DOCSLIB.ORG

A Construction of the Level 3 Modules for the Affine Lie

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Construction of the Level 3 Modules for the Affine Lie TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 348, Number 2, February 1996 A CONSTRUCTION OF THE LEVEL 3 MODULES FOR (2) THE AFFINE LIE ALGEBRA A2 AND A NEW COMBINATORIAL IDENTITY OF THE ROGERS-RAMANUJAN TYPE STEFANO CAPPARELLI Abstract. We obtain a vertex operator construction of level 3 standard rep- (2) resentations for the affine Lie algebra A2 . As a corollary, we also get new conbinatorial identities. 0. Introduction The identities of Rogers and Ramanujan have a famous and interesting history. They also seem to appear at the crossroads of various mathematical disciplines, as is symptomatic of an important piece of mathematics. These identities play an important role in the representation theory of affine algebras and have also made their appearance in an important work of R.J. Baxter on the hard hexagon model in statistical mechanics [Ba]. Some years ago, it was observed in [LM] that the character of certain repre- sentations of affine Kac-Moody algebras, when suitably specialized, coincided with one member of the classical Rogers-Ramanujan identities. The explanation of this fact was given by J. Lepowsky and R.L. Wilson in a series of papers [LW1-4] They (1) constructed the standard representations of the affine Lie algebra A1 . The rep- resentations of level 3, in particular, yielded the interpretation and proof of the Rogers-Ramanujan identities. The main new concept used in their work was that of “vertex operator” which has proved to be extremely fruitful, and has been also used, for example, in the construction of a natural representation of the Monster simple group (see [FLM]). Lepowsky and Wilson showed the remarkable interaction between the repre- sentation theory of affine Lie algebras and combinatorial identities and provided a general framework for the study of this relationship. This approach has been used successfully in a number of works, [LP1-2],[Mi1-3], [MP], [Ma]. Among these, the work of A.Meurman and M.Primc [MP], gave a construction of all standard (1) A1 -modules and a Lie-theoretic proof of the generalizations to all moduli of the Rogers-Ramanujan identities due to G.E. Andrews, B. Gordon, D. Bressoud, H. G¨olnitz, [A],[G], [Br], [G¨o]. This line of research, however, had not produced any combinatorial identities that were unknown to the specialists. In a previous work, [C2], we had stated certain identities which we believed to (2) underlie the standard representations of level 3 for the affine algebra A2 . The Lie Received by the editors January 12, 1994. 1991 Mathematics Subject Classification. Primary 17B65, 17B67, 05A19. c 1996 American Mathematical Society 481 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 482 STEFANO CAPPARELLI theoretic proof of such identities is the object of this paper. Here we give a vertex (2) operator construction of the level 3 standard A2 - modules and give a proof of the independence of the set of vectors provided in [C2]. While this work was in progress, Andrews gave an interesting combinatorial proof of one of these identities,[A2]. His method of proof is based on certain techniques that he developed in his study of Schur’s theorem. It is extremely interesting to note that Andrews’ motivation for such a study was the search of a combinatorial interpretation of the q-trinomial coefficients, that he and Baxter had developed during their research about certain generalizations of the hard hexagon model. It is clear that a whole infinite family of possibly new combinatorial identi- (2) ties corresponding to the higher standard A2 -modules remains to be investigated. This is presently under study. It will be important to investigate also the possible applications to the study of statistical mechanics models. Recently in [MP2] our combinatorial identities have appeared again in the study of a different specializa- (1) tion of the fundamental A1 -modules. 