DOCSLIB.ORG

Lie Groups and Lie Algebras June 28, 2006

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lie Groups and Lie Algebras June 28, 2006 Notes for Math 261A Lie groups and Lie algebras June 28, 2006 Contents Contents ......................... 1 How these notes came to be ................. 3 Lecture 1 ........................ 4 Lecture 2 ........................ 6 Tangent Lie algebras to Lie groups, 6 Lecture 3 ........................ 9 Lecture 4 ........................ 12 Lecture 5 ........................ 16 Simply Connected Lie Groups, 16 Lecture 6 - Hopf Algebras .................. 21 The universal enveloping algebra, 24 Lecture 7 ........................ 26 Universality of Ug, 26 Gradation in Ug, 27 Filtered spaces and algebras, 28 Lecture 8 - PBW theorem, Deformations of Associative Algebras . 31 Deformations of associative algebras, 32 Formal deformations of associative algebras, 34 Formal deformations of Lie algebras, 35 Lecture 9 ........................ 36 Lie algebra cohomology, 37 Lecture 10 ........................ 41 Lie algebra cohomology, 41 H2(g, g) and Deformations of Lie algebras, 42 Lecture 11 - Engel’s Theorem and Lie’s Theorem ......... 46 The radical, 51 Lecture 12 - Cartan Criterion, Whitehead and Weyl Theorems . 52 Invariant forms and the Killing form, 52 Lecture 13 - The root system of a semi-simple Lie algebra ..... 59 Irreducible finite dimensional representations of sl(2), 63 Lecture 14 - More on Root Systems .............. 65 Abstract Root systems, 66 The Weyl group, 68 Lecture 15 - Dynkin diagrams, Classification of root systems . 71 Isomorphisms of small dimension, 76 1 2 Lecture 16 - Serre’s Theorem................. 78 Lecture 17 - Constructions of Exceptional simple Lie Algebras . 82 Lecture 18 - Representations ................. 87 Representations, 87 Highest weights, 89 Lecture 19 - The Weyl character formula ............ 93 The character formula, 94 Lecture 20 ........................100 Compact Groups, 102 Lecture 21 ........................105 What does a general Lie group look like?, 105 Comparison of Lie groups and Lie algebras, 107 Finite groups and Lie groups, 108 Algebraic groups (over R) and Lie groups, 109 Lecture 22 - Clifford algebras and Spin groups ..........111 Clifford Algebras and Spin Groups, 112 Lecture 23 ........................117 Clifford Groups, 118 Lecture 24 ........................123 Spin representations of Spin and Pin groups, 123 Triality, 126 More about Orthogonal groups, 127 Lecture 25 - E8 ......................129 Lecture 26 ........................134 Construction of E8, 136 Lecture 27 ........................140 Construction of the Lie group of E8, 143 Real forms, 144 Lecture 28 ........................146 Working with simple Lie groups, 148 Every possible simple Lie group, 150 Lecture 29 ........................152 Lecture 30 - Irreducible unitary representations of SL2(R) . 159 Finite dimensional representations, 159 The Casimir operator, 162 Lecture 31 - Unitary representations of SL2(R) .. .. .. .165 Background about infinite-dimensional representations, 165 The group SL2(R), 166 Solutions to (some) Exercises.................171 References ........................176 Index ..........................178 3 How these notes came to be Among the Berkeley professors, there was once Alan Knutson, who would teach Math 261. But it happened that professor Knutson was on sabbati- cal elsewhere (I don’t know where for sure), and eventually went there (or somewhere else) for good. During this turbulent time, Maths 261AB were cancelled two years in a row. The last of these four semesters (Spring 2006), some graduate students gathered together and asked Nicolai Reshetikhin to teach them