DOCSLIB.ORG

University Microilms International 300 N

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
University Microilms International 300 N INFORMATION TO USERS This reproduction was made from a copy of a document sent to us for microfilming. While the most advanced technology has been used to photograph and reproduce this document, the quality of the reproduction is heavily dependent upon the quality of the material submitted. The following explanation of techniques is provided to help clarify markings or notations which may appear on this reproduction. 1.The sign or “target” for pages apparently lacking from the document photographed is “Missing Page(s)”. If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting through an image and duplicating adjacent pages to assure complete continuity. 2. When an image on the film is obliterated with a round black mark, it is an indication of either blurred copy because of movement during exposure, duplicate copy, or copyrighted materials that should not have been filmed. For blurred pages, a good image of the page can be found in the adjacent frame. If copyrighted materials were deleted, a target note will appear listing the pages in the adjacent frame. 3. When a map, drawing or chart, etc., is part of the material being photographed, a definite method of “sectioning” the material has been followed. It is customary to begin filming at the upper left hand comer of a large sheet and to continue from left to right in equal sections with small overlaps. If necessary, sectioning is continued again—beginning below the first row and continuing on until complete. 4. For illustrations that cannot be satisfactorily reproduced by xerographic means, photographic prints can be purchased at additional cost and inserted into your xerographic copy. These prints are available upon request from the Dissertations Customer Services Department. 5. Some pages in any document may have indistinct print. In all cases the best available copy has been filmed. University Microilms International 300 N. Zeeb Road Ann Arbor, Ml 48106 8510635 Singer, Phyllis E. KAC-MOODY ALGEBRAS WITH NONSYMMETRIZABLE CARTAN MATRICES The Ohio State University Ph.D. 1985 University Microfilms International300 N. Zeeb Road, Ann Arbor, Ml 48106 Copyright 1985 by Singer, Phyllis E. All Rights Reserved KAC-MOODY ALGEBRAS WITH NONSYMMETRIZABLE CARTAN MATRICES DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Phyllis E. Singer, B.S., M.S. ***** The Ohio State University 1985 Reading Committee: Approved by Harry P. Allen Robert B. Brown Joseph C. Ferrar <Xyzepk C 'Ijuuixn " Adviser Department of Mathematics Copyright by Phyllis E. Singer 1985 To my grandparents ACKNOWLEDGMENTS Many people have contributed to the production of this dissertation. My husband Ira, my parents Alan and Vivian Pound, my advisor Joseph Ferrar, and my friends Alayne Parson, Nora Hopkins, and Richard Charles provided moral support and encouragement. Ira, Joe, and Rich were especially patient and understanding. Charles Lodge introduced me to the wonders of abstract algebra, and Joseph Ferrar introduced me to the world of Lie algebras. Harry Allen, Robert Brown, Joseph Ferrar, and Aaron Meyerowitz carefully read the manuscript and offered many useful suggestions, for which I wish to thank them. Any remaining errors ore my responsibility. And, of course, I wish to thank Robert Moody and Victor Kac, whose pioneering articles laid the groundwork for this research. iii VITA March 16,1955 ............................. Born - Tachi kawa, Japan 1977 . ........................................... B.S., Michigan State University, East Lansing, Michigan 1977-1985 ................................... Graduate Teaching Associate, Department of Mathematics, The Ohio State University, Columbus, Ohio 1980 ............................................. M.S., The Ohio State University, Columbus, Ohio Autumn Semester, 1981 ............ Teaching Associate, University of Virginia, Charlottesville, Virginia FIELDS OF STUDY Area of Specialization: Lie Theory. Professor Joseph C. Ferrar Related Areas of Interest: Ring Theory. Combinatorics. TABLE OF CONTENTS DEDICATION ............................................................................................. ii ACKNOWLEDGMENTS ............................................................................... ill VITA ......................................................................................................... iv TABLE OF NOTATION AND TERMINOLOGY................................................vii TABLE OF REMARKS, PROPOSITIONS, LEMMAS, THEOREMS,COROLLARIES, AND EQUATIONS....................................................................................... xi CHAPTER I INTRODUCTION ........................................................................... 1 II SYMMETRIZABILITY .................................................................. 6 §1 Definitions and Properties .......................................... 6 §2 Another Necessary and Sufficient Condition for Symmetrizability ......................................................... 15 §3 Pitfalls on the Road to Symmetrizability ..................18 §4 Symmetrizing Nonsymmetrizable Matrices ..................21 III KAC-MOODY ALGEBRAS ..............................................................27 §5 Construction of the Lie Algebras ..................................27 §6 Serre Relations ......................................................... 34 v TABLE OF CONTENTS §7 Invariant Bilinear Forms on Algebras with (Very) Special Nonsymmetrizable Cartan M atrices 45 §8 iL^L.p.] when the Product Condition Fails ...............54 §9 The Weyl Group and Extending the Set of a for which dim [La,L_a l>1 ............................................................. 