We show that all commutative postlie algebra structures on loop algebras are. Infinitedimensional lie groups and algebras in mathematical. For the case of the infinite dimensional lie algebra, it leads to an infinite number of conserved classical quantities which generate hierarchy equations. Infinite dimensional lie algebras third edition infinite dimensional lie algebras third editionvictor g. When we specialize g to be a finite dimensional or affine lie algebra we obtain modules for toroidal lie algebras. The book is concerned with kac moody algebras, a particular class of infinitedimensional lie algebras, and their representations. David hernandez universite parisdiderot paris 7 quantum kac moody algebras and categorifications. The book is concerned with kac moody algebras, a particular class of infinite dimensional lie algebras, and their representations. The material covered ranges from basic definitions of lie groups to the classification of finite dimensional representations of semisimple lie algebras. Concerned with kac moody algebras, a particular class of infinite dimensional lie algebras, and their representations, this is the third revision of an important monograph. The book is concerned with kacmoody algebras, a particular class of infinite dimensional lie algebras, and their representations. While all the simple finitedimensional lie algebras belong to the considered class of. Kac moody lie algebras as bps lie algebras youtube. The aim of this course is to inform students about theories of lie algebras, vertex algebras and other structures related to these theories.
Infinite lie algebras and theta functions 127 the affine lie algebra g associated to 1 may be constructed as follows 14,31, 7,181. Infinitedimensional classical lie algebras mathoverflow. Multilinear algebra international winter school on gravity and light 2015 duration. Use features like bookmarks, note taking and highlighting while reading infinitedimensional lie algebras.
Characters of highest weight modules over affine lie algebras. Synopsis the third, substantially revised edition of a monograph concerned with kacmoody algebras, a particular class of infinitedimensional lie albegras, and their representations, based on courses given over a number of years at mit and in paris. Then you can start reading kindle books on your smartphone, tablet, or computer no. Commutative postlie algebra structures on kacmoody algebras. The three preceding examples all give the same lie algebra structure. Affine kacmoody algebras a natural problem is to generalize the theory of. In this paper, we launch a program to describe and classify modular invariant representations of infinitedimensional lie algebras and superalgebras. Thence we can build a real formofthewittalgebraas x. Introduction to finite and infinite dimensional lie super. Suppose v is a representation of a complex simple lie algebra that can be written as a tensor product of irreducible representations. Let l ct, t1 be the ring of laurent polynomials in t, and set lb l oc 0. Are infinite dimensional lie algebras related to unique lie.
Jun 29, 2010 representations of loop kac moody lie algebras. We will study the theory of semisimple and kacmoody lie algebras, and some other in. Universal enveloping algebra and the casimir operator. It is a kacmoody algebra for which the generalized cartan matrix is positive semidefinite and has corank 1. Finite and infinite dimensional lie algebras and applications in physics. Van est and korthagen 1964 gave perhaps the first example of what they called a nonenlargible lie algebra, having no corresponding lie group. Representation theory seminar at imsc on21 march 2012. The final classification of simple finite dimensional lie superalgebras is as follows section 4. Infinitedimensional lie algebras pdf free download epdf. We determine commutative postlie algebra structures on some infinitedimensional lie algebras. Classification and construction of semisimple lie algebras.
Lie symmetries, kacmoodyvirasoro algebras and integrability. Infinite dimensional lie algebras an introduction victor g. Some subalgebras of indefinite type kacmoody lie algebras. Contravariant form for infinitedimensional lie algebras and. Infinitedimensional lie algebras school of mathematics. The principal realization of the basic representation. While discussing all classes of finite and infinite dimensional lie algebras and lie superalgebras in terms of their different classes of root systems, the book focuses on kac moody algebras.
