Lie algebras of finite and affine type, volume , roger william carter, oct 27, 2005, mathematics, 632 pages. Untwisted affine lie algebras are associated with finitedimensional semisimple lie. Lie algebras of finite and affine type ebook, 2005. We will use this result as a definition of affine lie algebras. Affine lie algebras and affine root systems uvafnwi universiteit. Also, an excellent course on lie algebras could be constructed starting from the book. The first example is the finite cartan type b2, which is the cartan type of so5. The connected, standard 2endorigid algebras of finite representation type are exactly the standard 2cytilted algebras of finite representation type listed in 7, theorem 5. Those of finite type are the classical ones, while the affine lie algebras involve affine root systems and affine weyl groups as pointed out in comments here.
Antony wassermann, kacmoody and virasoro algebras, course notes 2011. If v is a hmodule, then b 0 a v is naturally a b 0 a hmodule. Lie algebras of finite and affine type semantic scholar. Those are some of the most intensively studied examples of freely generated vertex algebras and pbw generated associative algebras. Citeseerx double affine lie algebras and finite groups. Lie algebras of finite and affine type roger carter. Svv algebras walker, ruari donald, proceedings of the japan academy. The main novelty here is the definition in terms of an indefinite integral of the.
Search for library items search for lists search for contacts search for a library. Double affine lie algebras and finite groups request pdf. Robinson r1j defined a map from the symmetric group s to the set of pairs p,o of standard n. As in the latter case, lz is obtained as a generalized quasideterminant. Kantor note on lie algebras note on lie algebras, finite groups and finite geometries william m. This monograph provides a crystal clear exposition of the theory of finite dimensional simple lie algebras and a nice introduction to infinitedimensional kac moody algebras of affine type.
Symmetric quiver hecke algebras and r matrices of quantum affine algebras, ii kang, seokjin, kashiwara, masaki, and kim, myungho, duke mathematical journal, 2015. Pdf double affine lie algebras and finite groups david. Affine lie algebras from simple lie algebras definition. 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. Semantic scholar extracted view of lie algebras of finite and affine type. Introduction to finite and infinite dimensional lie algebras and superalgebras introduces the theory of lie superalgebras, their algebras, and their representations. This book provides a thorough but relaxed mathematical treatment of lie. Affine lie algebras and affine root systems a killingcartan type classi. Quantum tori with graded involution appear as coordinate algebras of extended affine lie algebras of type a 1 in 11, c in 2 and bc in 3. The list is ordered in a way to make identification easy, using only basis independent properties of the lie algebras. Lie algebras of finite and affine type by carter, roger. The material covered ranges from basic definitions of lie groups to the classification of finite dimensional representations of semisimple lie algebras. Then l is a finite dimensional semisimple lie algebra with csa spanned by.
Buy lie algebras of finite and affine type cambridge studies in advanced mathematics on. It is a kacmoody algebra for which the generalized cartan matrix is positive semidefinite and has corank 1. Proofs are given in detail and the only prerequisite is a sound knowledge of linear algebra. Everyday low prices and free delivery on eligible orders. In the course of constructing his simple group th, he considered the lie algebra lover ic of type es. Although the emphasis is on affine group schemes of finite type over a field, we also discuss more general objects, for example, affine group schemes not of finite type and base rings not fields. Explicit construction of finite dimensional lie algebras. If a is the cartan type of a finite semisimple lie group, then the dual cartan type is the type of the langland dual group. They also describe certain classes of nilpotent and solvable lie algebras of arbitrary finite dimensions for which complete or partial classification exists and discuss in detail their construction and properties. Central extensions will also be used for the structure theory of finite dimensional simple lie algebras in chapter 6, and in later chapters for the study of the monster. Lie algebras of finite and affine type cambridge studies. Basic means that we do not investigate the detailed structure of reductive groups using root data except in the final survey chapter. Higher genus affine lie algebras of krichever novikov type. Lie algebras of finite and affine type cambridge studies in.
Geometric representation theory and extended affine lie algebras. It turns out that they have a triangular decomposition p. Note on lie algebras, finite groups and finite geometries. Untwisted affine lie algebras are associated with finite dimensional semisimple lie.
Lie algebras of finite and affine type by carter, roger ebook. These algebras form a generalization of finite dimensional semisimple lie algebras, and many properties related to the structure of a lie algebra such. The goal of this paper is to present some results and more importantly state a number of conjectures suggesting that the representation theory of symplectic reflection algebras for wreath products categorifies certain structures in the representation theory of affine lie algebras namely, decompositions of the restriction of the basic representation to finite dimensional and affine. Basic means that we do not investigate the detailed. Lie algebras of finite and affine type book, 2005 worldcat. Lie algebras of finite and affine type by roger carter. The concept of cells originally came from combinatorial theory.
Let h be the coxeter number of any of the simple components of g. Mastery of linear algebra and of the methods of abstract algebra. Shafarevich finite groups of lie type finitedimensional division algebras berlin i. For a balanced order, we study imaginary semicuspidal. Any lie algebra over a general ring instead of a field is an example of a lie ring. Koenig and xi introduce the concept of a ne cellular algebras in 24. Kantor the subject of this note began with thompson thl,2. Lie algebras of finite and affine type pdf free download.