1. The algebra (2) A2 (2) We construct the affine algebra A2 using the results of [L1]. Let Φ be an A2 root system, ∆ = α1,α2 abaseofΦ,andLthe corresponding root lattice. Let , be a nondegenerate{ form} on L normalized so that α, α =2for α Φ. Let σ beh· the·i automorphism induced by the Dynkin diagram automorphismh i ∈ determined by σα1 = α2, σα2 = α1. Denote by σ1 the reflection with respect to α1 and consider (1.1) ν = σ1σ. Then ν has order 6 and is a “twisted Coxeter element” (see [Sp], [Fi]). It is known that twisted and untwisted Coxeter elements satisfy the properties (1.2) Σνpα =0, (1.3) Σp νpα, β 0 modulo M h i≡ for all α, β L, M the order of ν, summations ranging over ZM . In this setting∈ the central extension 6 (1.4) 1 κ κ =1 Lˆ − L 1 −→ h | i−→ −→ −→ determined by the commutator map Σνpα,β Σpνpα,β (1.5) C(α, β)=( 1)h iωh i − splits since C(α, β)=1,ωbeing a primitive sixth-root of 1. Hence Lˆ is the direct product L κ . Set ×h i (1.6) Q =(1 ν)L. − License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use (2) LEVEL 3 MODULES FOR THE AFFINE LIE ALGEBRA A2 483 1 Observe that since 1 ν = ν− we have Q = L. Now Theorem 6.2 in [L1] gives a − ˆ unique homomorphism ψ : L C∗ such that ψ(κ)=ωand L acts as the identity. → Set T = Cψ, the one-dimensional Lˆ-module affording ψ.Lethbe the complex- ification of L, i.e. (1.7) h = L Z C ⊗ and extend ν to h by linearity. Then h splits into two ν-eigenspaces (1.8) h = h(1) h( 1). ⊕ − Formtheaffinealgebra (1.9) h˜[ν]= h tn Cc Cd (n)⊗ ⊕ ⊕ n Z a∈ with brackets i j (1.10) [x t ,y t ]= x, y iδi+j,0c, ⊗ ⊗ h i [d, x ti]=ix ti ⊗ ⊗ for i, j Z,x h ,y h ,andccentral. Denote by s its commutator subalgebra ∈ ∈ (i) ∈ (j) (1.11) s = h tn Cc (n) ⊗ ⊕ n=0 a6 and consider the subalgebras n (1.12) s = h(n) t , ± n>0 ⊗ ±a (1.13) b = b[ν]=s Cc Cd +⊕ ⊕ Then we may observe that the commutator subalgebra of s is one-dimensional and coincides with Cc. Hence s is a Heisenberg Lie algebra. It is well-known that the induced h˜[ν]-module ˜ (1.14) (h[ν]) (b) C, U ⊗U where b acts on C trivially except for c which acts as the identity, is an irreducible representation of s. As vector spaces, (1.14) is isomorphic to (1.15) S = S(s ), − the symmetric algebra on s . − The action of d defines a grading on S.Forn Z,x h(n),writex(n)forthe n ∈ ∈ operator on S corresponding to x t .Forα hdenote by α(n) the projection of α onto h . Define, for α h, ⊗ ∈ (n) ∈ z n (1.16) E (α, z)=exp( α (n) − ) ± (n) n n>0 ±X License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 484 STEFANO CAPPARELLI where n Z, a pair of formal Laurent series with coefficients in End S.Set ∈ (1.17) VT = S C. ⊗ We can finally define, for any a Lˆ, the vertex operator ∈ a,¯ a¯ + (1.18) X(a, z)=6−h iσ(¯a)E−( a,¯ z)E ( a,¯ z)a − − where σ(¯a) is a (nonzero) normalization constant, α,α 1 να,α 2 ν2α,α (1.19) σ(α)=2−h i(1 ω− )h i(1 ω− )h i − − for α L. Also∈ set, for α L, ∈ n 1 (1.20) α(z)= α(n)z− − . n Z X∈ Letν ˆ denote the lifting to Lˆ of ν with the properties (1.21) (ˆνa)− = νa,¯ (1.22) νa¯ =¯a νaˆ = a ⇒ It is important to remark that (1.23) X(ˆνa,z) = lim X(a, z) z ω 1z → − 1 where the limit notation on the right hand side indicates the substitution of ω− z for z. This property is important for the following developments and it constitutes the motivation for the definition of Q in (1.6). We can now state the theorem that gives the commutation of two vertex opera- tors in this setting. It is a particular case of Theorem 8.1 in [L1]. Theorem 1.1. Let a Lˆ, such that a¯ L2 = α L; α, α =2 . Then ∈ ∈ { ∈ h i } 1 [X(a, z ),X(a, z )] = ε (ν2a,¯ a¯)X(ˆνa,z )δ(ω 2z /z ) 1 2 6 2 2 − 2 1 1 + ε (ν 2a,¯ a¯)X(ˆν 1a, z )δ(ω 4z /z ) 6 2 − − 2 − 2 1 1 + ε2( a,¯ a¯)Dδ( z2/z1) 62 − − 1 ε2( a,¯ a¯)z2a¯(z2)δ( z2/z1), − 6 − − where 1 2 1 2 ν− α+ν− α,β ν− α+2ν− α,β ε2(α, β)=( 1)h iωh i. − Remark. Notice that, because of (1.23), this result will determine the commutator of X(a, z1)andX(b, z2) for any pair a, b Lˆ2. ∈ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use (2) LEVEL 3 MODULES FOR THE AFFINE LIE ALGEBRA A2 485 Define a nonassociative algebra (g, [ , ]) over C as follows: · · (1.24) g = h Cxa, ⊕ a Lˆ2 a∈ where x is a set of symbols, subject only to the linear relation x = ωx , a a Lˆ2 κa a and { } ∈ [h,h]=0,[h, xa]= h, a¯ xa = [xa,h], h i − ε2(¯a, a¯)¯a if ab =1 − [xa,xb]= ε2(¯a, ¯b)xab if ab Lˆ2 ∈ 5 0ifab / Lˆ2 1,k,...,k ∈ ∪{ } for a, b Lˆ2,h h.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • 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.
