Notes for Lie Groups & Representations Instructor: Andrei Okounkov
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An Introduction to Affine Kac-Moody Algebras
An introduction to affine Kac-Moody algebras David Hernandez To cite this version: David Hernandez. An introduction to affine Kac-Moody algebras. DEA. Lecture notes from CTQM Master Class, Aarhus University, Denmark, October 2006, 2006. cel-00112530 HAL Id: cel-00112530 https://cel.archives-ouvertes.fr/cel-00112530 Submitted on 9 Nov 2006 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. AN INTRODUCTION TO AFFINE KAC-MOODY ALGEBRAS DAVID HERNANDEZ Abstract. In these lectures we give an introduction to affine Kac- Moody algebras, their representations, and applications of this the- ory. Contents 1. Introduction 1 2. Quick review on semi-simple Lie algebras 2 3. Affine Kac-Moody algebras 5 4. Representations of Lie algebras 8 5. Fusion product, conformal blocks and Knizhnik- Zamolodchikov equations 14 References 19 1. Introduction Affine Kac-Moody algebras ˆg are infinite dimensional analogs of semi-simple Lie algebras g and have a central role both in Mathematics (Modular forms, Geometric Langlands program...) and Mathematical Physics (Conformal Field Theory...). These lectures are an introduction to the theory of affine Kac-Moody algebras and their representations with basic results and constructions to enter the theory. -
Root Components for Tensor Product of Affine Kac-Moody Lie Algebra Modules
ROOT COMPONENTS FOR TENSOR PRODUCT OF AFFINE KAC-MOODY LIE ALGEBRA MODULES SAMUEL JERALDS AND SHRAWAN KUMAR 1. Abstract Let g be an affine Kac-Moody Lie algebra and let λ, µ be two dominant integral weights for g. We prove that under some mild restriction, for any positive root β, V(λ)⊗V(µ) contains V(λ+µ−β) as a component, where V(λ) denotes the integrable highest weight (irreducible) g-module with highest weight λ. This extends the corresponding result by Kumar from the case of finite dimensional semisimple Lie algebras to the affine Kac-Moody Lie algebras. One crucial ingredient in the proof is the action of Virasoro algebra via the Goddard-Kent-Olive construction on the tensor product V(λ) ⊗ V(µ). Then, we prove the corresponding geometric results including the higher cohomology vanishing on the G-Schubert varieties in the product partial flag variety G=P × G=P with coefficients in certain sheaves coming from the ideal sheaves of G-sub Schubert varieties. This allows us to prove the surjectivity of the Gaussian map. 2. Introduction Let g be a symmetrizable Kac–Moody Lie algebra, and fix two dominant inte- gral weights λ, µ 2 P+. To these, we can associate the integrable, highest weight (irreducible) representations V(λ) and V(µ). Then, the content of the tensor decom- position problem is to express the product V(λ)⊗V(µ) as a direct sum of irreducible components; that is, to find the decomposition M ⊕mν V(λ) ⊗ V(µ) = V(ν) λ,µ ; ν2P+ ν where mλ,µ 2 Z≥0 is the multiplicity of V(ν) in V(λ) ⊗ V(µ). -
Contemporary Mathematics 442
CONTEMPORARY MATHEMATICS 442 Lie Algebras, Vertex Operator Algebras and Their Applications International Conference in Honor of James Lepowsky and Robert Wilson on Their Sixtieth Birthdays May 17-21, 2005 North Carolina State University Raleigh, North Carolina Yi-Zhi Huang Kailash C. Misra Editors http://dx.doi.org/10.1090/conm/442 Lie Algebras, Vertex Operator Algebras and Their Applications In honor of James Lepowsky and Robert Wilson on their sixtieth birthdays CoNTEMPORARY MATHEMATICS 442 Lie Algebras, Vertex Operator Algebras and Their Applications International Conference in Honor of James Lepowsky and Robert Wilson on Their Sixtieth Birthdays May 17-21, 2005 North Carolina State University Raleigh, North Carolina Yi-Zhi Huang Kailash