Lie theory in a giant reading course. When the dust settled, there were two other professors willing to help in the instruction of Math 261A, Vera Serganova and Richard Borcherds. Thus Tag Team 261A was born. After a few lectures, professor Reshetikhin suggested that the students write up the lecture notes for the benefit of future generations. The first four lectures were produced entirely by the “editors”. The remaining lectures were LATEXed by Anton Geraschenko in class and then edited by the people in the following table. The columns are sorted by lecturer. Nicolai Reshetikhin Vera Serganova Richard Borcherds 1 Anton Geraschenko 11 Sevak Mkrtchyan 21 Hanh Duc Do 2 Anton Geraschenko 12 Jonah Blasiak 22 An Huang 3 Nathan George 13 Hannes Thiel 23 Santiago Canez 4 Hans Christianson 14 Anton Geraschenko 24 Lilit Martirosyan 5 Emily Peters 15 Lilit Martirosyan 25 Emily Peters 6 Sevak Mkrtchyan 16 Santiago Canez 26 Santiago Canez 7 Lilit Martirosyan 17 Katie Liesinger 27 Martin Vito-Cruz 8 David Cimasoni 18 Aaron McMillan 28 Martin Vito-Cruz 9 Emily Peters 19 Anton Geraschenko 29 Anton Geraschenko 10 Qingtau Chen 20 Hanh Duc Do 30 Lilit Martirosyan 31 Sevak Mkrtchyan Richard Borcherds then edited the last third of the notes. The notes were further edited by Crystal Hoyt, Sevak Mkrtchyan, and Anton Geraschenko. Send corrections and comments to [email protected]. Lecture 1 4 Lecture 1 Definition 1.1. A Lie group is a smooth manifold G with a group structure such that the multiplication µ : G G G and inverse map ι : G G are × → → smooth maps. ◮ Exercise 1.1. If we assume only that µ is smooth, does it follow that ι is smooth? n Example 1.2. The group of invertible endomorphisms of C , GLn(C), is a Lie group. The automorphisms of determinant 1, SLn(C), is also a Lie group. Example 1.3. If B is a bilinear form on Cn, then we can consider the Lie group A GL (C) B(Av, Aw)= B(v, w) for all v, w Cn . { ∈ n | ∈ } If we take B to be the usual dot product, then we get the group On(C). T 0 1m If we let n = 2m be even and set B(v, w) = v 1m 0 w, then we get Sp (C). − 2m Example 1.4. SU SL is a real form (look in lectures 27,28, and 29 for n ⊆ n more on real forms). Example 1.5. We’d like to consider infinite matrices, but the multiplication wouldn’t make sense, so we can think of GL GL via A A 0 , then n ⊆ n+1 7→ 0 1 define GL as n GLn. That is, invertible infinite matrices which look like ∞ the identity almost everywhere. S Lie groups are hard objects to work with because they have global charac- teristics, but we’d like to know about representations of them. Fortunately, there are things called Lie algebras, which are easier to work with, and representations of Lie algebras tell us about representations of Lie groups. Definition 1.6. A Lie algebra is a vector space V equipped with a Lie bracket [ , ] : V V V , which satisfies × → 1. Skew symmetry: [a, a] = 0 for all a V , and ∈ 2. Jacobi identity: [a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0 for all a, b, c V . ∈ A Lie subalgebra of a lie algebra V is a subspace W V which is closed ⊆ under the bracket: [W, W ] W . ⊆ Example 1.7. If A is a finite dimensional associative algebra, you can set [a, b] = ab ba. If you start with A = M , the algebra of n n matri- − n × ces, then you get the Lie algebra gl . If you let A M be the algebra n ⊆ n of matrices preserving a fixed flag V V V Cn, then you get 0 ⊂ 1 ⊂ ··· k ⊆ parabolicindexparabolic subalgebras Lie sub-algebras of gln. Lecture 1 5 Example 1.8. Consider the set of vector fields on Rn, Vect(Rn) = ℓ = i ∂ { Σe (x) [ℓ1,ℓ2]= ℓ1 ℓ2 ℓ2 ℓ1 . ∂xi | ◦ − ◦ } ◮ Exercise 1.2. Check that [ℓ1,ℓ2] is a first order differential operator. Example 1.9. If A is an associative algebra, we