Taylors formula and the differential of the exponential mapping. Pdf we study representations of the double affine lie algebra associated to a simple lie algebra. Infinite lie algebras and theta functions 127 the affine lie algebra g associated to 1 may be constructed as follows 14,31, 7,181. Lie groups, lie algebras, and their representations. Affine lie algebra an overview sciencedirect topics. Many concepts and results from the representation theory of finitedimensional lie groups and. The weyl character formula of the algebraic characters of the affine lie algebras generalizes to the weylkac character formula. Braid group action and quantum affine algebras springerlink. If time permit we shall also see some connections with theta functions. The links to ca, gt, ft, and ag in the pdf file will work if the files are. In view of the fact that entirely unrelated affine algebraic groups typically, vector groups and toroidal groups may have isomorphic lie algebras, this notion of algebraic lie algebra. Lecture notes introduction to lie groups mathematics. In particular, if x g is a lie group, then its tangent space at the identity element 1g is equipped with a lie algebra structure.
In view of the fact that entirely unrelated affine algebraic groups typically, vector groups and toroidal groups may have isomorphic lie algebras, this notion of algebraic lie algebra calls for some clarification. Character formula for the intergrable highest weight modules. More precisely we prove, in several cases, that the group. In these lectures the untwisted affine lie algebras will be constructed. Lectures on infinitedimensional lie algebra minoru wakimoto kyushu university, japan world scientific new jersey london singapore hong kong. There are various groups of people proved many results in this direction. Lie groups are smooth differentiable manifolds and as such can be studied using differential calculus, in contrast with the case of more general topological groups. Affine actions on lie groups and postlie algebra structures. Lie algebras of vector fields on smooth affine varieties. This lie algebra is a quite fundamental object, that crops up at. If g is a lie group, then the group autg of all continuous auto morphisms of g. The reader is assumed to be familiar with the theory of semisimple lie algebras, e. A ne lie algebras kevin wray january 16, 2008 abstract in these lectures the untwisted a ne lie algebras will be constructed.
Examples of lie algebras of polynomial vector fields on an ndimensional sphere, nonsingular hyperelliptic curves and linear algebraic groups are considered. It is a lie group if k is the real or complex field or quaternions. Victor kac, infinite dimensional lie algebras, lecture notes include. We also present new realizations of the twisted affine kacmoody algebras. The weyl group of an affine lie algebra can be written as a semidirect product of the weyl group of the zeromode algebra the lie algebra used to define the loop algebra and the coroot lattice.
David hernandez, an introduction to affine kacmoody algebras 2006 iain gordon, infinitedimensional lie algebras 20089 pdf. A lie algebra is said to be algebraic if it is isomorphic with the lie algebra of an affine algebraic group. Algebraic groups, lie groups, and their arithmetic subgroups this work has been replaced by the above three, and will not be revised or corrected. Looplike generators of the algebra are obtained which satisfy the relations of drinfelds new realization. We lift the lattice of translations in the extended affine weyl group to a braid group action on the quantum affine algebra. Let l ct, t1 be the ring of laurent polynomials in t, and set lb l oc 0. Intergrable representations of kacmoody lie algebras. We give an explicit construction of the affine lie algebraa 1 1 as an algebra of differential operators on. It is a lie algebra extension of the lie algebra of the lorentz group. Besides, the structure of the module of polynomial functions on an irreducible smooth affine variety over the lie algebra of vector fields is studied. This algebra is spanned by the creation and annihilation operators and by the homogeneous components of a certain exponential generating function which is strikingly similar to the vertex operator in the string model. In mathematics, the adjoint representation or adjoint action of a lie group g is a way of representing the elements of the group as linear transformations of the group s lie algebra, considered as a vector space.
Lie s ideas played a central role in felix kleins grand erlangen program to classify all possible geometries using group. Chevalleys theorem on the topology of solvable groups 2, the fact that the. Trigonometric lie algebras, affine lie algebras, and. Lie algebras alexei skorobogatov march 20, 2007 introduction for this course you need a very good understanding of linear algebra. Among the main results, we obtain a realization of trigonometric lie algebras as what were called the covariant algebras of the affine lie algebra a. In this paper, we explore natural connections among trigonometric lie algebras, general affine lie algebras, and vertex algebras. The links to gt, ca, ag, and ags in the pdf file will work if the files are. We give a new characterization of the affine kacmoody algebras in terms of extended affine lie algebras. The main tool for both methods is the semisimple splitting of a solvable lie algebra and its relation to the algebraic hull, which we also define on the level of lie algebras. The lie algebra su2 is the real span of the three tracefree antihermitian matrices x 1 p 1. At the end we construct affine central operators for each integer and they commute with the action of the affine lie algebra. In nearly every section, starting from affine lie algebras i also want to show the structures and connections beyond affine lie.
Coproduct formulas are given and a pbw type basis is constructed. Infinitedimensional lie algebras, theta functions and. Modules over affine lie algebras at critical level and. Readings introduction to lie groups mathematics mit. The categories of nitedimensional real lie algebras, and of connected simplyconnected lie. After that, one usually adjoins another basis element, which acts on g0as a derivation d. Construction of the affine lie algebra a 1 1 springerlink. N g the algebraic quotient of the adjoint action of g on n copies of its lie algebra g. Let c denote the onedimensional center of an affine lie algebra lg and let cc be the canonical central element 3. Lie algebras, algebraic groups, and lie groups james milne. The affine group of a lie group 353 chevalleys theorem on the topology of solvable groups 2, the fact that the universal covering of sl2, r is the only simple lie group homeomorphic to euclidean space, and the global leviwhitehead decomposition of g. One of the key ideas in the theory of lie groups is to replace the global object, the group, with its local or linearized version, which lie himself called its infinitesimal group and which has since become known as its lie algebra. Let us consider an affine representation of a lie group g.
We shall find it convenient to express the structure of an untwisted or twisted affine lie algebra in terms of formal. For example, if g is gln, k the lie group of nbyn invertible matrices over a fixed field, then the adjoint representation is the group homomorphism that sends any nbyn. Exponential mapping for a suitable affine connection on g. Auslander l we will refer to ag as the affine group of g, and regard it as a. This chapter is a basic introduction to affine lie algebras, preparing the stage for their application to conformal field theory. Simply transitive nilaffine actions of solvable lie groups. The lie algebras of affine group schemes semantic scholar. Abstract for a smooth manifold x equipped with a volume form, let. A ne lie algebras university of california, berkeley. Contents preface vii 1 preliminaries on afflne lie algebras 1 1. Affine actions on lie groups and post lie algebra structures article in linear algebra and its applications 4375 september 2011 with 49 reads how we measure reads.
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