2
6.5 Noncanonical Representations III: n þ1¼n . . . . . . . . . . . . . 184
1 2
6.6 Noncanonical Representations VI: n ¼n . . . . . . . . . . . . . . . . 193
1 2
6.7 Extensions of the Projective Representations . . . . . . . . . . . . . . 200
6.8 Projective Oscillator Representations . . . . . . . . . . . . . . . . . . . . 211
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
7 Representations of Even Orthogonal Lie Algebras . . . . . . . . . . . . . 217
7.1 Canonical Oscillator Representations . . . . . . . . . . . . . . . . . . . . 218
7.2 Noncanonical Oscillator Representations . . . . . . . . . . . . . . . . . 223
7.3 Extensions of the Conformal Representation. . . . . . . . . . . . . . . 231
7.4 Conformal Oscillator Representations. . . . . . . . . . . . . . . . . . . . 242
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
8 Representations of Odd Orthogonal Lie Algebras. . . . . . . . . . . . . . 253
8.1 Canonical Oscillator Representations . . . . . . . . . . . . . . . . . . . . 254
8.2 Noncanonical Oscillator Representations . . . . . . . . . . . . . . . . . 258
8.3 Extensions of the Conformal Representation. . . . . . . . . . . . . . . 272
8.4 Conformal Oscillator Representations. . . . . . . . . . . . . . . . . . . . 283
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
9 Representations of Symplectic Lie Algebras . . . . . . . . . . . . . . . . . . 293
9.1 Canonical Oscillator Representations . . . . . . . . . . . . . . . . . . . . 293
9.2 Noncanonical Oscillator Representations . . . . . . . . . . . . . . . . . 297
9.2.1 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
9.2.2 Proof of the Theorem When n ¼n . . . . . . . . . . . . . . 299
2
9.2.3 Proof of the Theorem When n \n \n. . . . . . . . . . . . 302
1 2
9.2.4 Proof of the Theorem When n ¼n \n . . . . . . . . . . . 308
1 2
9.3 Projective Oscillator Representations . . . . . . . . . . . . . . . . . . . . 317
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328