62 §10 When the Product Condition is Satisfied for all Nonrepeating Sequences of Length n, n = 4 .................. 68 §11 Computing dim La for Chordless Cycles ................. 72 §12 L((_2 0)), n s N .............................................................. 80 IV REMARKS, CONCLUSIONS, AND OPEN PROBLEMS .....................84 BIBLIOGRAPHY 86 TABLE OF NOTATION AND TERMINOLOGY generalized Cartan matrix (GCM) ........................................................... 1 symmetrizable ......................................................................................... 6 sign-symmetric ....................................................................................... 7 combinatorially symmetric ......................................................................7 combinatorially asymmetric ................................................................. 8 decomposable ..................... 9 indecomposable ......................................................................................... 9 indecomposable component ..................................................................... 9 digraph ......................................................................................................... 9 Gd(A) ........................................................................................................... 9 combinatorial structure ......................................................................... 9 G(A) ............................................................................................................. 9 adjacent ..................................................................................................... 9 path .............................................................................................................10 chain ...........................................................................................................10 product of the path .....................................................................................10 A(i,,...,ik) .....................................................................................................10 nonzero path ...............................................................................................10 closed path .................................................................................................10 cycle ...........................................................................................................10 simple cycle ...............................................................................................10 the product of the closed path is equal to its transpose product ... 10 i 1,...,ik satisfies the product condition ............................. 10 submatrix .....................................................................................................12 ^Aij)iJeP=Ap .............................................................................................*2 proper submatrix .......................................................................................12 r-submatrix ...............................................................................................12 vii TABLE OF NOTATION AND TERM IN0L06 V chordless cycle .........................................................................................12 chordless cycle of A .................................................................................13 c-cycle .......................................................................................................13 Hamiltonian p a th .......................................................................................13 g .........................................................................
Load more

Recommended publications
  • Kac-Moody Algebras and Applications
    Kac-Moody Algebras and Applications Owen Barrett December 24, 2014 Abstract This article is an introduction to the theory of Kac-Moody algebras: their genesis, their con- struction, basic theorems concerning them, and some of their applications. We first record some of the classical theory, since the Kac-Moody construction generalizes the theory of simple finite-dimensional Lie algebras in a closely analogous way. We then introduce the construction and properties of Kac-Moody algebras with an eye to drawing natural connec- tions to the classical theory. Last, we discuss some physical applications of Kac-Moody alge- bras,includingtheSugawaraandVirosorocosetconstructions,whicharebasictoconformal field theory. Contents 1 Introduction2 2 Finite-dimensional Lie algebras3 2.1 Nilpotency.....................................3 2.2 Solvability.....................................4 2.3 Semisimplicity...................................4 2.4 Root systems....................................5 2.4.1 Symmetries................................5 2.4.2 The Weyl group..............................6 2.4.3 Bases...................................6 2.4.4 The Cartan matrix.............................7 2.4.5 Irreducibility...............................7 2.4.6 Complex root systems...........................7 2.5 The structure of semisimple Lie algebras.....................7 2.5.1 Cartan subalgebras............................8 2.5.2 Decomposition of g ............................8 2.5.3 Existence and uniqueness.........................8 2.6 Linear representations of complex semisimple Lie algebras...........9 2.6.1 Weights and primitive elements.....................9 2.7 Irreducible modules with a highest weight....................9 2.8 Finite-dimensional modules............................ 10 3 Kac-Moody algebras 10 3.1 Basic definitions.................................. 10 3.1.1 Construction of the auxiliary Lie algebra................. 11 3.1.2 Construction of the Kac-Moody algebra................. 12 3.1.3 Root space of the Kac-Moody algebra g(A) ..............