say that ∂ : A A is a → derivation if ∂(ab) = (∂a)b + a∂b. Inner derivations are those of the form [d, ] for some d A; the others are called outer derivations. We denote the · ∈ set of derivations of A by (A), and you can verify that it is a Lie algebra. D Note that Vect(Rn) above is just (C (Rn)). D ∞ The first Hochschild cohomology, denoted H1(A, A), is the quotient (A)/ inner derivations . D { } Definition 1.10. A Lie algebra homomorphism is a linear map φ : L L → ′ that takes the bracket in L to the bracket in L′, i.e. φ([a, b]L) = [φ(a), φ(b)]L′ . A Lie algebra isomorphism is a morphism of Lie algebras that is a linear isomorphism.1 A very interesting question is to classify Lie algebras (up to isomorphism) of dimension n for a given n. For n = 2, there are only two: the trivial bracket [ , ] = 0, and [e1, e2] = e2. For n = 3, it can be done without too much trouble. Maybe n = 4 has been done, but in general, it is a very hard problem. If e is a basis for V , with [e , e ]= ck e (the ck are called the structure { i} i j ij k ij constants of V ), then the Jacobi identity is some quadratic relation on the k 3n cij, so the variety of Lie algebras is some quadratic surface in C . Given a smooth real manifold M n of dimension n, we can construct Vect(M n), the set of smooth vector fields on M n. For X Vect(M n), we ∈ can define the Lie derivative LX by (LX f)(m) = Xm f, so LX acts on n · · C∞(M ) as a derivation. ◮ Exercise 1.3. Verify that [L ,L ]= L L L L is of the form X Y X ◦ Y − Y ◦ X L for a unique Z Vect(M n). Then we put a Lie algebra structure on Z ∈ Vect(M n)= (C (M n)) by [X,Y ]= Z. D ∞ There is a theorem (Ado’s Theorem2) that any Lie algebra g is isomor- phic to a Lie subalgebra of gln, so if you understand everything about gln, you’re in pretty good shape. 1The reader may verify that this implies that the inverse is also a morphism of Lie algebras. 2Notice that if g has no center, then the adjoint representation ad : g → gl(g) is already faithful.
Load more

Recommended publications
  • Discussion on Some Properties of an Infinite Non-Abelian Group
    International Journal of Mathematics Trends and Technology (IJMTT) – Volume 48 Number 2 August 2017 Discussion on Some Then A.B = Properties of an Infinite Non-abelian = Group Where = Ankur Bala#1, Madhuri Kohli#2 #1 M.Sc. , Department of Mathematics, University of Delhi, Delhi #2 M.Sc., Department of Mathematics, University of Delhi, Delhi ( 1 ) India Abstract: In this Article, we have discussed some of the properties of the infinite non-abelian group of matrices whose entries from integers with non-zero determinant. Such as the number of elements of order 2, number of subgroups of order 2 in this group. Moreover for every finite group , there exists such that has a subgroup isomorphic to the group . ( 2 ) Keywords: Infinite non-abelian group, Now from (1) and (2) one can easily say that is Notations: GL(,) n Z []aij n n : aij Z. & homomorphism. det( ) = 1 Now consider , Theorem 1: can be embedded in Clearly Proof: If we prove this theorem for , Hence is injective homomorphism. then we are done. So, by fundamental theorem of isomorphism one can Let us define a mapping easily conclude that can be embedded in for all m . Such that Theorem 2: is non-abelian infinite group Proof: Consider Consider Let A = and And let H= Clearly H is subset of . And H has infinite elements. B= Hence is infinite group. Now consider two matrices Such that determinant of A and B is , where . And ISSN: 2231-5373 http://www.ijmttjournal.org Page 108 International Journal of Mathematics Trends and Technology (IJMTT) – Volume 48 Number 2 August 2017 Proof: Let G be any group and A(G) be the group of all permutations of set G.
