Some subalgebras of indefinite type kacmoody lie algebras. Thence we can build a real formofthewittalgebraas x. Hilbert schemes of points and infinite dimensional lie algebras. Van est and korthagen 1964 gave perhaps the first example of what they called a nonenlargible lie algebra, having no corresponding lie group. Let l ct, t1 be the ring of laurent polynomials in t, and set lb l oc 0. Representation theory seminar at imsc on21 march 2012. Lie symmetries, kacmoodyvirasoro algebras and integrability. Infinite dimensional lie algebras and superalgebras youtube. Multilinear algebra international winter school on gravity and light 2015 duration. We determine commutative postlie algebra structures on some infinitedimensional lie algebras. The automorphism theory is an important part of affine lie algebras and has. The material covered ranges from basic definitions of lie groups to the classification of finite dimensional representations of semisimple lie algebras. In this paper, we launch a program to describe and classify modular invariant representations of infinitedimensional lie algebras and superalgebras.
Infinitedimensional lie algebras pdf free download epdf. The final classification of simple finite dimensional lie superalgebras is as follows section 4. These sections were based on material in kac and rainas book \bombay lectures on highest weight representations of in nite dimensional lie algebras 4. Affine kacmoody algebras a natural problem is to generalize the theory of. Classification and construction of semisimple lie algebras. David hernandez universite parisdiderot paris 7 quantum kac moody algebras and categorifications. Each chapter begins with a motivating discussion and ends with a collection of exercises with hints to the more challenging problems. The book is concerned with kac moody algebras, a particular class of infinite dimensional lie algebras, and their representations. Universal enveloping algebra and the casimir operator. It will be shown that general polynomials of the kacmoody field can generate either a new infinite.
The weheraeus international winter school on gravity and light 77,051 views. When we specialize g to be a finite dimensional or affine lie algebra we obtain modules for toroidal lie algebras. Infinitedimensional lie algebras, theta functions and. The principal realization of the basic representation. 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. The aim of this course is to inform students about theories of lie algebras, vertex algebras and other structures related to these theories.
The book is concerned with kacmoody algebras, a particular class of infinite dimensional lie algebras, and their representations. We specialize to integrable representations of kacmoody algebras and free lie algebras. We prove a character formula for a large class of highest weight representations l. This extends the results for finitedimensional perfect lie algebras. Some subalgebras of indefinite type kacmoody lie algebras article in journal of pure and applied algebra 2125. Infinitedimensional lie groups and algebras in mathematical. Modular invariant representations of infinitedimensional lie. 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. Suppose v is a representation of a complex simple lie algebra that can be written as a tensor product of irreducible representations.
Introduction to finite and infinite dimensional lie superalgebras. 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. 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. Commutative postlie algebra structures on kacmoody algebras. On complete reducibility for infinitedimensional lie algebras. 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. 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. Application to the kdvtype hierarchies of nonlinear partial differential equations. Then you can start reading kindle books on your smartphone, tablet, or computer no. The book is concerned with kacmoody algebras, a particular class of infinitedimensional lie algebras, and their representations.
Introduction to finite and infinite dimensional lie super. Find all the books, read about the author, and more. Characters of highest weight modules over affine lie algebras. Use features like bookmarks, note taking and highlighting while reading infinitedimensional lie algebras. Jan 12, 2008 the book is concerned with kacmoody algebras, a particular class of infinitedimensional lie algebras, and their representations. Lie algebras, part 2, volume 7 1st edition elsevier. The book is concerned with kac moody algebras, a particular class of infinitedimensional lie algebras, and their representations.
From purely mathematical point of view, affine lie algebras are interesting because their representation theory, like. Needless to say, this kind of question largely depends on the precise definitions adopted for infinite dimensional lie groups and lie 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. It is a kacmoody algebra for which the generalized cartan matrix is positive semidefinite and has corank 1. 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. 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. Infinitedimensional lie algebras kindle edition by kac, victor g download it once and read it on your kindle device, pc, phones or tablets. Are infinite dimensional lie algebras related to unique lie. 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.
For the case of the infinite dimensional lie algebra, it leads to an infinite number of conserved classical quantities which generate hierarchy equations. Kac moody lie algebras as bps lie algebras youtube. Infinite lie algebras and theta functions 127 the affine lie algebra g associated to 1 may be constructed as follows 14,31, 7,181. Contravariant form for infinitedimensional lie algebras and. This is the third, substantially revised edition of this important monograph. Concerned with kacmoody algebras, a particular class of infinitedimensional lie algebras, and their representations, this is the third revision of an important monograph. Enter your mobile number or email address below and well send you a link to download the free kindle app. While all the simple finitedimensional lie algebras belong to the considered class of. In the case of the infinite dimensional algebras, we should be able to go infinitely far in this process. Infinitedimensional lie algebras school of mathematics. Each chapter begins with a motivating discussion and ends with a collection of exercises with. 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. Infinitedimensional classical lie algebras mathoverflow.
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