DOCSLIB.ORG

Oldandnewonthe Exceptionalgroupg2 Ilka Agricola

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Oldandnewonthe Exceptionalgroupg2 Ilka Agricola OldandNewonthe ExceptionalGroupG2 Ilka Agricola n a talk delivered in Leipzig (Germany) on product, the Lie bracket [ , ]; as a purely algebraic June 11, 1900, Friedrich Engel gave the object it is more accessible than the original Lie first public account of his newly discovered group G. If G happens to be a group of matrices, its description of the smallest exceptional Lie Lie algebra g is easily realized by matrices too, and group G2, and he wrote in the corresponding the Lie bracket coincides with the usual commuta- Inote to the Royal Saxonian Academy of Sciences: tor of matrices. In Killing’s and Lie’s time, no clear Moreover, we hereby obtain a direct defi- distinction was made between the Lie group and nition of our 14-dimensional simple group its Lie algebra. For his classification, Killing chose [G2] which is as elegant as one can wish for. a maximal set h of linearly independent, pairwise 1 [En00, p. 73] commuting elements of g and constructed base Indeed, Engel’s definition of G2 as the isotropy vectors Xα of g (indexed over a finite subset R of group of a generic 3-form in 7 dimensions is at elements α ∈ h∗, the roots) on which all elements the basis of a rich geometry that exists only on of h act diagonally through [ , ]: 7-dimensional manifolds, whose full beauty has been unveiled in the last thirty years. [H,Xα] = α(H)Xα for all H ∈ h. This article is devoted to a detailed historical In order to avoid problems when doing so he chose and mathematical account of G ’s first years, in 2 the complex numbers C as the ground field. The particular the contributions and the life of Engel’s dimension of the maximal abelian subalgebra h almost forgotten Ph. D. student Walter Reichel, (also called, somehow wrongly, a Cartan subalge- who worked out the details of this description in bra) is the rank of the Lie algebra. It is a general 1907. We will also give an introduction to mod- fact that all roots α ≠ 0 appear only once. If we ern G geometry and its relevance in theoretical 2 = C · physics (in particular, superstring theory). write gα : Xα (these are the root spaces), we obtain a decomposition of g as The Classification of Simple Lie Groups g = h M gα In 1887 Wilhelm Killing [Kil89] succeeded in classi- α∈R fying those transformation groups that are rightly and vectors in g , g for two roots α,β satisfy an called simple:bydefinition,thesearetheLiegroups α β ⊂ that are not abelian and do not have any nontrivial extremely easy multiplication rule: [gα, gβ] gα+β normal subgroups. if α + β is again a root, while otherwise it is zero. Every Lie group G has a Lie algebra g (the tangent Two families of complex simple Lie algebras space to the group manifold at the identity), which were well-known at that time: is a vector space endowed with a skew-symmetric (1) the Lie algebras so(n, C) consisting of skew-symmetric complex matrices, which Ilka Agricola is head of a Junior Research Group funded are the Lie algebras of the orthogonal by the Volkswagen Foundation at Humboldt-Universität groups SO(n, C) (n = 3 or n ≥ 5), zu Berlin. Her email address is agricola@mathematik. C hu-berlin.de. (2) the Lie algebras sl (n, ) consisting of 1In Engel’s own words: “Zudem ist hiermit eine direkte trace-free matrices, which are the Lie al- Definition unserer vierzehngliedrigen einfachen Gruppe gebras of the groups SL(n, C) of matrices gegeben, die an Eleganz nichts zu wünschen übrig lässt.” of determinant one (n ≥ 2). 