    [Show full text]
  • Group Theory in Particle Physics
    Group Theory in Particle Physics Joshua Albert Phy 205 http://en.wikipedia.org/wiki/Image:E8_graph.svg Where Did it Come From? Group Theory has it©s origins in: ● Algebraic Equations ● Number Theory ● Geometry Some major early contributers were Euler, Gauss, Lagrange, Abel, and Galois. What is a group? ● A group is a collection of objects with an associated operation. ● The group can be finite or infinite (based on the number of elements in the group. ● The following four conditions must be satisfied for the set of objects to be a group... 1: Closure ● The group operation must associate any pair of elements T and T© in group G with another element T©© in G. This operation is the group multiplication operation, and so we write: – T T© = T©© – T, T©, T©© all in G. ● Essentially, the product of any two group elements is another group element. 2: Associativity ● For any T, T©, T©© all in G, we must have: – (T T©) T©© = T (T© T©©) ● Note that this does not imply: – T T© = T© T – That is commutativity, which is not a fundamental group property 3: Existence of Identity ● There must exist an identity element I in a group G such that: – T I = I T = T ● For every T in G. 4: Existence of Inverse ● For every element T in G there must exist an inverse element T -1 such that: – T T -1 = T -1 T = I ● These four properties can be satisfied by many types of objects, so let©s go through some examples... Some Finite Group Examples: ● Parity – Representable by {1, -1}, {+,-}, {even, odd} – Clearly an important group in particle physics ● Rotations of an Equilateral Triangle – Representable as ordering of vertices: {ABC, ACB, BAC, BCA, CAB, CBA} – Can also be broken down into subgroups: proper rotations and improper rotations ● The Identity alone (smallest possible group).
    [Show full text]
  • Group Theory
    Appendix A Group Theory This appendix is a survey of only those topics in group theory that are needed to understand the composition of symmetry transformations and its consequences for fundamental physics. It is intended to be self-contained and covers those topics that are needed to follow the main text. Although in the end this appendix became quite long, a thorough understanding of group theory is possible only by consulting the appropriate literature in addition to this appendix. In order that this book not become too lengthy, proofs of theorems were largely omitted; again I refer to other monographs. From its very title, the book by H. Georgi [211] is the most appropriate if particle physics is the primary focus of interest. The book by G. Costa and G. Fogli [102] is written in the same spirit. Both books also cover the necessary group theory for grand unification ideas. A very comprehensive but also rather dense treatment is given by [428]. Still a classic is [254]; it contains more about the treatment of dynamical symmetries in quantum mechanics. A.1 Basics A.1.1 Definitions: Algebraic Structures From the structureless notion of a set, one can successively generate more and more algebraic structures. Those that play a prominent role in physics are defined in the following. Group A group G is a set with elements gi and an operation ◦ (called group multiplication) with the properties that (i) the operation is closed: gi ◦ g j ∈ G, (ii) a neutral element g0 ∈ G exists such that gi ◦ g0 = g0 ◦ gi = gi , (iii) for every gi exists an −1 ∈ ◦ −1 = = −1 ◦ inverse element gi G such that gi gi g0 gi gi , (iv) the operation is associative: gi ◦ (g j ◦ gk) = (gi ◦ g j ) ◦ gk.
