DOCSLIB.ORG

The Segal–Sugawara Construction

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Segal–Sugawara Construction The Segal–Sugawara construction Peter J. Haine January 5, 2020 Contents 1 Introduction 1 1.1 Reminders from previous talks ...................... 1 1.2 Talk overview ................................ 2 2 Loop algebras & Kac–Moody algebras 2 2.1 Loop algebras ................................ 3 2.2 Recollection on bilinear forms & semisimplicity ............. 3 2.3 Kac–Moody algebras ............................ 4 3 The Segal–Sugawara construction 5 3.1 Motivating case: the Heisenberg algebra ................. 6 3.2 The Casimir element ............................ 8 3.3 The general case .............................. 9 References 11 1 Introduction 1.1 Reminders from previous talks 1.1.1 Definition. The (complex) Witt algebra is the complex Lie algebra Witt of poly- 1 푖푚휃 d nomial vector fields on S . Explicitly, Witt has generators 퐿푚 ≔ 푖푒 d휃 for 푚 ∈ 퐙 with Lie bracket [퐿푚, 퐿푛] ≔ (푚 − 푛)퐿푚+푛 for all 푚, 푛 ∈ 퐙. 1.1.2. Ignoring regularity issues, the Witt algebra is the complexification of the Lie al- gebra of the group Diff +(S1) of orientation-preserving diffeomorphisms of the circle. 2 1.1.3. We have that HLie(Witt; 퐂) ≅ 퐂, so there is a 1-dimensional space of central extensions of the Witt algebra. 1 1.1.4 Definition. The (complex) Virasoro algebra Vir is the central extension 1 퐂chg Vir Witt 1 of Witt with generators 퐿푚 for 푚 ∈ 퐙 and a central element chg, and nontrivial Lie bracket given by 푚3 − 푚 [퐿 , 퐿 ] ≔ (푚 − 푛)퐿 + 훿 chg 푚 푛 푚+푛 푚,−푛 12 for all 푚, 푛 ∈ 퐙. We call the central element chg ∈ Vir the central charge. 1.1.5. Again, ignoring regularity issues, the Virasoro algebra is the complexification of the Lie algebra of the Virasoro group Diff̃ +(S1). 1.2 Talk overview 1.2.1. Let 퐺 be a simply-connected, simple, compact Lie group with Lie algebra 픤. Last time we looked at central extensions 1 S1 L̃퐺 L퐺 1 of the loop group L퐺 ≔ C∞(S1, 퐺). The group Diff +(S1) of orientation-preserving diffeomorphisms of the circle acts on L퐺 by precomposition. So we might expect an action of the Virasoro group Diff̃ +(S1) on L̃퐺. Last time we saw that even though there is not an action of Diff̃ +(S1) on L̃퐺, roughly, the Virasoro group acts on any positive energy representation of L̃퐺. However, the Virasoro action on positive energy representation of L̃퐺 is very inexplicit, and we can only guarantee the existence of the Virasoro action up to ‘essential equivalence’, which is not actually an equivalence relation. In particular, the Pressley–Segal Theorem [8, Theorem 13.4.3] does not explicitly explain how the central circle S1 ⊂ L̃퐺 acts. 1.2.2 Goal. The goal of this talk is to explain the Lie algebra version of the Pressley– Segal Theorem, which gives an explicit representation of the Virasoro algebra on any ‘positive energy’ representation of the Kac–Moody algebra L̃픤 associated to a simple Lie algebra 픤 (over the complex numbers). We’ll be able to do this by writing down explicit universal formulas for ‘elements’ of the universal enveloping algebra U(L̃픤) that satisfy the Virasoro relations. The catch is that these universal formulas involve infinite sums, so they do not actually make sense as elements of U(L̃픤), but they do make sense whenever we act on a representation where only finitely many of the terms don’t act by zero (this is the ‘positive energy’ condition). 