922 Notices of the AMS Volume 55, Number 8 It was Killing’s original intent to prove that these were the only simple complex Lie algebras.2 In fact, there exists a third family of simple algebras, namely the Lie algebras sp(2n, C) of the symplectic groups Sp(2n, C) for n ≥ 1, defined as invariance groups of non-degenerate 2-forms ω on C2n: Sp(2n, C) ={g ∈ GL(2n, C) : ω = g∗ω}. Oeuvres Complètes, vol. II; Around 1886, Sophus Lie and Friedrich Engel were aware of their existence, but they had not yet appeared in print anywhere [Ha00, p. 152]. To his big surprise, in May 18873 Killing discovered a completely unknown complex simple Lie algebra (Photo sources: Cartan, Engel, University Archive Greifswald.) of rank 2 and dimension 14, which is just the Élie Cartan Friedrich Engel (1869–1951) in the (1861–1941) around exceptional Lie algebra g2. By October 1887, he had basically completed his classification. He dis- year 1904. 1922. covered that besides g2 and the three families mentioned above, there exist four additional ex- First Results on G2 ceptional simple Lie algebras. In modern notation, We can only conjecture how Killing and his contem- they are: f , e , e , e , and they have dimensions 4 6 7 8 poraries felt aboutthe exceptionalLie algebras—as 52, 78, 133, and 248 respectively. disturbances to the symmetry or as exotic and There exist many real orthogonal Lie groups unique objects. But since Lie theory as a whole C with complexification SO(n, )—namely all or- was developed in these times, they were investi- thogonal groups SO(p, q) associated with scalar gated too, but not with high priority. G2 was the products with indefinite signature (p, q) such that first—and, for rather a long time, the only—Lie p + q = n; they are called real forms of SO(n, C) group for which further results were obtained. (similarly for the Lie algebras), and it is an easy This is natural for dimensional considerations, fact that SO(p, q) is compact only for q = 0. Just but we will see later that it has also much deeper as well, the complex Lie algebra g2 with complex reasons. Lie group G2 has two real forms that we are going From the weight lattice, which one obtains auto- c ∗ to denote by g2 and g2 ; of their (simply connected) matically during the classification, one can easily c ∗ Lie groups G2 and G2 , only the former is compact. determine the lowest dimensional representation Without doubt, the classification of complex of any simple Lie algebra. This Cartan did in the simple Lie algebras is one of the outstanding last section of his thesis, and he rightly observed results of nineteenth century mathematics (this that g2 admits a 7-dimensional complex represen- was not Killing’s point of view, however: his orig- tation, which furthermore possesses a symmetric inal aim had been a classification of all real Lie nondegenerate g2-invariant bilinear form [Ca94, algebras, he was unsatisfied with his own expo- p. 146]: sition and the incompleteness of results, so that 2 β := x0 + x1y1 + x2y2 + x3y3. he would not have published his results with- This scalar product has real coefficients, hence can out strong encouragement from Friedrich Engel). be interpreted over the reals as well; in that case Indeed, Killing’s formidable work contains some it has signature (4, 3), and one can understand gaps and mistakes:4 in his thesis (1894), Élie Cartan’s result as giving a real representation of Cartan gave a completely revised and polished ∗ the noncompact form g2 inside so(4, 3). presentation of the classification [Ca94], which At this stage, the question about explicit con- has therefore become the standard reference for structions of the exceptional Lie algebras becomes the result. pressing. Élie Cartan and Friedrich Engel obtained the first breakthrough, published in two simulta- neous notes to the Académie des Sciences de Paris 2Letter from W. Killing to Fr. Engel, April 12, 1886; see [Ca93], [En93]: For every point a ∈ C5, consider [Ha00, p. 153]. the 2-plane π in the tangent space T C5 that is 3Letter from W. Killing to Fr. Engel, May 23, 1887; see a a [Ha00, p. 161]. the zero set of the Pfaffian system 4 For example, he had found two exceptional Lie algebras dx3 = x1 dx2 − x2 