    [Show full text]
  • Computational Topics in Lie Theory and Representation Theory
    Utah State University DigitalCommons@USU All Graduate Theses and Dissertations Graduate Studies 5-2014 Computational Topics in Lie Theory and Representation Theory Thomas J. Apedaile Utah State University Follow this and additional works at: https://digitalcommons.usu.edu/etd Part of the Applied Statistics Commons Recommended Citation Apedaile, Thomas J., "Computational Topics in Lie Theory and Representation Theory" (2014). All Graduate Theses and Dissertations. 2156. https://digitalcommons.usu.edu/etd/2156 This Thesis is brought to you for free and open access by the Graduate Studies at DigitalCommons@USU. It has been accepted for inclusion in All Graduate Theses and Dissertations by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. COMPUTATIONAL TOPICS IN LIE THEORY AND REPRESENTATION THEORY by Thomas J. Apedaile A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Mathematics Approved: Ian Anderson Nathan Geer Major Professor Committee Member Zhaohu Nie Mark R McLellan Committee Member Vice President for Research and Dean of the School of Graduate Studies UTAH STATE UNIVERSITY Logan, Utah 2014 ii Copyright c Thomas J. Apedaile 2014 All Rights Reserved iii ABSTRACT Computational Topics in Lie Theory and Representation Theory by Thomas J. Apedaile, Master of Science Utah State University, 2014 Major Professor: Dr. Ian Anderson Department: Mathematics and Statistics The computer algebra system Maple contains a basic set of commands for work- ing with Lie algebras. The purpose of this thesis was to extend the functionality of these Maple packages in a number of important areas. First, programs for defining multiplication in several types of Cayley algebras, Jordan algebras and Clifford algebras were created to allow users to perform a variety of calculations.
    [Show full text]
  • The Chiral De Rham Complex of Tori and Orbifolds
    The Chiral de Rham Complex of Tori and Orbifolds Dissertation zur Erlangung des Doktorgrades der Fakult¨atf¨urMathematik und Physik der Albert-Ludwigs-Universit¨at Freiburg im Breisgau vorgelegt von Felix Fritz Grimm Juni 2016 Betreuerin: Prof. Dr. Katrin Wendland ii Dekan: Prof. Dr. Gregor Herten Erstgutachterin: Prof. Dr. Katrin Wendland Zweitgutachter: Prof. Dr. Werner Nahm Datum der mundlichen¨ Prufung¨ : 19. Oktober 2016 Contents Introduction 1 1 Conformal Field Theory 4 1.1 Definition . .4 1.2 Toroidal CFT . .8 1.2.1 The free boson compatified on the circle . .8 1.2.2 Toroidal CFT in arbitrary dimension . 12 1.3 Vertex operator algebra . 13 1.3.1 Complex multiplication . 15 2 Superconformal field theory 17 2.1 Definition . 17 2.2 Ising model . 21 2.3 Dirac fermion and bosonization . 23 2.4 Toroidal SCFT . 25 2.5 Elliptic genus . 26 3 Orbifold construction 29 3.1 CFT orbifold construction . 29 3.1.1 Z2-orbifold of toroidal CFT . 32 3.2 SCFT orbifold . 34 3.2.1 Z2-orbifold of toroidal SCFT . 36 3.3 Intersection point of Z2-orbifold and torus models . 38 3.3.1 c = 1...................................... 38 3.3.2 c = 3...................................... 40 4 Chiral de Rham complex 41 4.1 Local chiral de Rham complex on CD ...................... 41 4.2 Chiral de Rham complex sheaf . 44 4.3 Cechˇ cohomology vertex algebra . 49 4.4 Identification with SCFT . 49 4.5 Toric geometry . 50 5 Chiral de Rham complex of tori and orbifold 53 5.1 Dolbeault type resolution . 53 5.2 Torus .