We begin to study the lie theoretical analogs of symplectic reflection algebras for. Descargar lie algebras of finite and affine type en pdf. Representation theory of finite dimensional lie algebras july 21, 2015. They should be viewed as tangent lie algebras to the loop groups, with a correction term which is sometimes related to quantizationquantum anomaly these affine lie algebras appear in quantum field theory as the current algebras in the wzw model as well as in its chiral halfs. It includes topics such as geometric realizations of irreducible representations in three different approaches, combinatorics and geometry of canonical and crystal bases, finite \w\ algebras arising as the quantization of the transversal slice to a nilpotent orbit, structure theory of extended affine lie algebras, and representation theory of. This lie algebra is a quite fundamental object, that crops up at many places, and thus its representations are interesting in themselves. Root systems of levi type for lie algebras of affine type. This book provides a thorough but relaxed mathematical treatment of the subject, including both the cartankillingweyl theory of finite dimensional simple algebras and the more modern theory of kacmoody algebras. In mathematics, a kacmoody algebra named for victor kac and robert moody, who independently discovered them is a lie algebra, usually infinitedimensional, that can be defined by generators and relations through a generalized cartan matrix. Victor kac, infinite dimensional lie algebras, lecture notes include. Lie algebras, and the matrices of affine type are called the affine lie algebras, and. The second chapter brings the structure of the semisimple lie algebras. Lie algebras have many varied applications, both in mathematics and mathematical physics. Brundan, goodwin and independently kostant developed a theory of root system known as levi type root system when \a\ is a carlan matrix so that g\a\ is a finite dimensional semisimple lie algebra.
Structure and representation theory of infinitedimensional lie. Lie algebras of finite and a ne type, by roger carter. Daniel beltita, institute of mathematics, romanian academy, siam. This theory replicates much of the structure of usual root systems. Buy lie algebras of finite and affine type cambridge studies in advanced mathematics by roger carter isbn. Representation theory of finite dimensional lie algebras.
The second half introduces the theory of kacmoody algebras, concentrating particularly on those of affine type. The first half of the book deals with classification of the finite dimensional simple lie algebras and of their finite dimensional irreducible representations. Pdf vermatype modules for quantum affine lie algebras. In this paper, we explore natural connections among trigonometric lie algebras, general affine lie algebras, and vertex algebras. This extends the usual theory of nitedimensional cellular algebras to in nitedimensional algebras. Introduction to lie algebras, issn 16152085, 2006, 251. Application to the kdvtype hierarchies of nonlinear partial. Introduction to lie algebras, by karin erdmann and mark wildon. In view of these applications it seems clear that the theory of lie algebras, of both finite and affine types, will. Introduction to lie algebras algebras of finite and a ne. Affine lie algebras admit a concise characterization similar to that of simple finite dimensional lie algebras. Cellular algebras and diagram algebras in representation theory graham, j. We shall not need more that basic notions on linear algebra and.
Lie algebras of finite and affine type ebok roger carter. These lectures are intented to be a natural following of lectures on classical reprensentation theory. We introduce and begin to study lie theoretical analogs of symplectic reflection algebras for a finite cyclic group, which we call cyclic double affine lie algebra. Algebras of finite representation type arising from maximal. I corresponding to the division of the indecomposables into the preprojectives, the regulars, and the preinjectives.
David hernandez, an introduction to affine kacmoody algebras 2006 iain gordon, infinitedimensional lie algebras 20089 pdf. Introduction to finite and infinite dimensional lie super. If f is an indecomposable module having the property that every maximal submodule is indecomposable, we say that f is stable. Associated to every convex preorder on the set of positive roots is a system of cuspidal modules for the klr algebra. Lie algebras of finite and affine type by roger carter, b bollobas, w fulton, a katok, f kirwan, p sarnak and b simon topics. Representations of khovanovlaudarouquier algebras of affine.
If is a finite dimensional simple lie algebra, the corresponding affine lie algebra is constructed as a central extension of the infinitedimensional lie algebra. Sep 14, 2018 the operator lz is a quantum finite analogue of the operator of generalized adler type which we recently introduced in the classical affine setup. They include finitedimensional simple lie algebras as special cases but are. Those matrices of finite type correspond to the finitedimensional semisimple. Introduction to lie algebras and representation theory, by james e. The definition of finite groups of lie type due to chevalley involves restricting from a lie algebra over the complex numbers to a lie algebra over the integers, and the reducing modulo p to get a lie algebra over a finite field. Coordinate algebras of extended affine lie algebras of. Representations of khovanovlaudarouquier algebras of. A killingcartan type classification of affine lie algebras, reduced irreducible. It was recently given in 15, and it simplifies the original characterization contained in kac c. Affine kacmoody algebras a natural problem is to generalize the theory of.
Geometric representation theory and extended affine lie. Let g be a semisimple lie algebra over c with the following property. Finite order automorphisms of lie algebras mathematics. Carter mathematics institute university of warwick cambridge university press. An appendix gives a summary of the basic properties of each lie algebra of finite and affine type. A brief account of borcherds algebras is also included. Lie algebras alexei skorobogatov march 20, 2007 introduction for this course you need a very good understanding of linear algebra. The aim of this paper is to study the structure of the composition algebras of affine type. Our first result is about the family of prime representations introduced in hernandez and leclerc. Let a be an indecomposable gcm of finite or affine type.