dx1, of dimension 52 and overlooked that they are isomorphic, = − and his classification was based on three main theorems dx4 x2 dx3 x3 dx2, whose statements and proofs were partially wrong. dx5 = x3 dx1 − x1 dx3. September 2008 Notices of the AMS 923 5 The 14 vector fields on C whose local flows map gα stated before, one sees that the direct sum of the planes πa to each other satisfy the commu- h (corresponding, loosely speaking, to the origin), tator relations of the Lie algebra g2. Both authors the six positive root spaces, and the root space of then gave in the same papers a second geometric one negative of a simple root, span a subalgebra: realization of g2: Engel derived it from the first p := h ⊕ g g . by a contact transformation, while Cartan identi- i −αi M α α∈R+ fied g2 as the symmetries of the solution space of the system of second order partial differential Subalgebras of this kind are called parabolic sub- 5 equations (f = f (x, y)) algebras. The 9-dimensional groups P1 and P2 above are now exactly the subgroups of G with 4 3 2 2 fxx = (fyy ) , fxy = (fyy ) . 3 Lie algebras p1 and p2. By general results, the Both viewed their second realization as being space G2/Pi is a compact homogeneous variety, different from the one through the Pfaffian system. and it can be realized in the projectivization of the Of course, this is correct: stated in modern terms, representation space Vi with highest weight ωi (see figure) as the G2 orbit of some distinguished the complex Lie group G2 has two non-conjugate vector v ; but ω generates the 7-dimensional rep- 9-dimensional parabolic subgroups P1 and P2, and i 1 resentation (spanned by the six short roots and G2 acts on the two compact homogeneous spaces 5 zero with multiplicity one), while ω2 is the highest Mi := G2/Pi , i = 1, 2.
Load more

Recommended publications
  • Compactifying M-Theory on a G2 Manifold to Describe/Explain Our World – Predictions for LHC (Gluinos, Winos, Squarks), and Dark Matter
    Compactifying M-theory on a G2 manifold to describe/explain our world – Predictions for LHC (gluinos, winos, squarks), and dark matter Gordy Kane CMS, Fermilab, April 2016 1 OUTLINE • Testing theories in physics – some generalities - Testing 10/11 dimensional string/M-theories as underlying theories of our world requires compactification to four space-time dimensions! • Compactifying M-theory on “G2 manifolds” to describe/ explain our vacuum – underlying theory - fluxless sector! • Moduli – 4D manifestations of extra dimensions – stabilization - supersymmetry breaking – changes cosmology first 16 slides • Technical stuff – 18-33 - quickly • From the Planck scale to EW scale – 34-39 • LHC predictions – gluino about 1.5 TeV – also winos at LHC – but not squarks - 40-47 • Dark matter – in progress – surprising – 48 • (Little hierarchy problem – 49-51) • Final remarks 1-5 2 String/M theory a powerful, very promising framework for constructing an underlying theory that incorporates the Standard Models of particle physics and cosmology and probably addresses all the questions we hope to understand about the physical universe – we hope for such a theory! – probably also a quantum theory of gravity Compactified M-theory generically has gravity; Yang- Mills forces like the SM; chiral fermions like quarks and leptons; softly broken supersymmetry; solutions to hierarchy problems; EWSB and Higgs physics; unification; small EDMs; no flavor changing problems; partially observable superpartner spectrum; hidden sector DM; etc Simultaneously – generically Argue compactified M-theory is by far the best motivated, and most comprehensive, extension of the SM – gets physics relevant to the LHC and Higgs and superpartners right – no ad hoc inputs or free parameters Take it very seriously 4 So have to spend some time explaining derivations, testability of string/M theory Don’t have to be somewhere to test theory there – E.g.