    [Show full text]
  • Matrix Lie Groups
    Maths Seminar 2007 MATRIX LIE GROUPS Claudiu C Remsing Dept of Mathematics (Pure and Applied) Rhodes University Grahamstown 6140 26 September 2007 RhodesUniv CCR 0 Maths Seminar 2007 TALK OUTLINE 1. What is a matrix Lie group ? 2. Matrices revisited. 3. Examples of matrix Lie groups. 4. Matrix Lie algebras. 5. A glimpse at elementary Lie theory. 6. Life beyond elementary Lie theory. RhodesUniv CCR 1 Maths Seminar 2007 1. What is a matrix Lie group ? Matrix Lie groups are groups of invertible • matrices that have desirable geometric features. So matrix Lie groups are simultaneously algebraic and geometric objects. Matrix Lie groups naturally arise in • – geometry (classical, algebraic, differential) – complex analyis – differential equations – Fourier analysis – algebra (group theory, ring theory) – number theory – combinatorics. RhodesUniv CCR 2 Maths Seminar 2007 Matrix Lie groups are encountered in many • applications in – physics (geometric mechanics, quantum con- trol) – engineering (motion control, robotics) – computational chemistry (molecular mo- tion) – computer science (computer animation, computer vision, quantum computation). “It turns out that matrix [Lie] groups • pop up in virtually any investigation of objects with symmetries, such as molecules in chemistry, particles in physics, and projective spaces in geometry”. (K. Tapp, 2005) RhodesUniv CCR 3 Maths Seminar 2007 EXAMPLE 1 : The Euclidean group E (2). • E (2) = F : R2 R2 F is an isometry . → | n o The vector space R2 is equipped with the standard Euclidean structure (the “dot product”) x y = x y + x y (x, y R2), • 1 1 2 2 ∈ hence with the Euclidean distance d (x, y) = (y x) (y x) (x, y R2).
    [Show full text]
  • Cohomology Theory of Lie Groups and Lie Algebras
    COHOMOLOGY THEORY OF LIE GROUPS AND LIE ALGEBRAS BY CLAUDE CHEVALLEY AND SAMUEL EILENBERG Introduction The present paper lays no claim to deep originality. Its main purpose is to give a systematic treatment of the methods by which topological questions concerning compact Lie groups may be reduced to algebraic questions con- cerning Lie algebras^). This reduction proceeds in three steps: (1) replacing questions on homology groups by questions on differential forms. This is accomplished by de Rham's theorems(2) (which, incidentally, seem to have been conjectured by Cartan for this very purpose); (2) replacing the con- sideration of arbitrary differential forms by that of invariant differential forms: this is accomplished by using invariant integration on the group manifold; (3) replacing the consideration of invariant differential forms by that of alternating multilinear forms on the Lie algebra of the group. We study here the question not only of the topological nature of the whole group, but also of the manifolds on which the group operates. Chapter I is concerned essentially with step 2 of the list above (step 1 depending here, as in the case of the whole group, on de Rham's theorems). Besides consider- ing invariant forms, we also introduce "equivariant" forms, defined in terms of a suitable linear representation of the group; Theorem 2.2 states that, when this representation does not contain the trivial representation, equi- variant forms are of no use for topology; however, it states this negative result in the form of a positive property of equivariant forms which is of interest by itself, since it is the key to Levi's theorem (cf.
    [Show full text]
  • Lecture Notes on Compact Lie Groups and Their Representations
    Lecture Notes on Compact Lie Groups and Their Representations CLAUDIO GORODSKI Prelimimary version: use with extreme caution! September , ii Contents Contents iii 1 Compact topological groups 1 1.1 Topological groups and continuous actions . 1 1.2 Representations .......................... 4 1.3 Adjointaction ........................... 7 1.4 Averaging method and Haar integral on compact groups . 9 1.5 The character theory of Frobenius-Schur . 13 1.6 Problems.............................. 20 2 Review of Lie groups 23 2.1 Basicdefinition .......................... 23 2.2 Liealgebras ............................ 25 2.3 Theexponentialmap ....................... 28 2.4 LiehomomorphismsandLiesubgroups . 29 2.5 Theadjointrepresentation . 32 2.6 QuotientsandcoveringsofLiegroups . 34 2.7 Problems.............................. 36 3 StructureofcompactLiegroups 39 3.1 InvariantinnerproductontheLiealgebra . 39 3.2 CompactLiealgebras. 40 3.3 ComplexsemisimpleLiealgebras. 48 3.4 Problems.............................. 51 3.A Existenceofcompactrealforms . 53 4 Roottheory 55 4.1 Maximaltori............................ 55 4.2 Cartansubalgebras ........................ 57 4.3 Case study: representations of SU(2) .............. 58 4.4 Rootspacedecomposition . 61 4.5 Rootsystems............................ 63 4.6 Classificationofrootsystems . 69 iii iv CONTENTS 4.7 Problems.............................. 73 CHAPTER 1 Compact topological groups In this introductory chapter, we essentially introduce our very basic objects of study, as well as some fundamental examples. We also establish some preliminary results that do not depend on the smooth structure, using as little as possible machinery. The idea is to paint a picture and plant the seeds for the later development of the heavier theory. 1.1 Topological groups and continuous actions A topological group is a group G endowed with a topology such that the group operations are continuous; namely, we require that the multiplica- tion map and the inversion map µ : G G G, ι : G G × → → be continuous maps.
