On infinitedimensional 3 lie algebras lu ding, xiaoyu jia, and wei zhang citation. Infinite wedge space fand the diracpositrontheory 33 ix. The main result of this chapter is theorem 1, which gives a classification of contragredient lie algebras of finite growth. Any nitedimensional real lie algebra is isomorphic to the lie algebra of a lie group. For convenience, a 2cocycle in the lie algebras case is called a lie 2cocycle. Needless to say, this kind of question largely depends on the precise definitions adopted for infinitedimensional lie groups and lie algebras. Hybrid and metriplectic flows which combine hamiltonian and gradient components. Quantum probability, renormalization and infinite dimensional lie algebras. With numerous exercises and worked examples, it is ideal for graduate courses on lie groups and lie algebras. The natural filtrations of finitedimensional modular lie algebras of cartan type were proved to be invariant in 9, 10. The virasoro algebra is of paramount importance in string theory. Yuri bahturin memorial university of newfoundland, canada.
The book is concerned with kacmoody algebras, a particular class of infinitedimensional lie algebras, and their representations. Structure of representations with highest weight of. In the case of the infinitedimensional algebras, we should be able to go infinitely far. V c be a nondegenerate symmetric bilinear form on v. The automorphism groups are no longer differentiable manifolds, at least not modelled on finitedimensional euclidean spaces, nor even as far as i can tell on possiblyinfinitedimensional banach spaces.
Indeed, starting from s 1 being a single element set, every s i produced this way will be nite. Lie algebras, and also include another class of lie algebras called the af. Integrable highest weight modules over affine lie algebras. Yuri bahturin memorial university of newfoundland, canada vyacheslav futorny university of sao paulo, brazil mikhail kotchetov memorial university of newfoundland, canada ivan penkov jacobs university of bremen, germany schedule abstracts of talks bonne bay marine. Over the past years, associative algebras have come to play a major role in several areas of theoretical physics.
Continuing on, the main subjects of the book are the structure real and imaginary root systems of and the. The categories of nite dimensional real lie algebras, and of connected simplyconnected lie groups, are equivalent. Introduction to finite and infinite dimensional lie algebras and superalgebras introduces the theory of lie superalgebras, their algebras, and their representations. We will study the theory of semisimple and kacmoody lie algebras, and some other in. The moyal algebra is an infinitedimensional lie algebra that contains all classical lie algebras as subalgebras. Modular invariant representations of infinitedimensional.
Structure of representations with highest weight of infinite. On the other hand, using this idea, we either get a basis for v eventually or we get an increasing. In the case of finitedimensional modular lie superalgebras of cartan type, the invariance of the natural filtration was discussed in 12. Needless to say, this kind of question largely depends on the precise definitions adopted for infinite dimensional lie groups and lie algebras. In particular, we combine the results about stabilizers described above with some.
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. Infinite dimensional lie algebras in algebraic geometry 3 have an exact sequence 0. Combining the last two propositions gives us a complete description of the. In chapter iii we study the graded lie algebras corresponding to primitive pseudogroups of. 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.
Finitedimensional lie algebras and their representations for. A subquotient l of md has an induced weight decomposition l l. The paper 6 devoted to investigation of socalled 2local derivations on finite dimensional lie algebras and it is proved that every 2local derivation on a semisimple lie algebra l is a. Another simple case is given by constructing loops on a 1dimensional abelian lie algebra, and again making acentral extension. In this paper we describe a simple method for obtaining a classification of smalldimensional solvable lie algebras.
For the finitedimensional lie algebras this process terminates at some point, and new commutators are no longer producing new linearly independent of the ones obtained earlier elements. To every nite dimensional lie group see the denition below it is a special case of a continuous group all rotations around a xed center in r 3 gives an example one can assign a nite dimensional lie algebra also dened below. These groups will be treated as infinitedimensional lie groups. The idea now is to combine these two concepts and to use the existing tools. The main purpose is to give an impression of the enormous variety of in nite dimensional lie groups. We prove a character formula for a large class of highest weight representations l. This includes diffeomorphism groups and their natural subgroups like volumepreserving and symplectic transformations, as well as gauge groups and loop groups. D 1h that is also a lie homomorphism over k, commutes with the symbol maps and takes oi s to o s. Weak proregularity, weak stability, and the noncommutative mgm equivalence. The principal realization of the basic representation.
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. Introduction to finite and infinite dimensional lie super. On complete reducibility for infinitedimensional lie algebras. Affine lie algebras from simple lie algebras definition. If is a finite dimensional simple lie algebra, the corresponding affine lie algebra is constructed as a central extension of the infinitedimensional lie algebra. Modular invariant representations of infinitedimensional lie. In this thesis, we study some results on infinite dimensional lie algebras. Kacmoody algebras are a large class of infinitedimensional lie algebras whose structure is very similar to the finitedimensional cases above. The aim of this course is to inform students about theories of lie algebras, vertex algebras and other structures related to these theories. Pdf finite and infinite dimensional lie group structures on. It is also not true that infinite dimensional lie algebras are lie algebras of infinite dimensional lie groups in general see, for example, this mo question, and i have the impression that this is a genuine phenomenon and not just an artifact of working with too restrictive of a notion of infinite dimensional lie group, although i dont know. Unlike the finite dimensional simple complex lie algebras, which are described in terms of the classical matrix. The paper 6 devoted to investigation of socalled 2local derivations on finitedimensional lie algebras and it is proved that every 2local derivation on a semisimple lie algebra l is a. Are infinite dimensional lie algebras related to unique lie.
