This book gives a general introduction to the theory of representations of algebras. The rst part is based on previous notes by francesca mantese. Representation theory of finite dimensional lie algebras. These notes will be updated on a regular basis during the course. Famous examples are provided by universal enveloping algebras of lie algebras, quantum groups and weyl algebras. Enveloping algebras and geometric representation theory 5 a a lie lattice on xis any nitely generated ox submodule lof tx such that l. The category oof bernsteingelfandgelfand is a particularly wellstudied category in the representation theory of semisimple lie algebras, where the modules are weight modules and subject to certain niteness conditions. Representation theory of vertex operator algebras and. The structures involved can be generalized to quantum groups and kacmoody lie algebras. Pdf topological algebras with c enveloping algebras.
These results are in agreement with the investigation of related models in 22. An introduction to quantized lie groups and algebras. This paper is an introduction to cli ord algebras and a survey of some uses in representation theory. Introduction to representation theory mit opencourseware. Theorem 6 belongs to the province of representation theory. Readership graduate students and research mathematicians interested in group representations, algebra representations, commutative algebra, and category theory. In this program, the construction of a chiral conformal eld theory can be divided into the following steps. It can be an excellent textbook for a graduate course, as well as a very useful source of references in the theory of universal enveloping algebras, the area of mathematics that remains as important today as it was 20 years ago. Enveloping algebras of double poissonore extensions xingting wang department of mathematics, temple university, philadelphia special session on topics related to the interplay of noncommutative algebra and geometry, denton joint with jiafeng lu, seiqwon oh, xiaolan yu and guangbin zhuang september 9, 2017. A representation of an associative algebra aalso called a left amodule is a vector space v equipped with a homomorphism a endv, i. Representation theory of semisimple lie algebras 2009. Request pdf enveloping algebras and geometric representation theory since its inception in the early seventies, the study of enveloping algebras has undergone a significant and continuous. This is proved by showing that every unbounded operator representation. Adopting a panoramic viewpoint, this book offers an introduction to four different flavors of representation theory.
In particular, we emphasize the useful properties of the exponential mapping, which allows us to transition. Unbounded operator algebras and representation theory k. The plan of the course was to try to cope with two competing demands. A representation of an associative algebra a also called a left. Razmadze mathematical institute, georgian academy of sciences. Specifically, the finitedimensional irreducible representations are constructed as quotients of verma modules, and verma modules are constructed as quotients of the universal enveloping algebra. For the 1996 printing, the author updated the status of open problems and added some relevant references. Ideals in the enveloping algebra of the positive witt algebra. The universal enveloping algebra of a nitedimensional lie algebra are the associative. Then the representation theory of g with the pcharacter. 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. The main results are concerned with jordan algebras of semimatrix type, that is, algebras.
In the second part, many proofs are omitted or just. Thus, enveloping algebras is a vibrant area of research, with many mathematicians around the globe working and hundreds of papers published. Also, this representation is faithful on simple lie algebras, but if g has a nontrivial center, additional tricks must be played to get a faithful representation. A separate part of the book is devoted to each of these areas and they are all treated in sufficient depth to enable and hopefully entice the reader to pursue.
For infinitedimensional representations even the simplest noncommutative lie algebras admit an enormously compli cated representation theory. Other articles represent contributions to areas in and related to representation theory, such as noncommutative resolutions, twisted commutative algebras, and upper cluster algebras. Enveloping algebras and geometric representation theory. We show that the reduction functor is exact and sends an irreducible module to zero or an irreducible module at any level k moreover, we show that the character of each irreducible highest weight.
