An Introduction to Affine Kac-Moody Algebras
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3D Gauge Theories, Symplectic Duality and Knot Homology I
3d Gauge Theories, Symplectic Duality and Knot Homology I Tudor Dimofte Notes by Qiaochu Yuan December 8, 2014 We want to study some 3d N = 4 supersymmetric QFTs. They won't be topological, but the supersymmetry will still help. We'll look in particular at boundary conditions and compactifications to 2d. Most of this material is relatively old by physics standards (known since the '90s). A motivation for discussing these theories here is that they seem closely related to 1. geometric representation theory, 2. geometric constructions of knot homologies, and 3. symplectic duality. This story gives rise to some interesting mathematical predictions. On Tuesday we'll talk about what symplectic n-ality should mean. On Wednesday we'll give a new construction of projectives in variants of category O. On Thursday we'll talk about category O and knot homologies, and a new approach to both via 2d Landau-Ginzburg models. 1 Geometric representation theory For starters, let G = SL2. The universal enveloping algebra of its Lie algebra is U(sl2) = hE; F; H j [E; F ] = H; [H; E] = 2E; [H; F ] = −2F i: (1) The center is generated by the Casimir operator 1 χ = EF + FE + H2: (2) 2 The full category of U(sl2)-modules is hard to work with. Bernstein, Gelfand, and Gelfand proposed that we work in a nice subcategory O of highest weight representations (on which E acts locally finitely or locally nilpotently, depending on the definition) decomposing as a sum M M = Mµ (3) µ of generalized eigenspaces with respect to the action of the generator of the Cartan L subalgebra H. -
Parabolic Category O for Classical Lie Superalgebras
Parabolic category O for classical Lie superalgebras Volodymyr Mazorchuk Abstract We compare properties of (the parabolic version of) the BGG category O for semi-simple Lie algebras with those for classical (not necessarily simple) Lie superalgebras. 1 Introduction Category O for semi-simple complex finite dimensional Lie algebras, introduced in [BGG], is a central object of study in the modern representation theory (see [Hu]) with many interesting connections to, in particular, combinatorics, alge- braic geometry and topology. This category has a natural counterpart in the super- world and this super version O˜ of category O was intensively studied (mostly for some particular simple classical Lie superalgebra) in the last decade, see e.g. [Br1, Br2, Go2, FM2, CLW, CMW] or the recent books [Mu, CW] for details. The aim of the present paper is to compare some basic but general properties of the category O in the non-super and super cases for a rather general classical super-setup. We mainly restrict to the properties for which the non-super and super cases can be connected using the usual restriction and induction functors and the biadjunction (up to parity change) between these two functors is our main tool. The paper also complements, extends and gives a more detailed exposition for some results which appeared in [AM, Section 7]. The original category O has a natural parabolic version which first appeared in [RC]. We start in Section 2 by setting up an elementary approach (using root system geometry) to the definition of a parabolic category O for classical finite dimensional Lie superalgebras. -
Geometric Representation Theory, Fall 2005
GEOMETRIC REPRESENTATION THEORY, FALL 2005 Some references 1) J. -P. Serre, Complex semi-simple Lie algebras. 2) T. Springer, Linear algebraic groups. 3) A course on D-modules by J. Bernstein, availiable at www.math.uchicago.edu/∼arinkin/langlands. 4) J. Dixmier, Enveloping algebras. 1. Basics of the category O 1.1. Refresher on semi-simple Lie algebras. In this course we will work with an alge- braically closed ground field of characteristic 0, which may as well be assumed equal to C Let g be a semi-simple Lie algebra. We will fix a Borel subalgebra b ⊂ g (also sometimes denoted b+) and an opposite Borel subalgebra b−. The intersection b+ ∩ b− is a Cartan subalgebra, denoted h. We will denote by n, and n− the unipotent radicals of b and b−, respectively. We have n = [b, b], and h ' b/n. (I.e., h is better to think of as a quotient of b, rather than a subalgebra.) The eigenvalues of h acting on n are by definition the positive roots of g; this set will be denoted by ∆+. We will denote by Q+ the sub-semigroup of h∗ equal to the positive span of ∆+ (i.e., Q+ is the set of eigenvalues of h under the adjoint action on U(n)). For λ, µ ∈ h∗ we shall say that λ ≥ µ if λ − µ ∈ Q+. We denote by P + the sub-semigroup of dominant integral weights, i.e., those λ that satisfy hλ, αˇi ∈ Z+ for all α ∈ ∆+. + For α ∈ ∆ , we will denote by nα the corresponding eigen-space. -
Arxiv:1907.06685V1 [Math.RT]
CATEGORY O FOR TAKIFF sl2 VOLODYMYR MAZORCHUK AND CHRISTOFFER SODERBERG¨ Abstract. We investigate various ways to define an analogue of BGG category O for the non-semi-simple Takiff extension of the Lie algebra sl2. We describe Gabriel quivers for blocks of these analogues of category O and prove extension fullness of one of them in the category of all modules. 1. Introduction and description of the results The celebrated BGG category O, introduced in [BGG], is originally associated to a triangular decomposition of a semi-simple finite dimensional complex Lie algebra. The definition of O is naturally generalized to all Lie algebras admitting some analogue of a triangular decomposition, see [MP]. These include, in particular, Kac-Moody algebras and Virasoro algebra. Category O has a number of spectacular properties and applications to various areas of mathematics, see for example [Hu] and references therein. The paper [DLMZ] took some first steps in trying to understand structure and prop- erties of an analogue of category O in the case of a non-reductive finite dimensional Lie algebra. The investigation in [DLMZ] focuses on category O for the so-called Schr¨odinger algebra, which is a central extension of the semi-direct product of sl2 with its natural 2-dimensional module. It turned out that, for the Schr¨odinger algebra, the behavior of blocks of category O with non-zero central charge is exactly the same as the behavior of blocks of category O for the algebra sl2. At the same, block with zero central charge turned out to be significantly more difficult. -