C. Misra Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Andreas Blass Abel Klein 2000 Mathematics Subject Classification. Primary 17810, 17837, 17850, 17865, 17867, 17868, 17869, 81T40, 82823. Photograph of James Lepowsky and Robert Wilson is courtesy of Yi-Zhi Huang. Library of Congress Cataloging-in-Publication Data Lie algebras, vertex operator algebras and their applications : an international conference in honor of James Lepowsky and Robert L. Wilson on their sixtieth birthdays, May 17-21, 2005, North Carolina State University, Raleigh, North Carolina / Yi-Zhi Huang, Kailash Misra, editors. p. em. ~(Contemporary mathematics, ISSN 0271-4132: v. 442) Includes bibliographical references. ISBN-13: 978-0-8218-3986-7 (alk. paper) ISBN-10: 0-8218-3986-1 (alk. paper) 1. Lie algebras~Congresses. 2. Vertex operator algebras. 3. Representations of algebras~ Congresses. I. Leposwky, J. (James). II. Wilson, Robert L., 1946- III. Huang, Yi-Zhi, 1959- IV. -
A Localization Argument for Characters of Reductive Lie Groups: an Introduction and Examples Matvei Libine
This is page 375 Printer: Opaque this A localization argument for characters of reductive Lie groups: an introduction and examples Matvei Libine In Honor of Jacques Carmona ABSTRACT In this article I describe my recent geometric localization argument dealing with actions of noncompact groups which provides a geometric bridge between two entirely different character formulas for reductive Lie groups and answers the question posed in [Sch]. A corresponding problem in the compact group setting was solved by N. Berline, E. Getzler and M. Vergne in [BGV] by an application of the theory of equivariant forms and, particularly, the fixed point integral localization formula. This localization argument seems to be the first successful attempt in the direction of build- ing a similar theory for integrals of differential forms, equivariant with respect to actions of noncompact groups. I will explain how the argument works in the SL(2, R) case, where the key ideas are not obstructed by technical details and where it becomes clear how it extends to the general case. The general argument appears in [L]. Ihavemade every effort to present this article so that it is widely accessible. Also, although characteristic cycles of sheaves is mentioned, I do not assume that the reader is familiar with this notion. 1 Introduction For motivation, let us start with the case of a compact group. Thus we consider a connected compact group K and a maximal torus T ⊂ K . Let kR and tR denote the Lie algebras of K and T respectively, and k, t be their complexified Lie algebras. Let π be a finite-dimensional representation of K , that is π is a group homomorphism K → Aut(V ). -
NOTES on the WEYL CHARACTER FORMULA the Aim of These Notes
NOTES ON THE WEYL CHARACTER FORMULA The aim of these notes is to give a self-contained algebraic proof of the Weyl Character Formula. The necessary background results on modules for sl2(C) and complex semisimple Lie algebras are outlined in the first two sections. Some technical details are left to the exercises at the end; solutions are provided when the exercise is needed for the proof. 1. Representations of sl2(C) Define ! ! ! 1 0 0 1 0 0 h = ; e = ; f = 0 −1 0 0 1 0 2 and note that hh; e; fi = sl2(C). Let u; v be the canonical basis of E = C . Then each Symd E is irreducible with ud spanning the highest-weight space d of weight d and, up to isomorphism, Sym E is the unique irreducible sl2(C)- module with highest weight d. (See Exercises 1.1 and 1.2.) The diagram below shows the actions of h, e and f on the canonical basis of Symd E: loops show the action of h, arrows to the right show the action of e, arrow to the left show the action of f. d−2c−2 d−2c d−2c+2 d−c+1 d−c d−c−1 i • j * • j * • ) c−1 ud−c−1vc+1 c ud−cvc c+1 ud−c+1vc−1 In particular (a) the eigenvalues of h on Symd E are −d; −d + 2; : : : ; d − 2; d and each h-eigenspace is 1-dimensional, (b) if w 2 Symd E and h · w = (d − 2c)w then f · e · w = c(d − c + 1)w. -