    [Show full text]
  • Modular Forms and Affine Lie Algebras
    Modular Forms Theta functions Comparison with the Weyl denominator Modular Forms and Affine Lie Algebras Daniel Bump F TF i 2πi=3 e SF Modular Forms Theta functions Comparison with the Weyl denominator The plan of this course This course will cover the representation theory of a class of Lie algebras called affine Lie algebras. But they are a special case of a more general class of infinite-dimensional Lie algebras called Kac-Moody Lie algebras. Both classes were discovered in the 1970’s, independently by Victor Kac and Robert Moody. Kac at least was motivated by mathematical physics. Most of the material we will cover is in Kac’ book Infinite-dimensional Lie algebras which you should be able to access on-line through the Stanford libraries. In this class we will develop general Kac-Moody theory before specializing to the affine case. Our goal in this first part will be Kac’ generalization of the Weyl character formula to certain infinite-dimensional representations of infinite-dimensional Lie algebras. Modular Forms Theta functions Comparison with the Weyl denominator Affine Lie algebras and modular forms The Kac-Moody theory includes finite-dimensional simple Lie algebras, and many infinite-dimensional classes. The best understood Kac-Moody Lie algebras are the affine Lie algebras and after we have developed the Kac-Moody theory in general we will specialize to the affine case. We will see that the characters of affine Lie algebras are modular forms. We will not reach this topic until later in the course so in today’s introductory lecture we will talk a little about modular forms, without giving complete proofs, to show where we are headed.
    [Show full text]
  • Proquest Dissertations
    University of Alberta Descent Constructions for Central Extensions of Infinite Dimensional Lie Algebras by Jie Sun A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Department of Mathematical and Statistical Sciences Edmonton, Alberta Spring 2009 Library and Archives Bibliotheque et 1*1 Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington OttawaONK1A0N4 OttawaONK1A0N4 Canada Canada Your file Votre rSterence ISBN: 978-0-494-55620-7 Our file Notre rGterence ISBN: 978-0-494-55620-7 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library and permettant a la Bibliotheque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, preter, telecommunication or on the Internet, distribuer et vendre des theses partout dans le loan, distribute and sell theses monde, a des fins commerciales ou autres, sur worldwide, for commercial or non­ support microforme, papier, electronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in this et des droits moraux qui protege cette these. Ni thesis. Neither the thesis nor la these ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent etre im primes ou autrement printed or otherwise reproduced reproduits sans son autorisation.
    [Show full text]
  • Mathematics People
    people.qxp 10/18/95 11:55 AM Page 1543 Mathematics People Kac and Moody Receive Biography of Victor Kac Victor Kac was born on December 19, 1943, in Bugurus- Wigner Medal lan, USSR. He received his bachelor’s and master’s degrees (both in 1965) and his doctorate (1968) from Moscow State At the 20th Colloquium on Group Theoretical Methods in University. He was Physics, held in July 1994 in Osaka, Japan, Victor Kac of on the faculty of the Massachusetts Institute of Technology and Robert the Moscow Insti- Moody of the University of Alberta were named co-win- tute of Electronic ners of the Wigner Medal. Engineering from Though they did not work together, Kac and Moody in- 1968 to 1976. In dependently and simultaneously began in 1967 a series of 1977, he went to papers developing the theory of a new class of infinite-di- the Massachusetts mensional Lie algebras. Since then, the most important class Institute of Tech- nology, where he of such algebras has been called “Kac-Moody algebras”. is currently pro- “This was quite an event, because here was a whole new fessor of math- class of Lie algebras,” notes Morton Hamermesh, profes- ematics. Professor sor emeritus of physics at the University of Minnesota and Kac has presented a member of the committee awarding the medal. “It caught lectures in confer- the imagination of physicists because there are so many ences around the ways it could be applied to physics.” world, including An overview of the impact of Kac-Moody algebras on the following: In- physics—particularly particle physics, field theory, and ternational Con- string theory—is given in the article by L.