    [Show full text]
  • Higher AGT Correspondences, W-Algebras, and Higher Quantum
    Higher AGT Correspon- dences, W-algebras, and Higher Quantum Geometric Higher AGT Correspondences, W-algebras, Langlands Duality from M-Theory and Higher Quantum Geometric Langlands Meng-Chwan Duality from M-Theory Tan Introduction Review of 4d Meng-Chwan Tan AGT 5d/6d AGT National University of Singapore W-algebras + Higher QGL SUSY gauge August 3, 2016 theory + W-algebras + QGL Higher GL Conclusion Presentation Outline Higher AGT Correspon- dences, Introduction W-algebras, and Higher Quantum Lightning Review: A 4d AGT Correspondence for Compact Geometric Langlands Lie Groups Duality from M-Theory A 5d/6d AGT Correspondence for Compact Lie Groups Meng-Chwan Tan W-algebras and Higher Quantum Geometric Langlands Introduction Duality Review of 4d AGT Supersymmetric Gauge Theory, W-algebras and a 5d/6d AGT Quantum Geometric Langlands Correspondence W-algebras + Higher QGL SUSY gauge Higher Geometric Langlands Correspondences from theory + W-algebras + M-Theory QGL Higher GL Conclusion Conclusion 6d/5d/4d AGT Correspondence in Physics and Mathematics Higher AGT Correspon- Circa 2009, Alday-Gaiotto-Tachikawa [1] | showed that dences, W-algebras, the Nekrasov instanton partition function of a 4d N = 2 and Higher Quantum conformal SU(2) quiver theory is equivalent to a Geometric Langlands conformal block of a 2d CFT with W2-symmetry that is Duality from M-Theory Liouville theory. This was henceforth known as the Meng-Chwan celebrated 4d AGT correspondence. Tan Circa 2009, Wyllard [2] | the 4d AGT correspondence is Introduction Review of 4d proposed and checked (partially) to hold for a 4d N = 2 AGT conformal SU(N) quiver theory whereby the corresponding 5d/6d AGT 2d CFT is an AN−1 conformal Toda field theory which has W-algebras + Higher QGL WN -symmetry.
    [Show full text]
  • Infinite Groups with Large Balls of Torsion Elements and Small Entropy
    INFINITE GROUPS WITH LARGE BALLS OF TORSION ELEMENTS AND SMALL ENTROPY LAURENT BARTHOLDI, YVES DE CORNULIER Abstract. We exhibit infinite, solvable, abelian-by-finite groups, with a fixed number of generators, with arbitrarily large balls consisting of torsion elements. We also provide a sequence of 3-generator non-nilpotent-by-finite polycyclic groups of algebraic entropy tending to zero. All these examples are obtained by taking ap- propriate quotients of finitely presented groups mapping onto the first Grigorchuk group. The Burnside Problem asks whether a finitely generated group all of whose ele- ments have finite order must be finite. We are interested in the following related question: fix n sufficiently large; given a group Γ, with a finite symmetric generating subset S such that every element in the n-ball is torsion, is Γ finite? Since the Burn- side problem has a negative answer, a fortiori the answer to our question is negative in general. However, it is natural to ask for it in some classes of finitely generated groups for which the Burnside Problem has a positive answer, such as linear groups or solvable groups. This motivates the following proposition, which in particularly answers a question of Breuillard to the authors. Proposition 1. For every n, there exists a group G, generated by a 3-element subset S consisting of elements of order 2, in which the n-ball consists of torsion elements, and satisfying one of the additional assumptions: (1) G is solvable, virtually abelian, and infinite (more precisely, it has a free abelian normal subgroup of finite 2-power index); in particular it is linear.