2 Loop algebras & Kac–Moody algebras The first thing we need to explain in order to state the Segal–Sugawara construction is what the Kac–Moody algebra L̃픤 is. As the notation suggests, L̃픤 is the Lie algebra analog of the central extension L̃퐺 of the loop group L퐺 (with suitable finiteness hypotheses). Before talking about Kac–Moody algebras, we need to talk about loop algebras. 2 2.1 Loop algebras 2.1.1 Recollection. Let 픤 be a Lie algebra over a ring 푅, and let 푆 be an 푅-algebra. The basechange 픤 ⊗푅 푆 of 픤 to 푆 is the Lie algebra over 푆 with underlying 푆-module the basechange 픤 ⊗푅 푆 of the underlying 푅-module of 픤 to 푆 with Lie bracket extended from pure tensors from the formula [푋 ⊗ 푠 , 푋 ⊗ 푠 ] ≔ [푋 , 푋 ] ⊗ 푠 푠 . 1 1 2 2 픤⊗푅푆 1 2 픤 1 2 2.1.2 Definition. Let 픤 be a complex Lie algebra. The loop algebra L픤 of 픤 is the Lie algebra ±1 L픤 ≔ 픤 ⊗퐂 퐂[푡 ] , regarded as a Lie algebra over 퐂 (rather than 퐂[푡±1]). 2.1.3 Notation. Let 픤 be a complex Lie algebra, 푋 ∈ 픤, and 푚 an integer. We write 푋⟨푚⟩ ≔ 푋 ⊗ 푡푚 ∈ L픤 . 2.1.4. If {푢푖}푖∈퐼 is a Lie algebra basis for 픤, then {푢푖⟨푚⟩}(푖,푚)∈퐼×퐙 is a basis for L픤. 2.1.5 Remark. The loop algebra functor L∶ Lie퐂 → Lie퐂 preserves finite products. 2.1.6 Recollection. A finite dimensional Lie algebra 픤 is simple if 픤 is not abelian and the only ideals of 픤 are 픤 and 0. 2.1.7 Theorem (Garland [3, §§1 & 2]). If 픤 is a simple Lie algebra over 퐂, then 2 HLie(L픤; 퐂) ≅ 퐂 . In particular, if 픤 is simple there is a 1-dimensional space of central extensions of L픤. 2.2 Recollection on bilinear forms & semisimplicity 2.2.1 Notation. Let 픤 be a complex Lie algebra. We write ad∶ 픤 → End퐂(픤) for the adjoint representation, defined by ad(푋) ≔ [푋, −] . 2.2.2 Example. A Lie algebra 픤 is abelian if and only if the adjoint representation of 픤 is trivial. 3 2.2.3 Recollection (Killing form). Let 픤 be a finite dimensional Lie algebra. The Killing form on 픤 is the bilinear form K픤 ∶ 픤 × 픤 → 퐂 (푋, 푌) ↦ tr(ad(푋) ∘ ad(푌)) . The Killing form is symmetric and invariant in the sense that K픤([푋, 푌], 푍) = K픤(푋, [푌, 푍]) for all 푋, 푌, 푍 ∈ 픤. 2.2.4 Example. If 픤 is a simple Lie algebra, then every invariant symmetric bilinear form on 픤 is a 퐂-multiple of the Killing form K픤. See [2] for a nice exposition of this fact. 2.2.5 Example. Let 픞 be a finite dimensional abelian Lie algebra over 퐂. Then since the adjoint representation of 픞 is trivial, the Killing form of 픞 is identically zero. Also note that every bilinear form on the underlying vector space of 픞 is an invariant bilinear form on 픞. 2.2.6 Proposition ([9, Chapter ii, Theorems 2 & 4]). Let 픤 be a finite dimensional com- plex Lie algebra. The following conditions are equivalent: (2.2.6.1) The center of 픤 is trivial. (2.2.6.2) The only abelian ideal in 픤 is 0. (2.2.6.3) The Lie algebra 픤 is isomorphic to a product of simple Lie algebras. (2.2.6.4) Cartan–Killing criterion: the Killing form of 픤 is nondegenerate. If the equivalent conditions (2.2.6.2)–(2.2.6.4) are satisfied, we say that 픤 is semisimple. 2.3 Kac–Moody algebras Now we define the Lie algebra analogue of the central extension L̃퐺 of the loop group L퐺. In addition to the Lie algebra 픤, we need to input an invariant symmetric bilinear form on 픤; the Killing form provides a canonical choice. 