    [Show full text]
  • Deformations of G2-Structures, String Dualities and Flat Higgs Bundles Rodrigo De Menezes Barbosa University of Pennsylvania, [email protected]
    University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 2019 Deformations Of G2-Structures, String Dualities And Flat Higgs Bundles Rodrigo De Menezes Barbosa University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Mathematics Commons, and the Quantum Physics Commons Recommended Citation Barbosa, Rodrigo De Menezes, "Deformations Of G2-Structures, String Dualities And Flat Higgs Bundles" (2019). Publicly Accessible Penn Dissertations. 3279. https://repository.upenn.edu/edissertations/3279 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/3279 For more information, please contact [email protected]. Deformations Of G2-Structures, String Dualities And Flat Higgs Bundles Abstract We study M-theory compactifications on G2-orbifolds and their resolutions given by total spaces of coassociative ALE-fibrations over a compact flat Riemannian 3-manifold Q. The afl tness condition allows an explicit description of the deformation space of closed G2-structures, and hence also the moduli space of supersymmetric vacua: it is modeled by flat sections of a bundle of Brieskorn-Grothendieck resolutions over Q. Moreover, when instanton corrections are neglected, we obtain an explicit description of the moduli space for the dual type IIA string compactification. The two moduli spaces are shown to be isomorphic for an important example involving A1-singularities, and the result is conjectured to hold in generality. We also discuss an interpretation of the IIA moduli space in terms of "flat Higgs bundles" on Q and explain how it suggests a new approach to SYZ mirror symmetry, while also providing a description of G2-structures in terms of B-branes.
    [Show full text]
  • From Arthur Cayley Via Felix Klein, Sophus Lie, Wilhelm Killing, Elie Cartan, Emmy Noether and Superstrings to Cantorian Space–Time
    Available online at www.sciencedirect.com Chaos, Solitons and Fractals 37 (2008) 1279–1288 www.elsevier.com/locate/chaos From Arthur Cayley via Felix Klein, Sophus Lie, Wilhelm Killing, Elie Cartan, Emmy Noether and superstrings to Cantorian space–time L. Marek-Crnjac Institute of Mathematics, Physics and Mechanics, Jadranska ulica 19, P.O. Box 2964, SI-1001 Ljubljana, Slovenia Abstract In this work we present a historical overview of mathematical discoveries which lead to fundamental developments in super string theory, super gravity and finally to E-infinity Cantorian space–time theory. Cantorian space–time is a hierarchical fractal-like semi manifold with formally infinity many dimensions but a finite expectation number for these dimensions. The idea of hierarchy and self-similarity in science was first entertain by Right in the 18th century, later on the idea was repeated by Swedenborg and Charlier. Interestingly, the work of Mohamed El Naschie and his two contra parts Ord and Nottale was done independently without any knowledge of the above starting from non- linear dynamics and fractals. Ó 2008 Published by Elsevier Ltd. 1. Introduction Many of the profound mathematical discovery and dare I say also inventions which were made by the mathemati- cians Arthur Cayley, Felix Klein, Sophus Lie, Wilhelm Killing, Elie Cartan and Emmy Noether [1] are extremely important for high energy particles in general [2] as well as in the development of E-infinity, Cantorian space–time the- ory [3,4]. The present paper is dedicated to the historical background of this subject. 2. Arthur Cayley – beginner of the group theory in the modern way Arthur Cayley was a great British mathematician.
    [Show full text]
  • Modular Invariance of Characters of Vertex Operator Algebras
    JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 9, Number 1, January 1996 MODULAR INVARIANCE OF CHARACTERS OF VERTEX OPERATOR ALGEBRAS YONGCHANG ZHU Introduction In contrast with the finite dimensional case, one of the distinguished features in the theory of infinite dimensional Lie algebras is the modular invariance of the characters of certain representations. It is known [Fr], [KP] that for a given affine Lie algebra, the linear space spanned by the characters of the integrable highest weight modules with a fixed level is invariant under the usual action of the modular group SL2(Z). The similar result for the minimal series of the Virasoro algebra is observed in [Ca] and [IZ]. In both cases one uses the explicit character formulas to prove the modular invariance. The character formula for the affine Lie algebra is computed in [K], and the character formula for the Virasoro algebra is essentially contained in [FF]; see [R] for an explicit computation. This mysterious connection between the infinite dimensional Lie algebras and the modular group can be explained by the two dimensional conformal field theory. The highest weight modules of affine Lie algebras and the Virasoro algebra give rise to conformal field theories. In particular, the conformal field theories associated to the integrable highest modules and minimal series are rational. The characters of these modules are understood to be the holomorphic parts of the partition functions on the torus for the corresponding conformal field theories. From this point of view, the role of the modular group SL2(Z)ismanifest. In the study of conformal field theory, physicists arrived at the notion of chi- ral algebras (see e.g.