    [Show full text]
  • On the Third-Degree Continuous Cohomology of Simple Lie Groups 2
    ON THE THIRD-DEGREE CONTINUOUS COHOMOLOGY OF SIMPLE LIE GROUPS CARLOS DE LA CRUZ MENGUAL ABSTRACT. We show that the class of connected, simple Lie groups that have non-vanishing third-degree continuous cohomology with trivial ℝ-coefficients consists precisely of all simple §ℝ complex Lie groups and of SL2( ). 1. INTRODUCTION ∙ Continuous cohomology Hc is an invariant of topological groups, defined in an analogous fashion to the ordinary group cohomology, but with the additional assumption that cochains— with values in a topological group-module as coefficient space—are continuous. A basic refer- ence in the subject is the book by Borel–Wallach [1], while a concise survey by Stasheff [12] provides an overview of the theory, its relations to other cohomology theories for topological groups, and some interpretations of algebraic nature for low-degree cohomology classes. In the case of connected, (semi)simple Lie groups and trivial real coefficients, continuous cohomology is also known to successfully detect geometric information. For example, the situ- ation for degree two is well understood: Let G be a connected, non-compact, simple Lie group 2 ℝ with finite center. Then Hc(G; ) ≠ 0 if and only G is of Hermitian type, i.e. if its associated symmetric space of non-compact type admits a G-invariant complex structure. If that is the case, 2 ℝ Hc(G; ) is one-dimensional, and explicit continuous 2-cocycles were produced by Guichardet– Wigner in [6] as an obstruction to extending to G a homomorphism K → S1, where K<G is a maximal compact subgroup. The goal of this note is to clarify a similar geometric interpretation of the third-degree con- tinuous cohomology of simple Lie groups.
    [Show full text]
  • Arxiv:1109.4101V2 [Hep-Th]
    Quantum Open-Closed Homotopy Algebra and String Field Theory Korbinian M¨unster∗ Arnold Sommerfeld Center for Theoretical Physics, Theresienstrasse 37, D-80333 Munich, Germany Ivo Sachs† Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA 02138, USA and Arnold Sommerfeld Center for Theoretical Physics, Theresienstrasse 37, D-80333 Munich, Germany (Dated: September 28, 2018) Abstract We reformulate the algebraic structure of Zwiebach’s quantum open-closed string field theory in terms of homotopy algebras. We call it the quantum open-closed homotopy algebra (QOCHA) which is the generalization of the open-closed homo- topy algebra (OCHA) of Kajiura and Stasheff. The homotopy formulation reveals new insights about deformations of open string field theory by closed string back- grounds. In particular, deformations by Maurer Cartan elements of the quantum closed homotopy algebra define consistent quantum open string field theories. arXiv:1109.4101v2 [hep-th] 19 Oct 2011 ∗Electronic address: [email protected] †Electronic address: [email protected] 2 Contents I. Introduction 3 II. Summary 4 III. A∞- and L∞-algebras 7 A. A∞-algebras 7 B. L∞-algebras 10 IV. Homotopy involutive Lie bialgebras 12 A. Higher order coderivations 12 B. IBL∞-algebra 13 C. IBL∞-morphisms and Maurer Cartan elements 15 V. Quantum open-closed homotopy algebra 15 A. Loop homotopy algebra of closed strings 17 B. IBL structure on cyclic Hochschild complex 18 C. Quantum open-closed homotopy algebra 19 VI. Deformations and the quantum open-closed correspondence 23 A. Quantum open string field theory 23 B. Quantum open-closed correspondence 24 VII.
    [Show full text]