2.3.1 Definition. Let 픤 be a Lie algebra over 퐂 with invariant symmetric bilinear form 퐵∶ 픤 × 픤 → 퐂. The Kac–Moody algebra of 픤 with respect to the form 퐵 is the central extension ̃ 1 퐂푐 L퐵픤 L픤 1 with central element 푐 and with Lie bracket extended from the relation [푋⟨푚⟩, 푌⟨푛⟩]̃ ≔ [푋⟨푚⟩, 푌⟨푛⟩] + 훿 푚퐵(푋, 푌)푐 L퐵픤 L픤 푚,−푛 = [푋, 푌]픤⟨푚 + 푛⟩ + 훿푚,−푛푚퐵(푋, 푌)푐 for all 푋, 푌 ∈ 픤. 4 2.3.2. If {푢푖}푖∈퐼 is a Lie algebra basis for 픤, then {푢푖⟨푚⟩}(푖,푚)∈퐼×퐙 ∪ {푐} ̃ is a basis for L퐵픤. 2.3.3 Remark. The Kac–Moody algebra L̃픤 is usually denoted by 픤̂ and is also known as the affine Lie algebra of 픤. 2.3.4 Remark. Let 픤1 and 픤2 be complex Lie algebras equipped with invariant symmet- ric bilinear forms 퐵1 ∶ 픤1 × 픤1 → 퐂 and 퐵2 ∶ 픤2 × 픤2 → 퐂 . Write 퐵 for the bilinear form on the product Lie algebra 픤1 × 픤2 defined by 퐵((푥1, 푥2), (푦1, 푦2)) ≔ 퐵1(푥1, 푦1) + 퐵(푥2, 푦2) . Then we have a canonical isomorphism L̃ (픤 × 픤 ) ≅ L̃ 픤 × L̃ 픤 . 퐵 1 2 퐵1 1 퐵2 2 3 The Segal–Sugawara construction We now have enough of of the background on Lie algebras to give a vague statement of the Segal–Sugawara construction. 3.0.1 Definition. Let 픤 be a Lie algebra over 퐂 and 퐵 an invariant symmetric bilinear ̃ form on 픤. A representation 휌∶ L퐵픤 → End퐂(푉) has positive energy if for all 푣 ∈ 푉 and 푋 ∈ 픤 there exists an integer 푚 > 0 such that 휌(푋⟨푚⟩)푣 = 0 . 3.0.2 Remark. In the theory of Kac–Moody algebras, positive energy representations are more often called admissible. We have chosen the term ‘positive energy’ to align with the loop group terminology. 3.0.3 Theorem (Segal–Sugawara construction, vague formulation). Let 픤 be an abelian or simple Lie algebra over 퐂 and let 퐵∶ 픤 × 픤 → 퐂 be a nondegenerate invariant symmetric bilinear form on 픤.
Load more

Recommended publications
  • Invariant Differential Operators 1. Derivatives of Group Actions
    (October 28, 2010) Invariant differential operators Paul Garrett [email protected] http:=/www.math.umn.edu/~garrett/ • Derivatives of group actions: Lie algebras • Laplacians and Casimir operators • Descending to G=K • Example computation: SL2(R) • Enveloping algebras and adjoint functors • Appendix: brackets • Appendix: proof of Poincar´e-Birkhoff-Witt We want an intrinsic approach to existence of differential operators invariant under group actions. n n The translation-invariant operators @=@xi on R , and the rotation-invariant Laplacian on R are deceptively- easily proven invariant, as these examples provide few clues about more complicated situations. For example, we expect rotation-invariant Laplacians (second-order operators) on spheres, and we do not want to write a formula in generalized spherical coordinates and verify invariance computationally. Nor do we want to be constrained to imbedding spheres in Euclidean spaces and using the ambient geometry, even though this succeeds for spheres themselves. Another basic example is the operator @2 @2 y2 + @x2 @y2 on the complex upper half-plane H, provably invariant under the linear fractional action of SL2(R), but it is oppressive to verify this directly. Worse, the goal is not merely to verify an expression presented as a deus ex machina, but, rather to systematically generate suitable expressions. An important part of this intention is understanding reasons for the existence of invariant operators, and expressions in coordinates should be a foregone conclusion. (No prior acquaintance with Lie groups or Lie algebras is assumed.) 1. Derivatives of group actions: Lie algebras For example, as usual let > SOn(R) = fk 2 GLn(R): k k = 1n; det k = 1g act on functions f on the sphere Sn−1 ⊂ Rn, by (k · f)(m) = f(mk) with m × k ! mk being right matrix multiplication of the row vector m 2 Rn.