    [Show full text]
  • On Mirror Maps for Manifolds of Exceptional Holonomy JHEP10(2019)204 ]) 2 and 2 G ] (See Also [ 1 21 31 ]
    Published for SISSA by Springer Received: August 23, 2019 Accepted: October 2, 2019 Published: October 21, 2019 On mirror maps for manifolds of exceptional holonomy JHEP10(2019)204 Andreas P. Braun,a Suvajit Majumdera and Alexander Ottob;c aMathematical Institute, University of Oxford Woodstock Road, Oxford, OX2 6GG, U.K. bPerimeter Institute for Theoretical Physics, 31 Caroline St N, Waterloo, ON N2L 2Y5, Canada cDepartment of Physics, University of Waterloo, Waterloo, ON N2L 3G1, Canada E-mail: [email protected], [email protected], [email protected] Abstract: We study mirror symmetry of type II strings on manifolds with the exceptional holonomy groups G2 and Spin(7). Our central result is a construction of mirrors of Spin(7) manifolds realized as generalized connected sums. In parallel to twisted connected sum G2 manifolds, mirrors of such Spin(7) manifolds can be found by applying mirror symmetry to the pair of non-compact manifolds they are glued from. To provide non-trivial checks for such geometric mirror constructions, we give a CFT analysis of mirror maps for Joyce orbifolds in several new instances for both the Spin(7) and the G2 case. For all of these models we find possible assignments of discrete torsion phases, work out the action of mirror symmetry, and confirm the consistency with the geometrical construction. A novel feature appearing in the examples we analyse is the possibility of frozen singularities. Keywords: String Duality, Conformal Field Models in String Theory, Superstring Vacua ArXiv ePrint: 1905.01474 Open Access, c The Authors. https://doi.org/10.1007/JHEP10(2019)204 Article funded by SCOAP3.
    [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]
  • From String Theory and Moonshine to Vertex Algebras
    Preample From string theory and Moonshine to vertex algebras Bong H. Lian Department of Mathematics Brandeis University [email protected] Harvard University, May 22, 2020 Dedicated to the memory of John Horton Conway December 26, 1937 – April 11, 2020. Preample Acknowledgements: Speaker’s collaborators on the theory of vertex algebras: Andy Linshaw (Denver University) Bailin Song (University of Science and Technology of China) Gregg Zuckerman (Yale University) For their helpful input to this lecture, special thanks to An Huang (Brandeis University) Tsung-Ju Lee (Harvard CMSA) Andy Linshaw (Denver University) Preample Disclaimers: This lecture includes a brief survey of the period prior to and soon after the creation of the theory of vertex algebras, and makes no claim of completeness – the survey is intended to highlight developments that reflect the speaker’s own views (and biases) about the subject. As a short survey of early history, it will inevitably miss many of the more recent important or even towering results. Egs. geometric Langlands, braided tensor categories, conformal nets, applications to mirror symmetry, deformations of VAs, .... Emphases are placed on the mutually beneficial cross-influences between physics and vertex algebras in their concurrent early developments, and the lecture is aimed for a general audience. Preample Outline 1 Early History 1970s – 90s: two parallel universes 2 A fruitful perspective: vertex algebras as higher commutative algebras 3 Classification: cousins of the Moonshine VOA 4 Speculations The String Theory Universe 1968: Veneziano proposed a model (using the Euler beta function) to explain the ‘st-channel crossing’ symmetry in 4-meson scattering, and the Regge trajectory (an angular momentum vs binding energy plot for the Coulumb potential).