    [Show full text]
  • Notes for Lie Groups & Representations Instructor: Andrei Okounkov
    Notes for Lie Groups & Representations Instructor: Andrei Okounkov Henry Liu May 2, 2017 Abstract These are my live-texed notes for the Spring 2017 offering of MATH GR6344 Lie Groups & Repre- sentations. There are known omissions. Let me know when you find errors or typos. I'm sure there are plenty. 1 Kac{Moody Lie Algebras 1 1.1 Root systems and Weyl groups . 1 1.2 Reflection groups . 2 1.3 Regular polytopes and Coxeter groups . 4 1.4 Kac{Moody Lie algebras . 5 1.5 Examples of Kac{Moody algebras . 7 1.6 Category O . 8 1.7 Gabber{Kac theorem . 10 1.8 Weyl{Kac character formula . 11 1.9 Weyl character formula . 12 1.10 Affine Kac{Moody Lie algebras . 15 2 Equivariant K-theory 18 2.1 Equivariant sheaves . 18 2.2 Equivariant K-theory . 20 3 Geometric representation theory 22 3.1 Borel{Weil . 22 3.2 Localization . 23 3.3 Borel{Weil{Bott . 25 3.4 Hecke algebras . 26 3.5 Convolution . 27 3.6 Difference operators . 31 3.7 Equivariant K-theory of Steinberg variety . 32 3.8 Quantum groups and knots . 33 a Chapter 1 Kac{Moody Lie Algebras Given a semisimple Lie algebra, we can construct an associated root system, and from the root system we can construct a discrete group W generated by reflections (called the Weyl group). 1.1 Root systems and Weyl groups Let g be a semisimple Lie algebra, and h ⊂ g a Cartan subalgebra. Recall that g has a non-degenerate bilinear form (·; ·) which is preserved by the adjoint action, i.e.
    [Show full text]
  • Branching Functions for Admissible Representations of Affine Lie
    axioms Article Branching Functions for Admissible Representations of Affine Lie Algebras and Super-Virasoro Algebras Namhee Kwon Department of Mathematics, Daegu University, Gyeongsan, Gyeongbuk 38453, Korea; [email protected] Received: 2 May 2019; Accepted: 17 July 2019; Published: 19 July 2019 Abstract: We explicitly calculate the branching functions arising from the tensor product decompositions between level 2 and principal admissible representations over slb2. In addition, investigating the characters of the minimal series representations of super-Virasoro algebras, we present the tensor product decompositions in terms of the minimal series representations of super-Virasoro algebras for the case of principal admissible weights. Keywords: branching functions; admissible representations; characters; affine Lie algebras; super-Virasoro algebras 1. Introduction One of the basic problems in representation theory is to find the decomposition of a tensor product between two irreducible representations. In fact, the study of tensor product decompositions plays an important role in quantum mechanics and in string theory [1,2], and it has attracted much attention from combinatorial representation theory [3]. In addition, recent studies reveal that tensor product decompositions are also closely related to the representation theory of Virasoro algebra and W-algebras [4–6]. In [6], the authors extensively study decompositions of tensor products between integrable representations over affine Lie algebras. They also investigate relationships among tensor products, branching functions and Virasoro algebra through integrable representations over affine Lie algebras. In the present paper we shall follow the methodology appearing in [6]. However, we will focus on admissible representations of affine Lie algebras. Admissible representations are not generally integrable over affine Lie algebras, but integrable with respect to a subroot system of the root system attached to a given affine Lie algebra.