    [Show full text]
  • Group of Diffeomorphisms of the Unite Circle As a Principle U(1)-Bundle
    Group of diffeomorphisms of the unite circle as a principle U(1)-bundle Irina Markina, University of Bergen, Norway Summer school Analysis - with Applications to Mathematical Physics Gottingen¨ August 29 - September 2, 2011 Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 1/49 String Minkowski space-time moving in time woldsheet string string Worldsheet as an imbedding of a cylinder C into the Minkowski space-time with the induced metric g. 2 Nambu-Gotô action SNG = −T dσ | det gαβ| ZC p T is the string tension. Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 2/49 Polyakov action Change to the imbedding independent metric h on worldsheet. 2 αβ Polyakov action SP = −T dσ | det hαβ|h ∂αx∂βx, ZC p 0 1 δSP α, β = 0, 1, x = x(σ ,σ ). Motion satisfied δhαβ = 0. −1 δSP Energy-momentum tensor Tαβ = αβ T | det hαβ| δh p Tαβ = 0 and SP = SNG, whereas in general SP ≥ SNG Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 3/49 Gauges The metric h has 3 degrees of freedom - gauges that one need to fix. • Global Poincaré symmetries - invariance under Poincare group in Minkowski space • Local invariance under the reparametrizaition by 2D-diffeomorphisms dσ˜2 | det h˜| = dσ2 | det h| • Local Weyl rescaling p p α β ρ(σ0,σ1) α β hαβdσ dσ → e hαβdσ dσ ρ(σ0,σ1) 2 αβ µ ν hαβ = e ηαβ, SP = −T dσ η ηµν∂αx ∂βx ZC Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p.
    [Show full text]
  • 7 International Centre for Theoretical Physics
    IC/93/173 -n I 'r.2_. \ INTERNATIONAL CENTRE FOR 7 THEORETICAL PHYSICS IIK;IIKK SPIN EXTENSIONS OF THE VIRASORO ALCE1IRA, AREA-PRESERVING AL(;EURAS AND THE MATRIX ALGEBRA M. Zakkuri INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION MIRAMARE-TRIESTE IC/93/173 Infinite dimensional algebras have played a central role in the development of string theories and two dimensional conformal field theory. These symmetries have been shown International Atomic Energy Agency to be related to the Virasoro algebra, its supersymmetric extensions and the Kac-Moody and one. Together, they are generated by conformal spin ,s currents with s <2. United Nations Educational Scientific and Cultural Organization However, some years ago, Zamolodchikov discovered a new extension, W3, involving besides the usual spin 2 conformal current, a conformal spin 3 current [1]. The obtained INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS structure is called non-linear Lie algebra. Recently, much interest has been made in the understanding of these higher conformal spin extensions of the Virasoro algebra [2.3,4}. In this paper, we study these algebraic structures using the infinite matrix representa- tion. Actually the present work may be viewed as a generalization of the infinite matrix HIGHER SPIN EXTENSIONS OF THE VIRASORO ALGEBRA, realization of the Virasoro algebra obtained by Kac et al. [5|. Some results have been AREA-PRESERVING ALGEBRAS AND THE MATRIX ALGEBRA described in our first papers [6,7]. However, we give here some precisions on the generality of above results. Moreover, we extend the above construction to include the algebra of area preserving diffeomorphism on the 2-dimensional torus [8], Actually, this result is k 1 expected to be generalizable for higher dimensions [10], namely for the torus T , k > 1.