Affine Lie Algebras 8
Affine Lie Algebras Kevin Wray January 16, 2008 Abstract In these lectures the untwisted affine Lie algebras will be constructed. The reader is assumed to be familiar with the theory of semisimple Lie algebras, e.g. that he or she knows a big part of James E. Humphreys' Introduction to Lie algebras and repre- sentation theory [1]. The notations used in these notes will be taken from [1]. These lecture notes are based on the course Affine Lie Algebras given by Prof. Dr. Johan van de Leur at the Mathematical Research Insitute in Utrecht (The Netherlands) during the fall of 2007. 1 Semisimple Lie Algebras 1.1 Root Spaces Recall some basic notions from [1]. Let L be a semisimple Lie algebra, H a Cartan subalgebra (CSA), and κ(x; y) = T r(ad (x) ad (y)) the Killing form on L. Then the Killing form is symmetric, non-degenerate (since L is semisimple and using theorem 5.1 page 22 z [1]), and associative; i.e. κ : L × L ! F is bilinear on L and satisfies κ ([x; y]; z) = κ (x; [y; z]) : The restriction of the Killing form to the CSA, denoted by κjH (·; ·), is non-degenerate (Corollary 8.2 page 37 [1]). This allows for the identification of H with H∗ (see [1] x8.2: ∗ to φ 2 H there corresponds a unique element tφ 2 H satisfying φ(h) = κ(tφ; h) for all h 2 H). This makes it possible to define a symmetric, non-degenerate bilinear form, ∗ ∗ ∗ (·; ·): H × H ! F, given on H as ∗ (α; β) = κ(tα; tβ)(8 α; β 2 H ) : ∗ Let Φ ⊂ H be the root system corresponding to L and ∆ = fα1; : : : ; α`g a fixed basis of Φ (∆ is also called a simple root system). -
Title: Algebraic Group Representations, and Related Topics a Lecture by Len Scott, Mcconnell/Bernard Professor of Mathemtics, the University of Virginia
Title: Algebraic group representations, and related topics a lecture by Len Scott, McConnell/Bernard Professor of Mathemtics, The University of Virginia. Abstract: This lecture will survey the theory of algebraic group representations in positive characteristic, with some attention to its historical development, and its relationship to the theory of finite group representations. Other topics of a Lie-theoretic nature will also be discussed in this context, including at least brief mention of characteristic 0 infinite dimensional Lie algebra representations in both the classical and affine cases, quantum groups, perverse sheaves, and rings of differential operators. Much of the focus will be on irreducible representations, but some attention will be given to other classes of indecomposable representations, and there will be some discussion of homological issues, as time permits. CHAPTER VI Linear Algebraic Groups in the 20th Century The interest in linear algebraic groups was revived in the 1940s by C. Chevalley and E. Kolchin. The most salient features of their contributions are outlined in Chapter VII and VIII. Even though they are put there to suit the broader context, I shall as a rule refer to those chapters, rather than repeat their contents. Some of it will be recalled, however, mainly to round out a narrative which will also take into account, more than there, the work of other authors. §1. Linear algebraic groups in characteristic zero. Replicas 1.1. As we saw in Chapter V, §4, Ludwig Maurer thoroughly analyzed the properties of the Lie algebra of a complex linear algebraic group. This was Cheval ey's starting point. -
Introduction to Vertex Operator Algebras I 1 Introduction