Starting with this knowledge, it surveys most of the known results and theories of the hilbert schemes of points, implying infinite dimensional lie algebras and their actions. Infinite dimensional lie algebras occurring in algebraic. The weheraeus international winter school on gravity and light 77,051 views. Infinitedimensional lie algebras kindle edition by kac, victor g download it once and read it on your kindle device, pc, phones or tablets. In the case of the infinite dimensional algebras, we should be able to go infinitely far in this process. The book is far from elementary and is suitable for researchers and graduate students with a good knowledge of intersection theory and geometric invariant theory. Lie algebras, part 2, volume 7 1st edition elsevier. We specialize to integrable representations of kacmoody algebras and free lie algebras. Victor kac, infinite dimensional lie algebras, 3rd edition, cambridge university press 1990 isbn 0521466938 antony wassermann, lecture notes on kacmoody and virasoro algebras hazewinkel, michiel, ed.
Infinite dimensional lie algebras occurring in algebraic geometry spring 2015 content of the course this course is set up as an introduction to the theory of conformal blocks wesszuminowitten theory, whose ultimate goal is described below. Needless to say, this kind of question largely depends on the precise definitions adopted for infinite dimensional lie groups and lie algebras. Introduction to finite and infinite dimensional lie superalgebras. Inner ideals were completely classi ed for nite dimensional simple lie algebras and recently this classi cation was extended to the nitary simple lie algebras by lopez, garcia. The book is concerned with kacmoody algebras, a particular class of infinitedimensional lie algebras, and their representations. Modular invariant representations of infinitedimensional lie.
Infinitedimensional lie algebras, theta functions and. Concerned with kacmoody algebras, a particular class of infinitedimensional lie algebras, and their representations, this is the third revision of an important monograph. For the finite dimensional lie algebras this process terminates at some point, and new commutators are no longer producing new linearly independent of the ones obtained earlier elements. In the early 90s it was proposed by several people feiginfrenkel, finkelberg, lusztig that there should exist a category of sheavesdmodules on the semi infinite flag manifold, though of as the quotient of the loop group gt by its subgroup nttt. Infinite dimensional lie algebras and superalgebras youtube. Each chapter begins with a motivating discussion and ends with a collection of exercises with hints to the more challenging problems. Application to the kdvtype hierarchies of nonlinear partial differential equations. It will be shown that general polynomials of the kacmoody field can generate either a new infinite. Some subalgebras of indefinite type kacmoody lie algebras article in journal of pure and applied algebra 2125. From purely mathematical point of view, affine lie algebras are interesting because their representation theory, like.
These sections were based on material in kac and rainas book \bombay lectures on highest weight representations of in nite dimensional lie algebras 4. Each chapter begins with a motivating discussion and ends with a collection of exercises with. On complete reducibility for infinitedimensional lie algebras. This extends the results for finitedimensional perfect lie algebras. It is based on courses given over a number of years at mit and in paris, and is sufficiently selfcontained and detailed to be used for graduate courses. Enter your mobile number or email address below and well send you a link to download the free kindle app. Infinitedimensional lie algebras, theta functions and modular forms author links open overlay panel victor g kac dale h peterson. Bombay lectures on highest weight representations if infinite dimensional lie algebras subject.
Infinite dimensional lie algebras 3rd edition algebra cambridge. The automorphism theory is an important part of affine lie algebras and has. Jan 12, 2008 the book is concerned with kacmoody algebras, a particular class of infinitedimensional lie algebras, and their representations. We show that the characters of all highest weight modules over an affine lie algebra with the highest weight away from the critical hyperplane are meromorphic functions in the positive half of the cartan subalgebra, their singularities being at most simple poles at zeros of real roots. In mathematics, an affine lie algebra is an infinitedimensional lie algebra that is constructed in a canonical fashion out of a finitedimensional simple lie algebra. We prove a character formula for a large class of highest weight representations l.
1159 1056 561 647 1081 438 787 68 1595 1250 740 681 1151 931 1584 443 1104 562 821 260 1440 476 901 429 1445 524 78 147 1181 629 280 457 377 1020 633 282 427 490 486 637 858 919 681 515 909 1347 1375