    [Show full text]
  • Minimal Dark Matter Models with Radiative Neutrino Masses
    Master’s thesis Minimal dark matter models with radiative neutrino masses From Lagrangians to observables Simon May 1st June 2018 Advisors: Prof. Dr. Michael Klasen, Dr. Karol Kovařík Institut für Theoretische Physik Westfälische Wilhelms-Universität Münster Contents 1. Introduction 5 2. Experimental and observational evidence 7 2.1. Dark matter . 7 2.2. Neutrino oscillations . 14 3. Gauge theories and the Standard Model of particle physics 19 3.1. Mathematical background . 19 3.1.1. Group and representation theory . 19 3.1.2. Tensors . 27 3.2. Representations of the Lorentz group . 31 3.2.1. Scalars: The (0, 0) representation . 35 1 1 3.2.2. Weyl spinors: The ( 2 , 0) and (0, 2 ) representations . 36 1 1 3.2.3. Dirac spinors: The ( 2 , 0) ⊕ (0, 2 ) representation . 38 3.2.4. Majorana spinors . 40 1 1 3.2.5. Lorentz vectors: The ( 2 , 2 ) representation . 41 3.2.6. Field representations . 42 3.3. Two-component Weyl spinor formalism and van der Waerden notation 44 3.3.1. Definition . 44 3.3.2. Correspondence to the Dirac bispinor formalism . 47 3.4. The Standard Model . 49 3.4.1. Definition of the theory . 52 3.4.2. The Lagrangian . 54 4. Component notation for representations of SU(2) 57 4.1. SU(2) doublets . 59 4.1.1. Basic conventions and transformation of doublets . 60 4.1.2. Dual doublets and scalar product . 61 4.1.3. Transformation of dual doublets . 62 4.1.4. Adjoint doublets . 63 4.1.5. The question of transpose and conjugate doublets .
    [Show full text]
  • Strings, Symmetries and Representations
    Strings, symmetries and representations C. Schweigert, I. Runkel LPTHE, Universite´ Paris VI, 4 place Jussieu, F – 75 252 Paris Cedex 05 J. Fuchs Institutionen for¨ fysik, Universitetsgatan 5, S – 651 88 Karlstad Abstract. Several aspects of symmetries in string theory are reviewed. We discuss the roleˆ of symmetries both of the string world sheet and of the target space. We also show how to obtain string scattering amplitudes with the help of structures familiar from the representation theory of quantum groups. 1. Strings and conformal field theory Symmetries of various types, and consequently representation theoretic tools, play an important roleˆ in string theory and conformal field theory. The present contribution aims at reviewing some of their aspects, the choice of topics being influenced by our personal taste. After a brief overview of string theory and conformal field theory, we first discuss orbifolds and duality symmetries. We then turn to D-branes and theories of open strings, which we investigate in the final section using Frobenius algebras in representation categories. A minimalistic point of view on (perturbative) string theory is to regard it as a pertur- bative quantization of a field theory, with the perturbation expansion being organized not in terms of graphs, i.e. one-dimensional objects, but rather in terms of surfaces. What makes this perturbation expansion particularly interesting is that it even covers the quantization of theories that include a gravitational sector. The configuration space of a classical string is given by the embeddings of its two- dimensional world sheet Σ, with local coordinates τ and σ, into a target space M with coordinates Xµ.
    [Show full text]
  • Lecture 7 - Complete Reducibility of Representations of Semisimple Algebras
    Lecture 7 - Complete Reducibility of Representations of Semisimple Algebras September 27, 2012 1 New modules from old A few preliminaries are necessary before jumping into the representation theory of semisim- ple algebras. First a word on creating new g-modules from old. Any Lie algebra g has an action on a 1-dimensional vector space (or F itself), given by the trivial action. Second, any action on spaces V and W can be extended to an action on V ⊗ W by forcing the Leibnitz rule: for any basis vector v ⊗ w 2 V ⊗ W we define x:(v ⊗ w) = x:v ⊗ w + v ⊗ x:w (1) One easily checks that x:y:(v ⊗ w) − y:x:(v ⊗ w) = [x; y]:(v ⊗ w). Assuming g has an action on V , it has an action on its dual V ∗ (recall V ∗ is the vector space of linear functionals V ! F), given by (v:f)(x) = −f(x:v) (2) for any functional f : V ! F in V ∗. This is in fact a version of the \forcing the Leibnitz rule." That is, recalling that we defined x:(f(v)) = 0, we define x:f 2 V ∗ implicitly by x: (f(v)) = (x:f)(v) + f(x:v): (3) For any vector spaces V , W , we have an isomorphism Hom(V; W ) ≈ V ∗ ⊗ W; (4) so Hom(V; W ) is a g-module whenever V and W are. This can be defined using the above rules for duals and tensor products, or, equivalently, by again forcing the Leibnitz rule: for F 2 Hom(V; W ), we define x:F 2 Hom(V; W ) implicitly by x:(F (v)) = (x:F )(v) + F (x:v): (5) 1 2 Schur's lemma and Casimir elements Theorem 2.1 (Schur's Lemma) If g has an irreducible representation on gl(V ) and if f 2 End(V ) commutes with every x 2 g, then f is multiplication by a constant.