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. These possibly infinitedimensional lie algebras have found applica. Infinite dimensional lie algebra, infinite dimensional lie group, inte. 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. His method uses gr obner bases and a computer algebra.
Let l ct, t1 be the ring of laurent polynomials in t, and set lb l oc 0. Irreducible representations of infinitedimensional lie algebras. Pdf quantum probability, renormalization and infinite. Hilbert schemes of points and infinite dimensional lie. Such choice is possible because the field c is uncountably infinite. The similar result was obtained for the infinitedimensional case. Representations of infinitedimensional lie algebras benjamin j. It is a kacmoody algebra for which the generalized cartan matrix is positive semidefinite and has corank 1. Infinite lie algebras and theta functions 127 the affine lie algebra g associated to 1 may be constructed as follows 14,31, 7,181. Using this method, we obtain the classification of three and fourdimensional solvable lie algebras over fields of any characteristic. Lie 2algebras, obtaining the former from strict lie 2groups and using the latter to classify lie 2algebras in terms of 3rd cohomology classes in lie algebra cohomology.
Van est and korthagen 1964 gave perhaps the first example of what they called a nonenlargible lie algebra, having no corresponding lie group. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. Along with the underlying eld assumptions algebraic closure and characteristic 0 this enables. It is also not true that infinitedimensional lie algebras are lie algebras of infinitedimensional lie groups in general see, for example, this mo question, and i have the impression that this is a genuine phenomenon and not just an artifact of working with too restrictive of a notion of infinitedimensional lie group, although i dont know. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. Such a structure come with a multiweight w w i i2ki.
Infinitedimensional lie algebras, theta functions and. Infinite dimensional lie algebras third edition infinite dimensional lie algebras third editionvictor g. In section 9 and section 10 we treat two more typical examples of soliton equations, the 2 dimensional toda lattice. On the other hand, using this idea, we either get a basis for v eventually or we get an increasing collection of linearly independent sets s 1. Infinitedimensional lie groups and algebras in mathematical. Bombay lectures on highest weight representations if infinite dimensional lie algebras subject. The starting point of this development was the realization by witten that there is an intimate connection between three dimensional chernsimons theory and two dimensional current algebra 1. Lie superalgebras are a natural generalization of lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. Infinitedimensional lie theory for gauge groups fachbereich. To every nitedimensional lie group see the denition below it is a special case of a continuous group all rotations around a xed center in r 3 gives an example one can assign a nitedimensional lie algebra also dened below. Included are the infinite dimensional orthogonal or symplectic lie algebras b, c, d and the kacmoody lie algebras of euclidean type. Lie theory more difficult and also richer than the finitedimensional theory.
Pdf infinitedimensional symmetries of twodimensional. Some studies on infinitedimensional liesuper algebra. Finitedimensional lie algebras and their representations. This picture is different for infinitedimensional groups, and how bad it becomes. Infinitedimensional lie algebras pdf free download epdf.
In this paper, we launch a program to describe and classify modular invariant representations of infinitedimensional lie algebras and superalgebras. In the case of the infinite dimensional algebras, we should be able to go infinitely far in this process. Is there an infinite dimensional lie group associated to. Structure and representation theory of infinitedimensional lie. Pdf infinitedimensional groups and their representations. As in the lie algebras case, onedimensional leibniz central extensions of a leibniz algebra.
Infinite dimensional lie algebras occurring in algebraic. Applications include fluid dynamics, maxwells equations, and plasma physics. Furthermore, a lie algebra isomorphism is a homomorphism that is both injective and surjective. Irreducible representations of infinitedimensional lie. Finally, we combine results to prove the central result of this section, which completes. As a particular case we obtain an extention of the abovementioned results of 9 and 4. Infinitedimensional classical lie algebras mathoverflow.
I will also discuss the classes of integrable modules and bounded modules. Are infinite dimensional lie algebras related to unique. This classi cation allows us to construct for any nitedimensional lie algebra g a canonical 1parameter family of lie 2algebras g which reduces to g at 0. Infinitedimensional symmetries of twodimensional generalized burgers equations article pdf available in journal of mathematical physics 517. It is known from linear algebra that over c all nondegenerate symmetric bilinear forms. The book is concerned with kacmoody algebras, a particular class of infinite dimensional lie algebras, and their representations. Infinitedimensional lie algebras 101 all submodules of ma are homogeneous with respect to the weight decomposition. Any nite dimensional real lie algebra is isomorphic to the lie algebra of a lie group. An easy check shows that g is a lie subalgebra of glv. Before leaving the reals behind us, note the following theorems we will not prove. This is the third, substantially revised edition of this important monograph.
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