Introduction to representation theory pavel etingof, oleg golberg, sebastian hensel, tiankai liu, alex schwendner, dmitry vaintrob, and elena yudovina. Compact lie groups and representation theory lecture notes. An abstract lie algebra g is a real or complex vector. Lectures 1 to 11 are formal, lectures 12 to 16 are rather meant as an inspiration for self study. His main research interests are in group representation theory and harmonic analysis, both pure and applied. Introduction to representation theory mit opencourseware. Very basic lie theory roger howe department of mathematics, yale university, new haven, ct 06520 lie theory, the theory of lie groups, lie algebras and their applications, is a fundamental part of mathematics. Lie algebras, their representation theory and gl minor thesis. Teichmuller space is a fundamental space that is important in many areas of mathematics and physics. The essential phenomenon of lie theory, to be explicated in the rest of this paper, is that one may associate in a natural way to a lie group g its lie algebra g. Representation theory investigates the different ways in which a given algebraic objectsuch as a group or a lie algebracan act on a vector space. One of the most important applications of lie algebra representations is to the representation theory of real reductive lie group. Lie groups, lie algebras, and their representations university of.
In addition to the above mentioned, one should add representations over nonarchimedian local. Representation theory of lie superalgebras and related topics generalized teichmuller spaces, spin structures, and ptolemy transformations ivan chiho, ip kyoto university abstract. Besides being a subject of great intrinsic beauty, the theory enjoys the additional benefit of having applications in myriad contexts outside pure mathematics, including quantum field theory and. Two other recommendable texts which only discuss lie algebras are the books \introduction to lie algebras and representation theory by j. Algebras and representation theory features carefully refereed papers relating, in its broadest sense, to the structure and representation theory of algebras, including lie algebras and superalgebras, rings of differential operators, group rings and algebras, c algebras and hopf algebras, with particular emphasis on quantum groups. In this theory, one considers representations of the group algebra a cg of a. Pdf this primer was written during my last visit to dublin, march 2016. The theory as we know it today began with the ground breaking work of the norwegian mathematician sophus lie, who introduced the notion of continuous transformation groups and showed the crucial role that lie algebras play in their classi cation and representation theory. A lie algebra is a vector space over a eld fendowed with a bracket operation l l. The proof of the peterweyl theorem in chapter v was strongly inspired by taos online notes 34 and 33. A very efficient introduction to lie algebra representation theory. We then show that lie 2algebras can be classi ed up to equivalence in terms of a lie algebra g, a representation of g on a vector space v, and an element of the lie. Since world war ii it has been the focus of a burgeoning research effort, and is. May 31, 2000 topics include definitions and examples of lie groups and lie algebras, the relationship between lie groups and lie algebras via the exponential mapping, the basics of representations theory, the bakercampbellhausdorff formula, a detailed study of the representations of su3, and a brief survey of the representation theory of general.
This lecture is part of a tutorial on lie algebras. We first study the example of sl2c and then provide. His main research interests are combinatorics and representation theory, especially the representation theory of the symmetric group. Lie algebras and ados theorem 3 of german mathematician wilhem karl joseph killing 18471923, who had no access to the scandinavian journals that lie published in and consequently invented the theory of lie algebras on his own for which he received much scorn from lie. Any associative algebra can be considered as a lie algebra if we let the lie bracket be the commutator. A nice short text is the book \lectures on lie groups and lie algebras by. Lecture notes to the graduate course finite dimensional algebra during spring 2019 at. Lecture notes introduction to representation theory. C, is the complex vector space of all linear operators on mthat are antisymmetric with respect to the bilinear form on m cn. Keywords abelian group algebra cohomology cohomology group finite group group action homology lie algebra lie group representation theory vector space. This book is designed to introduce the reader to the theory of semisimple lie algebras over an algebraically closed field of characteristic 0, with emphasis on representations. A lie algebra l is a vector space over a eld f together with a binary operation.
Lie algebras are an essential tool in studying both algebraic groups and lie groups. A sentimental journey through representation theory. Representation theory princeton math princeton university. Restricted cohomology and representation theory 5 is a restricted lie algebra. The lie algebra g is first of all a vector space and secondly is endowed with a bilinear nonassociative product called the lie bracket or commutator and usually denoted. As simple as is the definition of representation theory given above, it fragments considerably when we try to get more specific. Notes on lie algebras cornell department of mathematics.