    [Show full text]
  • An Identity Crisis for the Casimir Operator
    An Identity Crisis for the Casimir Operator Thomas R. Love Department of Mathematics and Department of Physics California State University, Dominguez Hills Carson, CA, 90747 [email protected] April 16, 2006 Abstract 2 P ij The Casimir operator of a Lie algebra L is C = g XiXj and the action of the Casimir operator is usually taken to be C2Y = P ij g XiXjY , with ordinary matrix multiplication. With this defini- tion, the eigenvalues of the Casimir operator depend upon the repre- sentation showing that the action of the Casimir operator is not well defined. We prove that the action of the Casimir operator should 2 P ij be interpreted as C Y = g [Xi, [Xj,Y ]]. This intrinsic definition does not depend upon the representation. Similar results hold for the higher order Casimir operators. We construct higher order Casimir operators which do not exist in the standard theory including a new type of Casimir operator which defines a complex structure and third order intrinsic Casimir operators for so(3) and so(3, 1). These opera- tors are not multiples of the identity. The standard theory of Casimir operators predicts neither the correct operators nor the correct num- ber of invariant operators. The quantum theory of angular momentum and spin, Wigner’s classification of elementary particles as represen- tations of the Poincar´eGroup and quark theory are based on faulty mathematics. The “no-go theorems” are shown to be invalid. PACS 02.20S 1 1 Introduction Lie groups and Lie algebras play a fundamental role in classical mechan- ics, electrodynamics, quantum mechanics, relativity, and elementary particle physics.
    [Show full text]
  • Geometric Approach to Kac–Moody and Virasoro Algebras
    Journal of Geometry and Physics 62 (2012) 1984–1997 Contents lists available at SciVerse ScienceDirect Journal of Geometry and Physics journal homepage: www.elsevier.com/locate/jgp Geometric approach to Kac–Moody and Virasoro algebrasI E. Gómez González, D. Hernández Serrano ∗, J.M. Muñoz Porras ∗, F.J. Plaza Martín ∗ Departamento de Matemáticas, Universidad de Salamanca, Plaza de la Merced 1-4, 37008 Salamanca, Spain IUFFYM, Instituto Universitario de Física Fundamental y Matemáticas, Universidad de Salamanca, Plaza de la Merced s/n, 37008 Salamanca, Spain article info a b s t r a c t Article history: In this paper we show the existence of a group acting infinitesimally transitively on the Received 17 March 2011 moduli space of pointed-curves and vector bundles (with formal trivialization data) and Received in revised form 12 January 2012 whose Lie algebra is an algebra of differential operators. The central extension of this Accepted 1 May 2012 Lie algebra induced by the determinant bundle on the Sato Grassmannian is precisely a Available online 8 May 2012 semidirect product of a Kac–Moody algebra and the Virasoro algebra. As an application of this geometric approach, we give a local Mumford-type formula in terms of the cocycle MSC: associated with this central extension. Finally, using the original Mumford formula we primary 14H60 14D21 show that this local formula is an infinitesimal version of a general relation in the secondary 22E65 Picard group of the moduli of vector bundles on a family of curves (without any formal 22E67 trivialization). 22E47 ' 2012 Elsevier B.V. All rights reserved.