Quantizations of Regular Functions on Nilpotent Orbits 3
QUANTIZATIONS OF REGULAR FUNCTIONS ON NILPOTENT ORBITS IVAN LOSEV Abstract. We study the quantizations of the algebras of regular functions on nilpo- tent orbits. We show that such a quantization always exists and is unique if the orbit is birationally rigid. Further we show that, for special birationally rigid orbits, the quan- tization has integral central character in all cases but four (one orbit in E7 and three orbits in E8). We use this to complete the computation of Goldie ranks for primitive ideals with integral central character for all special nilpotent orbits but one (in E8). Our main ingredient are results on the geometry of normalizations of the closures of nilpotent orbits by Fu and Namikawa. 1. Introduction 1.1. Nilpotent orbits and their quantizations. Let G be a connected semisimple algebraic group over C and let g be its Lie algebra. Pick a nilpotent orbit O ⊂ g. This orbit is a symplectic algebraic variety with respect to the Kirillov-Kostant form. So the algebra C[O] of regular functions on O acquires a Poisson bracket. This algebra is also naturally graded and the Poisson bracket has degree −1. So one can ask about quantizations of O, i.e., filtered algebras A equipped with an isomorphism gr A −→∼ C[O] of graded Poisson algebras. We are actually interested in quantizations that have some additional structures mir- roring those of O. Namely, the group G acts on O and the inclusion O ֒→ g is a moment map for this action. We want the G-action on C[O] to lift to a filtration preserving action on A. -
Classifying the Representation Type of Infinitesimal Blocks of Category
Classifying the Representation Type of Infinitesimal Blocks of Category OS by Kenyon J. Platt (Under the Direction of Brian D. Boe) Abstract Let g be a simple Lie algebra over the field C of complex numbers, with root system Φ relative to a fixed maximal toral subalgebra h. Let S be a subset of the simple roots ∆ of Φ, which determines a standard parabolic subalgebra of g. Fix an integral weight ∗ µ µ ∈ h , with singular set J ⊆ ∆. We determine when an infinitesimal block O(g,S,J) := OS of parabolic category OS is nonzero using the theory of nilpotent orbits. We extend work of Futorny-Nakano-Pollack, Br¨ustle-K¨onig-Mazorchuk, and Boe-Nakano toward classifying the representation type of the nonzero infinitesimal blocks of category OS by considering arbitrary sets S and J, and observe a strong connection between the theory of nilpotent orbits and the representation type of the infinitesimal blocks. We classify certain infinitesimal blocks of category OS including all the semisimple infinitesimal blocks in type An, and all of the infinitesimal blocks for F4 and G2. Index words: Category O; Representation type; Verma modules Classifying the Representation Type of Infinitesimal Blocks of Category OS by Kenyon J. Platt B.A., Utah State University, 1999 M.S., Utah State University, 2001 M.A, The University of Georgia, 2006 A Thesis Submitted to the Graduate Faculty of The University of Georgia in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Athens, Georgia 2008 c 2008 Kenyon J. Platt All Rights Reserved Classifying the Representation Type of Infinitesimal Blocks of Category OS by Kenyon J. -
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). -
Introuduction to Representation Theory of Lie Algebras
Introduction to Representation Theory of Lie Algebras Tom Gannon May 2020 These are the notes for the summer 2020 mini course on the representation theory of Lie algebras. We'll first define Lie groups, and then discuss why the study of representations of simply connected Lie groups reduces to studying representations of their Lie algebras (obtained as the tangent spaces of the groups at the identity). We'll then discuss a very important class of Lie algebras, called semisimple Lie algebras, and we'll examine the repre- sentation theory of two of the most basic Lie algebras: sl2 and sl3. Using these examples, we will develop the vocabulary needed to classify representations of all semisimple Lie algebras! Please email me any typos you find in these notes! Thanks to Arun Debray, Joakim Færgeman, and Aaron Mazel-Gee for doing this. Also{thanks to Saad Slaoui and Max Riestenberg for agreeing to be teaching assistants for this course and for many, many helpful edits. Contents 1 From Lie Groups to Lie Algebras 2 1.1 Lie Groups and Their Representations . .2 1.2 Exercises . .4 1.3 Bonus Exercises . .4 2 Examples and Semisimple Lie Algebras 5 2.1 The Bracket Structure on Lie Algebras . .5 2.2 Ideals and Simplicity of Lie Algebras . .6 2.3 Exercises . .7 2.4 Bonus Exercises . .7 3 Representation Theory of sl2 8 3.1 Diagonalizability . .8 3.2 Classification of the Irreducible Representations of sl2 .............8 3.3 Bonus Exercises . 10 4 Representation Theory of sl3 11 4.1 The Generalization of Eigenvalues . -
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) .............. -
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. -
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). -
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.