    [Show full text]
  • SECOND COHOMOLOGY of Aff(1) ACTING on N-ARY DIFFERENTIAL OPERATORS
    Bull. Korean Math. Soc. 56 (2019), No. 1, pp. 13{22 https://doi.org/10.4134/BKMS.b170774 pISSN: 1015-8634 / eISSN: 2234-3016 SECOND COHOMOLOGY OF aff(1) ACTING ON n-ARY DIFFERENTIAL OPERATORS Imed Basdouri, Ammar Derbali, and Soumaya Saidi Abstract. We compute the second cohomology of the affine Lie algebra n aff(1) on the dimensional real space with coefficients in the space Dλ,µ of n-ary linear differential operators acting on weighted densities where λ = (λ1; : : : ; λn). We explicitly give 2-cocycles spanning these cohomology. 1. Introduction In mathematical deformation theory one studies how an object in a certain category of spaces can be varied in dependence on the points of a parameter space. In other words, deformation theory thus deals with the structure of families of objects like varieties, singularities, vector bundles, coherent sheaves, algebras or differentiable maps. Deformation problems appear in various areas of mathematics, in particular in algebra, algebraic and analytic geometry, and mathematical physics. Cohomology is a useful tool in Poisson Geometry, plays an important role in Deformation and Quantization Theory, and attracts more and more interest among algebraists. In the theory of Lie groups, Lie algebras and their representation theory, a Lie algebra extension is an enlargement of a given Lie algebra g by another Lie algebra h. Extensions arise in several ways. There is the trivial extension obtained by taking a direct sum of two Lie algebras. Other types are the split extension and the central extension. Extensions may arise naturally, for instance, when forming a Lie algebra from projective group representations.
    [Show full text]
  • Infinite-Dimensional Lie Algebras
    Infinite-Dimensional Lie Algebras Part III Essay by Nima Amini Mathematical Tripos 2013-2014 Department of Pure Mathematics and Mathematical Statistics University of Cambridge 1 Contents 1 Kac-Moody Algebras 6 1.1 Summary of finite-dimensional semisimple Lie algebras . .7 1.2 Definitions . 12 1.3 Examples . 22 1.4 Symmetrizable Kac-Moody algebras . 24 1.5 The Weyl group . 28 1.6 Classification of generalized Cartan matrices . 29 2 Affine Lie Algebras 33 2.1 Realizations of affine Lie algebras . 34 2.2 Finite-dimensional representations . 37 3 Quantum Affine Algebras 44 3.1 Quantized enveloping algebras . 45 3.2 Drinfeld presentation . 50 3.3 Quantum loop algebras . 52 3.4 Chari-Pressley theorem . 56 3.5 Evaluation representations of Uq(slc2)............... 60 3.6 q-characters . 63 A Hopf Algebras 71 B Universal Enveloping Algebras 72 2 \ It is a well kept secret that the theory of Kac-Moody algebras has been a disaster. True, it is a generalization of a very important object, the simple finite-dimensional Lie algebras, but a generalization too straightforward to ex- pect anything interesting from it. True, it is remarkable how far one can go with all these ei's, fi's and hi's. Practically all, even most difficult results of finite-dimensional theory, such as the theory of characters, Schubert calculus and cohomology theory, have been extended to the general set-up of Kac-Moody algebras. But the answer to the most important question is missing: what are these algebras good for? Even the most sophisticated results, like the connections to the theory of quivers, seem to be just scratching the surface.