Projective representations of generalized reduced enveloping. This book, which is the first systematic exposition of the algebraic approach to representations of lie groups via representations of or modules over the corresponding universal enveloping algebras, turned out to be so well written that even today it remains one of the main textbooks and reference books on the subject. Enveloping algebras and geometric representation theory organised by iain gordon, edinburgh bernard leclerc, caen wolfgang soergel, freiburg 10 may 16 may 2015 workshop. Introduction to lie algebras and representation theory an excellent introduction to the subject, ideal for a one semester graduate course. Dec 01, 2018 enveloping algebras and geometric representation theory. In the structure theory of quantized enveloping algebras, the algebra isomorphisms determined by lusztig led to the first general construction of pbw bases of these algebras. Hopf algebra structure on the universal enveloping algebra of a leibniz algebra. Representations of algebras, locally compact groups, and banach algebraic bundles, volume 1 1st edition basic representation theory of groups and algebras. They find many applications in differential geometry and mathematical physics, and are indispensable to many directions of research in representation theory of lie groups and algebras. Elements of the representation theory of associative algebras. Universal enveloping algebras recall that, for an associative algebra a with unity 1, a lie algebra structure on a. Sep 23, 2010 we apply the new theory of cluster algebras of fomin and zelevinsky to study some combinatorial problems arising in lie theory. Besides being useful in many parts of mathematics and physics, the theory of semisimple lie algebras is inherently attractive, combining as it does a certain amount of depth and a satisfying degree of completeness in its basic results.
The aim of this workshop is to discuss the current state of research in the area of enveloping algebras and their applications, primarily in representation theory. With spela spenko, michaela vancliff, padmini veerapen, and. In representation theory they appear as the images of the. Introduction to representation theory by pavel etingof. In technical terms, representation theory studies representations of associative algebras.
Representation theory of finite groups and associative algebras by curtis, charles w. I hope the book will also enable the reader to enter into the more advanced phases of the theory. An associative unitary respectively, lie algebra is said to be. Stone 23, theorem l has shown that a boolean ring with unit is the set of all open and closed sets in a compact bicompact zerodimensional space. Karin erdmanns research focus lies on representation theory of finite groups, and finitedimensional algebras. This carefully written textbook provides an accessible introduction to the representation theory of algebras, including representations of quivers. The interplay of representation theory and combinatorics builds on a long tradition. Algebras and representation theory springer undergraduate. The aims of this course are to introduce techniques for understanding finite and infinite dimensional representations, and to understand the structure of the enveloping algebra using representation theory. I believe in a certain program of obtaining a theory of leibniz groups along these lines.
In section2, we introduce a theory of verma modules, and study the associated unitarization problem. Notes to lie algebras and representation theory zhengyaowu abstract. Weyl algebras are the simplest noncommutative deformations of polynomials relevant to many elds of mathematics and physics. Universal enveloping algebras of leibniz algebras and co. Introduction to lie algebras and representation theory. A category is a set of objects together with morphisms functions between them. Integrable representations of enveloping algebras springerlink. The counterpart of the associative law in the theory of lie algebras is the jacobi. Unbounded operator algebras and representation theory.
The basics for the representation theory of lie antialgebras and their relation to lie superalgebras was also developed in 11. Representations of lie algebras, casimir operators and their applications. Lie algebras and representation theory download book. Representation type of jordan algebras sciencedirect. Enveloping algebras of double poissonore extensions.
Introduction to lie algebras and representation theory graduate texts in mathematics series by j. Humphreys book introduction to lie algebras and representation theory should be more than enough background for these lectures. The subject is presented from the perspective of linear representations of quivers and homological algebra. Universal enveloping algebra is an important object in the represen. In representation theory they appear as the images of the associated representations of the lie algebras or of the enveloping algebras on the garding domain and in quantum field theory they occur as the vector space of field operators or the algebra generated by them.
Introduction to clifford algebras and uses in representation theory ryan simeon abstract. A frechet algebraa has ac enveloping algebra if and only if every operator representation ofa mapsa into bounded operators. International workshop enveloping algebras and representation. Enveloping algebras graduate studies in mathematics. She has written many research articles, and is the author of a research monograph and a textbook. The present paper is a representation theoretic approach to lie antialgebras in continuation of 11. The universal enveloping algebra plays an important role in the representation theory of semisimple lie algebras, described above. N cor respond to the two chiralities in the theory of current algebras. Introduction to the representation theory of algebras. Using the hopf algebra structure of ul, it is shown that over. Organizers iain gordon, edinburgh bernard leclerc, caen wolfgang soergel, freiburg. We will discuss a classi cation of simple weight modules for the weyl algebras of in nite rank and its application for the representation theory of a ne kacmoody algebras. Representation theory of algebras an introduction lidia angeleri university of verona master program mathematics 201617 updated on january 20, 2017 important.