数理解析研究所講究録 904 巻 1995 年 1-25 1 Introduction to vertex operator algebras I Chongying Dong1 Department of Mathematics, University of California, Santa Cruz, CA 95064 1 Introduction The theory of vertex (operator) algebras has developed rapidly in the last few years. These rich algebraic structures provide the proper formulation for the moonshine module construction for the Monster group ([BI-B2], [FLMI], [FLM3]) and also give a lot of new insight into the representation theory of the Virasoro algebra and affine Kac-Moody algebras (see for instance [DL3], [DMZ], [FZ], [W]). The modern notion of chiral algebra in conformal field theory [BPZ] in physics essentially corresponds to the mathematical notion of vertex operator algebra; see e.g. [MS]. This is the first part of three consecutive lectures by Huang, Li and myself. In this part we are mainly concerned with the definitions of vertex operator algebras, twisted modules and examples. The second part by Li is about the duality and local systems and the third part by Huang is devoted to the contragradient modules and geometric interpretations of vertex operator algebras. (We refer the reader to Li and Huang’s lecture notes for the related topics.) So many exciting topics are not covered in these three lectures. The book [FHL] is an excellent introduction to the subject. There are also existing papers [H1], [Ge] and [P] which review the axiomatic definition of vertex operator algebras, geometric interpretation of vertex operator algebras, the connection with conformal field theory, Borcherds algebras and the monster Lie algebra. Most work on vertex operator algebras has been concentrated on the concrete exam- ples of vertex operator algebras and the representation theory. -
A Recursive Formula for Characters of Simple Lie Algebras
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 137, 126-144 (1991) A Recursive Formula for Characters of Simple Lie Algebras STEVEN N. KASS* Centre de recherches mathkmatiques, UniversitP de MontrPal C. P. 6128, &cc. “A”, MontrPal, Qutbec, Canada H3C 357 Communicated by Robert Steinberg Received July 20, 1988; accepted July 18, 1989 In 1964, Antoine and Speiser published succinct and elegant formulae for the characters of the irreducible highest weight modules for the Lie algebras A, and B,. (The result for A, may have been known as early as 1957.) These can be derived from a recursive formula for characters valid for all simple complex Kac-Moody Lie algebras for which the Weyl-Kac character formula holds. 0 1991 Academic Press. Inc. 1. INTRODUCTION To avoid the encumbrance of technicalities, we present the results first for simple finite-dimensional Lie algebras over C. We use the following notation, generally following Humphreys [H]. Let L be a finite-dimensional complex simple Lie algebra of rank 1. We denote by H a Cartan subalgebra of L, by ai (1 Q i < I) a choice of simple roots, by Q the integral lattice they generate, and by R, R+, and R- the root system, positive roots, and negative roots with respect to H and the ai, respectively. Let A be the weight lattice for L, and li (1~ i,< I) the fundamental weights corresponding to the choice of simple roots. The set of dominant weights will be denoted A +. -
Lie Algebras by Shlomo Sternberg
Lie algebras Shlomo Sternberg April 23, 2004 2 Contents 1 The Campbell Baker Hausdorff Formula 7 1.1 The problem. 7 1.2 The geometric version of the CBH formula. 8 1.3 The Maurer-Cartan equations. 11 1.4 Proof of CBH from Maurer-Cartan. 14 1.5 The differential of the exponential and its inverse. 15 1.6 The averaging method. 16 1.7 The Euler MacLaurin Formula. 18 1.8 The universal enveloping algebra. 19 1.8.1 Tensor product of vector spaces. 20 1.8.2 The tensor product of two algebras. 21 1.8.3 The tensor algebra of a vector space. 21 1.8.4 Construction of the universal enveloping algebra. 22 1.8.5 Extension of a Lie algebra homomorphism to its universal enveloping algebra. 