    [Show full text]
  • Affine Lie Algebras
    Groups and Algebras for Theoretical Physics Masters course in theoretical physics at The University of Bern Spring Term 2016 R SUSANNE REFFERT Contents Contents 1 Complex semi-simple Lie Algebras 2 1.1 Basic notions . .2 1.2 The Cartan–Weyl basis . .3 1.3 The Killing form . .5 1.4 Weights . .6 1.5 Simple roots and the Cartan matrix . .7 1.6 The Chevalley basis . .9 1.7 Dynkin diagrams . .9 1.8 The Cartan classification for finite-dimensional simple Lie algebras . 10 1.9 Fundamental weights and Dynkin labels . 12 1.10 The Weyl group . 14 1.11 Normalization convention . 16 1.12 Examples: rank 2 root systems and their symmetries . 17 1.13 Visualizing the root system of higher rank simple Lie algebras . 19 1.14 Lattices . 19 1.15 Highest weight representations . 22 1.16 Conjugate representations . 26 1.17 Remark about real Lie algebras . 27 1.18 Characteristic numbers of simple Lie algebras . 27 1.19 Relevance for theoretical physics . 27 2 Generalizations and extensions: Affine Lie algebras 30 2.1 From simple to affine Lie algebras . 30 2.2 The Killing form . 32 2.3 Simple roots, the Cartan matrix and Dynkin diagrams . 34 2.4 Classification of the affine Lie algebras . 35 2.5 A remark on twisted affine Lie algebras . 38 2.6 The Chevalley basis . 38 2.7 Fundamental weights . 39 2.8 The affine Weyl group . 41 2.9 Outer automorphisms . 45 2.10 Visualizing the root systems of affine Lie algebras . 47 2.11 Highest weight representations . 50 3 Advanced topics: Beyond affine Lie algebras 55 3.1 The Virasoro algebra .
    [Show full text]
  • Affine Kac-Moody Algebras, Integrable Systems and Their Deformations
    AFFINE KAC-MOODY ALGEBRAS, INTEGRABLE SYSTEMS AND THEIR DEFORMATIONS EDWARD FRENKEL Representation theory of affine Kac-Moody algebras at the critical level contains many intricate structures, in particular, the hamiltonian structures of the KdV and modified KdV hierarchies and the Miura transformation between them. In this talk I will describe these structures and their deformations which will lead us to the deformed Virasoro and W–algebras and the integrable hierarchies associated to them. I will also discuss briefly the relation of these matters to the geometric Langlands correspondence. It is a great honor for me to give this talk as the first recipient of the Hermann Weyl Prize. Weyl was a pioneer of applications of symmetry in quantum physics, a scientist who truly appreciated the beauty of mathematics. He once said: My work has always tried to unite the true with the beautiful and when I had to choose one or the other, I usually chose the beautiful. 1. Affine Kac-Moody algebras Let g be a simple Lie algebra over C. Fix an invariant inner product κ on g and let bgκ denote the one-dimensional central extension of g ⊗ C((t)) with the commutation relations (1) [A ⊗ f(t),B ⊗ g(t)] = [A, B] ⊗ f(t)g(t) − (κ(A, B) Res fdg)K, where K is the central element. The Lie algebra bgκ is the affine Kac-Moody algebra associated to g. A representation of bgκ on a complex vector space V is called smooth if for any vector N v ∈ V there exists N ∈ Z+ such that g⊗t C[[t]]·v = 0.
    [Show full text]
  • Lecture 20: Wess-Zumino-Witten CFT and Fusion Algebras
    Review: Affine Kac-Moody Lie Algebras WZW Theories Lecture 20: Wess-Zumino-Witten CFT and Fusion Algebras Daniel Bump January 1, 2020 Review: Affine Kac-Moody Lie Algebras WZW Theories Overview We have so far emphasized the BPZ minimal models, in which c takes particular values < 1. These are special in that the Hilbert space H decomposes into a finite number of Vir ⊕ Vir modules. For c > 1, in order to obtain such a finite decomposition it is necessary to enlarge Vir. If this can be done, we speak of a rational CFT: see Lecture 19 for a more careful definition of this concept, due to Moore and Seiberg. One natural such extension enlarges Vir to include an affine Lie algebra. This leads to the Wess-Zumino-Witten (WZW) models. A further generalization of this theory, the coset models of Goddard-Kent-Olive ([GKO]) contains both the minimal models and the WZW models. Review: Affine Kac-Moody Lie Algebras WZW Theories Lecture 6 review: A note on notation References: [DMS] Chapter 14 Kac, Infinite-Dimensional Lie algebras, Chapters 6,7,12,13 Kac and Peterson, Adv. in Math. 53 (1984). We begin by reviewing the construction of affine Lie algebras from Lecture 6, changing the notation slightly. Let g be a finite-dimensional semisimple complex Lie algebra with Cartan subalgebra h. We will define the untwisted affine Lie algebra bg by first making a central extension, then adjoining a derivation d. Finally we will make a semidirect product of bg with Vir in which d is identified with -L0. Review: Affine Kac-Moody Lie Algebras WZW Theories Review: central extensions Let g be a Lie algebra and a an abelian Lie algebra.