    [Show full text]
  • Historical Review of Lie Theory 1. the Theory of Lie
    Historical review of Lie Theory 1. The theory of Lie groups and their representations is a vast subject (Bourbaki [Bou] has so far written 9 chapters and 1,200 pages) with an extraordinary range of applications. Some of the greatest mathematicians and physicists of our times have created the tools of the subject that we all use. In this review I shall discuss briefly the modern development of the subject from its historical beginnings in the mid nineteenth century. The origins of Lie theory are geometric and stem from the view of Felix Klein (1849– 1925) that geometry of space is determined by the group of its symmetries. As the notion of space and its geometry evolved from Euclid, Riemann, and Grothendieck to the su- persymmetric world of the physicists, the notions of Lie groups and their representations also expanded correspondingly. The most interesting groups are the semi simple ones, and for them the questions have remained the same throughout this long evolution: what is their structure, where do they act, and what are their representations? 2. The algebraic story: simple Lie algebras and their representations. It was Sopus Lie (1842–1899) who started investigating all possible (local)group actions on manifolds. Lie’s seminal idea was to look at the action infinitesimally. If the local action is by R, it gives rise to a vector field on the manifold which integrates to capture the action of the local group. In the general case we get a Lie algebra of vector fields, which enables us to reconstruct the local group action.
    [Show full text]
  • Structure of Indefinite Quasi-Hyperbolic Kac-Moody
    Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 1821–1833 © Research India Publications http://www.ripublication.com/gjpam.htm Structure of Indefinite Quasi-Hyperbolic (3) Kac-Moody Algebras QH2 A. Uma Maheswari Associate Professor & Head, Department of Mathematics, Quaid-E-Millath Government College for Women (Autonomous), Chennai-600 002, Tamil Nadu, India. M. Priyanka Research Scholar, Department of Mathematics, Quaid-E-Millath Government College for Women (Autonomous), Chennai-600 002, Tamil Nadu, India. Abstract The paper discusess the indefinite type of quasi-hyperbolic Kac-Moody algebras (3) QH2 which are extended from the indefinite class of hyperbolic Kac-Moody al- (3) gebras H2 . These Quasi hyperbolic algebras are realized as graded Lie algebra of Kac-Moody type. Using the homological approach and Hochschild-Serre spectral sequences theory, the structure of the components of the maximal graded ideals upto level five is determined. AMS subject classification: 17B67. Keywords: Kac-Moody algebra, indefinite, Quasi- hyperbolic, homology modules, maximal graded ideal, spectral sequences. 1822 A. Uma Maheswari and M. Priyanka 1. Introduction Victor Kac and Robert Moody independently introduced Kac-Moody Lie algebras in 1968. Since then, Kac-Moody algebras have grown into an important field with appli- cations in physics and many areas of mathematics. The structure of the Kac-Moody algebras of finite type and affine type are well understood. However, determining the structure of indefinite type Kac-Moody algebras is an open problem. In [1], root multiplicities upto level 2 were computed for the hyperbolic Kac-Moody (1) Lie algebra HA1 . Using homological techniques of graded Lie algebras and spectral sequence theory, the root multiplicities for the hyperbolic type of Kac-Moody Lie algebra (1) (n) HA1 and HA1 upto level five have been computed in [2–5].
    [Show full text]
  • Topics in Representation Theory: Cultural Background 1 Some History
    Topics in Representation Theory: Cultural Background This semester we will be covering various topics in representation theory, see the separate syllabus for a detailed list of topics, including some that we may or may not get to. Much of the material will be rather standard for a course of this kind, but an attempt will be made to explain some of the relations between representation theory and quantum mechanics. The notion of “quantization” has motivated several different approaches to the construction of group repre- sentations and some of the most important applications of representation theory are to the solution of quantum mechanical problems. Some important things we won’t be covering include the representation the- ory of non-compact Lie groups such as SL(2, R), and of arithmetically interest- ing groups such as GL(n, Qp). Our approach to representation theory will be quite geometric, but in the context of differential geometry over the real num- bers, not that of algebraic geometry over more general fields, i.e. the theory of “algebraic groups”. 1 Some History It often helps to have some idea of the history of a field while trying to learn it, so we’ll begin with a short outline of the history of important ideas in the subject, as well as the names of the mathematicians most responsible for their development. For more details of this history, two excellent recent references are [1] and [5]. • 1873: Lie groups. Sophus Lie (1842-1899) • 1888: Classification of Lie algebras. Wilhelm Killing (1847-1923) • 1896: Representations of finite groups, characters.