In many ways, this book is my ticket out of attending lectures. Lie groups, lie algebras, and their representations. If time permits we may study lie algebras over the field of real numbers or look at jordan algebras. A problem one meets right in the beginning of such a course is to motivate the notion of a lie algebra and to indicate the importance of representation. Then i alhebras to humphreys introduction to lie algebras and representation theory which has already been mentioned and is the absolute best. Jul 21, 2012 this lecture is part of a tutorial on lie algebras. Topics include definitions and examples of lie groups and lie algebras, the relationship between lie groups and lie algebras via the exponential mapping. By a representation we mean a set of linear transformations that is, matrices tx, ty, and tz with the same commutation relations as the ts. The content of the first four chapters is closely paralleled by an introductory graduate level course in lie algebra and representation theory at mit although the instructor did not explicitly declare this as class text. The aim of this course is to develop the basic general theory of lie algebras to give a rst insight into the basics of the structure theory and representation theory of semisimple lie algebras.
The complex orthogonal lie algebra on cn, written l son. We then show that lie 2algebras can be classi ed up to equivalence in terms of a lie algebra g, a representation of g on a vector space v, and an element of the lie algebra cohomology group h3g. Here we give denition of lie algebra and its representation. Lie theory, the theory of lie groups, lie algebras and their applications, is a fundamental part. The notes cover a number of standard topics in representation theory of groups, lie algebras, and quivers. In technical terms, representation theory studies representations of associative algebras. In recent times, a more general theory, namely that of locally c algebras 7 and of locally w algebras 3, 9, is developed. Hall, lie groups, lie algebras, and representations. A good knowledge of linear algebra including eigenvalues, bilinear forms, euclidean spaces, and tensor products of vector spaces is presupposed, as well as some acquaintance with the methods of abstract algebra. Representation theory of lie superalgebras and related topics. The main sources for these notes are the books 6 and 8. Humphreys introduction to lie algebras and representation theory an excellent introduction to the subject, ideal for a one semester graduate course. The socalled lie algebras of classical type are psubalgebras of glnf for some positive integer n satisfying certain conditions, e. After a general discussion of su3 and the related algebra su3, the information about the weight.
Lie, abstract form every irreducible representation of a finite dimensional complex solvable lie algebra is 1dimensional. Another important reference is the book 6, introduction to lie algebras and representation theory, by james e. These notes give an elementary introduction to lie groups, lie algebras, and their representations. Lie algebras alexei skorobogatov march 20, 2007 introduction for this course you need a very good understanding of linear algebra. Representation theory of complex semisimple lie algebras. We let r be endowed with the euclidean topology in this example. Representation theory of lie algebras classication of reductive algebraic groups workshop on group theory 18 dec 2006 5 jan 2007, bangalore anupam singh chapter 1. Representations of automorphism groups on the homology of matroids. For any vector space v we have slv, the special linear lie algebra of v, consisting of the operators on v.
Chapter i develops the basic theory of lie algebras, including the fundamental theorems of engel, lie, cartan, weyl, ado, and poincarebirkhoffwitt. For any vector space v we have slv, the special linear lie algebra of v, consisting of the operators on v of trace 0. We will develop the basic theory of lie algebras and later well establish how they arise from lie groups and essentially motivate their existence. The american mathematical monthly exceptionally well written and ideally suited either for independent reading or as a text for an introduction to lie algebras and their representations. Generally, the symmetries of physical systems is described by lie groups, but as it is shown in this paper, the representation theory of groups and algebras are directly related to each other. Representation theory of finite dimensional lie algebras. Let qbe another distinct point in r, we will show that r is hausdor.
These free notes by alistair savage are an excellent introduction based on stillwells and halls books. An introduction to lie groups, lie algebras and their. Lie algebra representation theory su3representations in. Suppose that g is the lie algebra of a lie group g. Finally, chapter vi was my attempt to specialize highest weight theory to the lie group sun and the complex lie algebra sl n. Lie algebras and ados theorem princeton university. The special linear lie algebra sln,r consists of all n. Representation theory of lie algebras deferential geometry. See the playlist lie algebras and their representations for more information. Although not necessary, we will use covering theory within the. Since lie groups are real manifolds, g is a real lie algebra. The best references for lie theory are the three volumes 1, lie groups and lie algebras, chapters, 2, lie groups and lie algebras, chapters 46, and 3, lie groups and lie algebras, chapters 79, all by nicolas bourbaki. Notes to lie algebras and representation theory zhengyaowu abstract.
Introduction to lie algebras and representation theory an excellent introduction to the subject, ideal for a one semester graduate course. There is a theory of complex lie groups we mostly will not develop. Ivanov syllabus of the course recommended textbooks. The third chapter brings the theory of the finite dimensional representations of a semisimple lie. An elementary introduction for the second half of the course, chapters 14. Basic representation theory, spring 2020 math in moscow.
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