    [Show full text]
  • Toroidal Fact.Pdf
    TOROIDAL PREFACTORIZATION ALGEBRAS ASSOCIATED TO HOLOMORPHIC FIBRATIONS AND A RELATIONSHIP TO VERTEX ALGEBRAS MATT SZCZESNY, JACKSON WALTERS, AND BRIAN WILLIAMS ABSTRACT. Let X be a complex manifold, π : E → X a locally trivial holomorphic fi- bration with fiber F, and (g, h•, •i) a Lie algebra with an invariant symmetric form. We associate to this data a holomorphic prefactorization algebra Fg,π on X in the formalism of Costello-Gwilliam. When X = C, g is simple, and F is a smooth affine variety, we extract from Fg,π a vertex algebra which is a vacuum module for the universal central extension of the Lie algebra g ⊗ H0(F, O)[z, z−1]. As a special case, when F is an algebraic torus (C∗)n, we obtain a vertex algebra naturally associated to an (n + 1)–toroidal algebra, generalizing the affine vacuum module. 1. Introduction 2 2. Lie algebras and vertex algebras 6 3. (Pre)factorization algebras and examples 14 4. Factorization algebras from holomorphic fibrations 21 5. X = C and vertex algebras 27 References 39 arXiv:1904.03176v1 [math.QA] 5 Apr 2019 1. INTRODUCTION Affine Kac-Moody algebras play an important role in many areas of mathematics and physics, specifically representation theory and conformal field theory. The affine Kac- Moody algebra g associated to a complex Lie algebra g is a central extension of the loop algebra Lg = g[z, z−1]. Toroidal algebras provide a generalization of affine Kac-Moody algebras to includeb “multi-dimensional” loops. The (n + 1)-toroidal algebra is defined as a central extension of the Lie algebra n+1 ± ± L g = g[z, z1 ..., zn ].
    [Show full text]
  • Lecture 6: Affine Lie Algebras
    Kac-Moody Lie Algebras (Untwisted) Affine Kac-Moody Lie Algebras Representations and modular forms Sage methods Lecture 6: Affine Lie Algebras Daniel Bump January 1, 2020 Kac-Moody Lie Algebras (Untwisted) Affine Kac-Moody Lie Algebras Representations and modular forms Sage methods References Kac, Infinite-Dimensional Lie Algebras DMS, Chapter 14 Sage has considerable algorithms for computing with affine Lie algebras, including methods computing with highest weight integrable representations, which are the most important class. Below are links to the Thematic Tutorial, Lie Methods and Related Combinatorics in Sage by Bump, Salisbury and Schilling. Note that this includes a section on affine root systems that is of interest independent of the Sage material. Affine Root System Basics (Web link to Sage documentation) Integrable Highest Weight Representations of Affine Lie algebras Kac-Moody Lie Algebras (Untwisted) Affine Kac-Moody Lie Algebras Representations and modular forms Sage methods Kac-Moody Lie Algebras Kac-Moody Lie algebras are generalizations of finite-dimensional simple Lie algebras. They include finite-dimensional simple Lie algebras as special cases but are usually infinite-dimensional. Many concepts and results from the representation theory of finite-dimensional Lie groups and Lie algebras extend to Kac-Moody Lie algebras. This includes the root system, Weyl group, weight lattice, the parametrization of important representations (the integrable highest weight ones) by dominant weights and the Weyl character formula for these representations. Among Kac-Moody Lie algebras, affine Lie algebras are an important infinite-dimensional class. Each affine Lie algebra g is related to a finite-dimensional Lie algebra g0. We will only consider untwisted affine Lie algebras.
    [Show full text]