    [Show full text]
  • 2003 Annual Report to Members
    2003 Annual Report to Members June 13, 2004 2003 Annual Report to Members Table of Contents President’s Report . .2 Forum 2003 Report . .4 Executive Director's report . .6 Treasurer’s Report . .8 Committee Reports Advancement of Mathematics . .14 Education . .16 Electronic Services . .17 Endowment Grants . .18 Finance Committee . .19 International Affairs . .20 Mathematical Competitions . .21 Nominating . .23 Publications . .24 Research . .25 Student . .27 Women in Mathematics . .29 Editorial Board . .30 Contributors . .31 Executive Committee . .32 Board of Directors . .32 Executive Office . .33 1 Canadian Mathematical Society 2003 Annual Report to Members President’s Report Christiane Rousseau (University of Montreal) 2003, a splendid year for mathematics in Canada presence of Mahdu Sudan reminded us of the time when we honoured him 2003 was an exciting year for mathematics in Canada. In March there was at the Canadian Embassy in Beijing as the recipient of the Nevanlinna Prize the inauguration of Banff International Research Station (BIRS). BIRS during ICM 2002. The participants also had their choice among fourteen provides mathematicians with an exceptional facility to concentrate on diverse symposia including one in Education and one in History of research and exchange ideas in a splendid environment. In May the Mathematics. At the banquet we honoured our four prize winners: Andy Canadian School Mathematics Forum took place in Montreal: it raised a lot Liu, winner of the 2003 Adrien Pouliot Prize in mathematical education, of enthusiasm for mathematics education in Canada and follow-up James Arthur, winner of the G. de B. Robinson Prize for the best article “A activities are taking place across Canada.
    [Show full text]
  • Spin Covers of Maximal Compact Subgroups of Kac-Moody Groups and of Weyl Groups
    Spin Covers of Maximal Compact Subgroups of Kac-Moody Groups and of Weyl Groups by Davoud Ghatei A thesis submitted to The University of Birmingham for the degree of Doctor of Philosophy School of Mathematics The University of Birmingham April 2014 University of Birmingham Research Archive e-theses repository This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder. Dedicated to my father Prof. M. A. Ghatei with love and admiration. 1 Abstract The maximal compact subgroup SO(n; R) of SL(n; R) admits two double covers Spin(n; R) and O(n; R): In this thesis we show that in fact given any simply-laced diagram ∆; with associated split real Kac-Moody group G(∆); the `maximal compact' subgroup K(∆) also admits a double cover Spin(∆): We then describe a subgroup of Spin(∆) which is a cover of the Weyl group W (∆) associated to g(∆): This answers a conjecture of Damour and Hillmann. Moreover, the group D which extends W is used in an attempt to classify the images of the so-called generalised spin representations. Acknowledgements First and foremost my gratitude and debt are owed to my supervisors Professor Ralf K¨ohl and Professor Sergey Shpectorov.
    [Show full text]
  • Rigidity of Microsphere Heaps
    Abstract ˆ RAHMOELLER, MARGARET LYNN . On Demazure Crystals for the Quantum Affine AlgebraUq s l (n) . (Under the direction of Dr. Kailash Misra.) Lie algebras and their representations have been an important area of mathematical research due to their relevance in several areas of mathematics and physics. It is known that the representation of a Lie algebra can be studied by considering the representations of its universal enveloping al- gebra, which is an associative algebra. In 1968, Victor Kac and Robert Moody defined a class of infinite dimensional Lie algebras called affine Lie algebras. An affine Lie algebra can be viewed as the universal central extension of the Lie algebra of polynomial maps from the unity circle to ˆ a finite dimensional simple Lie algebra. In this thesis, we consider the affine Lie algebra s l (n) associated with the finite dimensional simple Lie algebra s l (n) of n n trace zero matrices over the field of complex numbers. We consider certain representations of× the quantum affine Lie algebra ˆ ˆ ˆ Uq (s l (n)), which is a q -deformation of the universal enveloping algebra U (s l (n)) of s l (n) intro- duced by Michio Jimbo and Vladimir Drinfeld in 1985. In 1988, it was shown by George Lusztig that ˆ ˆ the q -deformation of a representation ofUq (s l (n)) parallels the representation of s l (n) for generic q . It is known that for any dominant integral weight λ there is a unique (up to isomorphism) irreducible ˆ Uq (s l (n))- module V (λ).
    [Show full text]