22 1.8.6 Universal enveloping algebra of a direct sum. 22 1.8.7 Bialgebra structure. 23 1.9 The Poincar´e-Birkhoff-Witt Theorem. 24 1.10 Primitives. 28 1.11 Free Lie algebras . 29 1.11.1 Magmas and free magmas on a set . 29 1.11.2 The Free Lie Algebra LX ................... 30 1.11.3 The free associative algebra Ass(X). 31 1.12 Algebraic proof of CBH and explicit formulas. 32 1.12.1 Abstract version of CBH and its algebraic proof. 32 1.12.2 Explicit formula for CBH. 32 2 sl(2) and its Representations. 35 2.1 Low dimensional Lie algebras. 35 2.2 sl(2) and its irreducible representations. 36 2.3 The Casimir element. 39 2.4 sl(2) is simple. -
Weyl Character Formula for U(N)
THE WEYL CHARACTER FORMULA–I PRAMATHANATH SASTRY 1. Introduction This is the first of two lectures on the representations of compact Lie groups. In this lecture we concentrate on the group U(n), and classify all its irreducible representations. The methods give the paradigm and set the template for study- ing representations of a large class of compact Lie groups—a class which includes semi-simple compact groups. The climax is the character formula of Weyl, which classifies all irreducible representations for this class, up to unitary equivalence. The typing was done in a hurry (last night and this morning, to be precise), and there may well be more serious errors than mere typos. Typos of course abound. 2. Fourier Analysis on abelian groups It is well-known, and easy to prove, that a compact ableian Lie group 1 is nec- essarily a torus Tn = S1 × . × S1 (n factors). A character on T = Tn is a Lie T C∗ 2 m1 mn group map χ : → . The maps χm1,... ,mn ((t1,...,tn) 7→ t1 ...tn ) lists n all characters on T as (m1,...,mn) varies over Z . This is a fairly elementary result—indeed, the image of a character χ must be a compact connected subgroup of C∗, forcing χ to take values in S1 ⊂ C∗. From there to seeing χ must equal a χm1,... ,mn is easy (try the case n = 1 !). T has an invariant Borel measure dt = dt1 ... dtn (the product of the “arc length” measures on the various S1 factors), the so called Haar measure on T 3. -
Branching Rules of Classical Lie Groups in Two Ways
Branching Rules of Classical Lie Groups in Two Ways by Lingxiao Li Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Bachelor of Science in Mathematics with Honors at Stanford University June 2018 Thesis advisor: Daniel Bump Contents 1 Introduction 2 2 Branching Rule of Successive Unitary Groups 4 2.1 Weyl Character Formula . .4 2.2 Maximal Torus and Root System of U(n) .....................5 2.3 Statement of U(n) # U(n − 1) Branching Rule . .6 2.4 Schur Polynomials and Pieri’s Formula . .7 2.5 Proof of U(n) # U(n − 1) Branching Rule . .9 3 Branching Rule of Successive Orthogonal Groups 13 3.1 Maximal Tori and Root Systems of SO(2n + 1) and SO(2n) .......... 13 3.2 Branching Rule for SO(2n + 1) # SO(2n) ..................... 14 3.3 Branching Rule for SO(2n) # SO(2n − 1) ..................... 17 4 Branching Rules via Duality 21 4.1 Algebraic Setup . 21 4.2 Representations of Algebras . 22 4.3 Reductivity of Classical Groups . 23 4.4 General Duality Theorem . 25 4.5 Schur-Weyl Duality . 27 4.6 Weyl Algebra Duality . 28 4.7 GL(n; C)-GL(k; C) Duality . 31 4.8 O(n; C)-sp(k; C) Duality . 33 4.9 Seesaw Reciprocity . 36 4.10 Littlewood-Richardson Coefficients . 37 4.11 Stable Branching Rule for O(n; C) × O(n; C) # O(n; C) ............. 39 1 Acknowledgements I would like to express my great gratitude to my honor thesis advisor Daniel Bump for sug- gesting me the interesting problem of branching rules, for unreservedly and patiently sharing his knowledge with me on Lie theory, and for giving me valuable instruction and support in the process of writing this thesis over the recent year.