    [Show full text]
  • A Quantum Group in the Yang-Baxter Algebra
    A Quantum Group in the Yang-Baxter Algebra Alexandros Aerakis December 8, 2013 Abstract In these notes we mainly present theory on abstract algebra and how it emerges in the Yang-Baxter equation. First we review what an associative algebra is and then introduce further structures such as coalgebra, bialgebra and Hopf algebra. Then we discuss the con- struction of an universal enveloping algebra and how by deforming U[sl(2)] we obtain the quantum group Uq[sl(2)]. Finally, we discover that the latter is actually the Braid limit of the Yang-Baxter alge- bra and we use our algebraic knowledge to obtain the elements of its representation. Contents 1 Algebras 1 1.1 Bialgebra and Hopf algebra . 1 1.2 Universal enveloping algebra . 4 1.3 The quantum Uq[sl(2)] . 7 2 The Yang-Baxter algebra and the quantum Uq[sl(2)] 10 1 Algebras 1.1 Bialgebra and Hopf algebra To start with, the most familiar algebraic structure is surely that of an asso- ciative algebra. Definition 1 An associative algebra A over a field C is a linear vector space V equipped with 1 • Multiplication m : A ⊗ A ! A which is { bilinear { associative m(1 ⊗ m) = m(m ⊗ 1) that pictorially corresponds to the commutative diagram A ⊗ A ⊗ A −−−!1⊗m A ⊗ A ? ? ? ?m ym⊗1 y A ⊗ A −−−!m A • Unit η : C !A which satisfies the axiom m(η ⊗ 1) = m(1 ⊗ η) = 1: =∼ =∼ A ⊗ C A C ⊗ A ^ m 1⊗η > η⊗1 < A ⊗ A By reversing the arrows we get another structure which is called coalgebra.
    [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]
  • Arxiv:2009.00393V2 [Hep-Th] 26 Jan 2021 Supersymmetric Localisation and the Conformal Bootstrap
    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 007, 38 pages Harmonic Analysis in d-Dimensional Superconformal Field Theory Ilija BURIC´ DESY, Notkestraße 85, D-22607 Hamburg, Germany E-mail: [email protected] Received September 02, 2020, in final form January 15, 2021; Published online January 25, 2021 https://doi.org/10.3842/SIGMA.2021.007 Abstract. Superconformal blocks and crossing symmetry equations are among central in- gredients in any superconformal field theory. We review the approach to these objects rooted in harmonic analysis on the superconformal group that was put forward in [J. High Energy Phys. 2020 (2020), no. 1, 159, 40 pages, arXiv:1904.04852] and [J. High Energy Phys. 2020 (2020), no. 10, 147, 44 pages, arXiv:2005.13547]. After lifting conformal four-point functions to functions on the superconformal group, we explain how to obtain compact expressions for crossing constraints and Casimir equations. The later allow to write superconformal blocks as finite sums of spinning bosonic blocks. Key words: conformal blocks; crossing equations; Calogero{Sutherland models 2020 Mathematics Subject Classification: 81R05; 81R12 1 Introduction Conformal field theories (CFTs) are a class of quantum field theories that are interesting for several reasons. On the one hand, they describe the critical behaviour of statistical mechanics systems such as the Ising model. Indeed, the identification of two-dimensional statistical systems with CFT minimal models, first suggested in [2], was a celebrated early achievement in the field. For similar reasons, conformal theories classify universality classes of quantum field theories in the Wilsonian renormalisation group paradigm. On the other hand, CFTs also play a role in the description of physical systems that do not posses scale invariance, through certain \dualities".
    [Show full text]
  • Quantum Mechanics 2 Tutorial 9: the Lorentz Group
    Quantum Mechanics 2 Tutorial 9: The Lorentz Group Yehonatan Viernik December 27, 2020 1 Remarks on Lie Theory and Representations 1.1 Casimir elements and the Cartan subalgebra Let us first give loose definitions, and then discuss their implications. A Cartan subalgebra for semi-simple1 Lie algebras is an abelian subalgebra with the nice property that op- erators in the adjoint representation of the Cartan can be diagonalized simultaneously. Often it is claimed to be the largest commuting subalgebra, but this is actually slightly wrong. A correct version of this statement is that the Cartan is the maximal subalgebra that is both commuting and consisting of simultaneously diagonalizable operators in the adjoint. Next, a Casimir element is something that is constructed from the algebra, but is not actually part of the algebra. Its main property is that it commutes with all the generators of the algebra. The most commonly used one is the quadratic Casimir, which is typically just the sum of squares of the generators of the algebra 2 2 2 2 (e.g. L = Lx + Ly + Lz is a quadratic Casimir of so(3)). We now need to say what we mean by \square" 2 and \commutes", because the Lie algebra itself doesn't have these notions. For example, Lx is meaningless in terms of elements of so(3). As a matrix, once a representation is specified, it of course gains meaning, because matrices are endowed with multiplication. But the Lie algebra only has the Lie bracket and linear combinations to work with. So instead, the Casimir is living inside a \larger" algebra called the universal enveloping algebra, where the Lie bracket is realized as an explicit commutator.
    [Show full text]