The aim of this note is to develop the basic general theory of lie algebras to give a first insight into the basics of the structure theory and representation theory of semi simple lie algebras. Similarly, a representation of a lie algebra g on a vector space v is a lie algebra homomorphism. I have tried to make all arguments as simple and direct as i. There is more than one version of what should be the universal enveloping algebra of a leibniz algebra. The importance of casimir operators for the representation theory of complex lie algebras comes from the schurs lemma, theorem 2. Hopf algebra structure on the universal enveloping algebra of. For example, verma modules can be constructed as quotients of the universal enveloping algebra. A representation of an associative algebra aalso called a left amodule is a vector space v equipped with a homomorphism a. Much more than the material in the lectures is explained in jantzens book einhullendealgebren halbeinfacher liealgebren. Two other recommendable texts which only discuss lie algebras are the books \introduction to lie algebras and representation theory by j.
Representations of algebras, locally compact groups, and. It starts with examples of classification problems of matrices under linear transformations, explaining the three common setups. At the moment research focusses on the representation theory. Lie theory, universal enveloping algebras, and the poincar. Quantum grothendieck ring isomorphisms for quantum affine algebras of type a and b, conference on algebraic representation theory 2018 reference slides, tongji university, november 2018. The universal enveloping algebra and verma modules. Cli ord algebras are a a generalization of the complex numbers that have important uses in mathematical physics. The representation theory of semisimple lie algebras over the field of complex numbers has been developed by cartan and weyl. Enveloping algebras and geometric representation theory table of contents konstantin ardakov joint with simon wadsley.
Quantum grothendieck ring isomorphisms for quantum affine algebras of type a and b, representation theory seminar, rims, december 2018. Representation theory princeton math princeton university. Universal enveloping algebras are used in the representation theory of lie groups and lie algebras. There is a survey in english by borho a survey on enveloping algebras of semisimple lie. Associating geometry to the lie superalgebra sl11 and to the color lie algebra sl c 2,k. In mathematics, a universal enveloping algebra is the most general unital, associative algebra that contains all representations of a lie algebra.
Categories, functors, and the universal enveloping algebra. A representation of an associative algebra a also called a left amodule is a vector space v equipped with a homomorphism. The notions of diagram of a jordan algebra and of jordan tensor algebra of a bimodule are introduced and a mapping qui is constructed which associates to the diagram of a jordan algebra j the quiver of its universal associative enveloping algebra s j. In particular the study of algebras that admit a notion of highest weight modules has turned out to be remarkably fruitful. 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. We emphasize various interactions between w algebras and universal enveloping algebras. There is also a discussion of the compact form and other real forms of a complex semisimple lie algebra, and a section on automorphisms. Affine niltemperleylieb algebras and generalized weyl. The final part of a threevolume set providing a modern account of the representation theory of finite dimensional associative algebras over an algebraically closed field.
This chapter deals with representations of enveloping algebras. An associative unitary respectively, lie algebra is said to be proper in case the intersection of all of its cofinite twosided respectively, lie ideals is zero. A subrepresentation of a representation vis a subspace u. Representations of reduced enveloping algebras and cells in the. Though some of the considerations and of the main results e. These algebras play an important role in the representation theory of restricted lie algebras since their module categories approximate to a certain degree to be made precise below the category of modules for the correspondinglie algebra, and methods. However some of cartans proofs see 22 make explicit use of the classifica tion of semisimple lie algebras and in fact require a verification of the as.
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. 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. A nice short text is the book \lectures on lie groups and lie algebras by. Hopf algebra structure on the universal enveloping algebra. Algebras and representation theory aims to play a unifying role in this, presenting to its reader both uptodate information about progress within the field of rings, algebras and their representations as well as clarifying relationships with other fields. Not only enveloping algebra can be contructed there, but also the appropriate internal hopf analogues of gln unpublished work of mine and the internal weyl algebras work of a student of mine. Lecture notes to the graduate course finite dimensional algebra. Representations of lie algebras, casimir operators and. Lecture notes to the graduate course finite dimensional algebra during spring 2019 at. A finite w algebra is an associative algebra constructed from a semisimple lie algebra and its nilpotent element.