    [Show full text]
  • Super-Higgs in Superspace
    Article Super-Higgs in Superspace Gianni Tallarita 1,* and Moritz McGarrie 2 1 Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez, Santiago 7941169, Chile 2 Deutsches Elektronen-Synchrotron, DESY, Notkestrasse 85, 22607 Hamburg, Germany; [email protected] * Correspondence: [email protected] or [email protected] Received: 1 April 2019; Accepted: 10 June 2019; Published: 14 June 2019 Abstract: We determine the effective gravitational couplings in superspace whose components reproduce the supergravity Higgs effect for the constrained Goldstino multiplet. It reproduces the known Gravitino sector while constraining the off-shell completion. We show that these couplings arise by computing them as quantum corrections. This may be useful for phenomenological studies and model-building. We give an example of its application to multiple Goldstini. Keywords: supersymmetry; Goldstino; superspace 1. Introduction The spontaneous breakdown of global supersymmetry generates a massless Goldstino [1,2], which is well described by the Akulov-Volkov (A-V) effective action [3]. When supersymmetry is made local, the Gravitino “eats” the Goldstino of the A-V action to become massive: The super-Higgs mechanism [4,5]. In terms of superfields, the constrained Goldstino multiplet FNL [6–12] is equivalent to the A-V formulation (see also [13–17]). It is, therefore, natural to extend the description of supergravity with this multiplet, in superspace, to one that can reproduce the super-Higgs mechanism. In this paper we address two issues—first we demonstrate how the Gravitino, Goldstino, and multiple Goldstini obtain a mass. Secondly, by using the Spurion analysis, we write down the most minimal set of new terms in superspace that incorporate both supergravity and the Goldstino multiplet in order to reproduce the super-Higgs mechanism of [5,18] at lowest order in M¯ Pl.
    [Show full text]
  • Lie Group and Geometry on the Lie Group SL2(R)
    INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR Lie group and Geometry on the Lie Group SL2(R) PROJECT REPORT – SEMESTER IV MOUSUMI MALICK 2-YEARS MSc(2011-2012) Guided by –Prof.DEBAPRIYA BISWAS Lie group and Geometry on the Lie Group SL2(R) CERTIFICATE This is to certify that the project entitled “Lie group and Geometry on the Lie group SL2(R)” being submitted by Mousumi Malick Roll no.-10MA40017, Department of Mathematics is a survey of some beautiful results in Lie groups and its geometry and this has been carried out under my supervision. Dr. Debapriya Biswas Department of Mathematics Date- Indian Institute of Technology Khargpur 1 Lie group and Geometry on the Lie Group SL2(R) ACKNOWLEDGEMENT I wish to express my gratitude to Dr. Debapriya Biswas for her help and guidance in preparing this project. Thanks are also due to the other professor of this department for their constant encouragement. Date- place-IIT Kharagpur Mousumi Malick 2 Lie group and Geometry on the Lie Group SL2(R) CONTENTS 1.Introduction ................................................................................................... 4 2.Definition of general linear group: ............................................................... 5 3.Definition of a general Lie group:................................................................... 5 4.Definition of group action: ............................................................................. 5 5. Definition of orbit under a group action: ...................................................... 5 6.1.The general 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]
  • What Is the Group of Conjugate Symplectic Matrices? Howard E
    What is the group of conjugate symplectic matrices? Howard E. Haber Santa Cruz Institute for Particle Physics University of California, Santa Cruz, CA 95064, USA June 26, 2017 Abstract The group of complex symplectic matrices, Sp(n,C) is defined as the set of 2n 2n T 0 In × complex matrices, M GL(2n, C) M JM = J , where J I , and In is the { ∈ | } ≡ − n 0 n n identity matrix. If the transpose is replaced by hermitian conjugate, then one obtains × the group of conjugate symplectic matrices, M GL(2n, C) M †JM = J . In these { ∈ | } notes, we demonstrate that the group of conjugate symplectic matrices is isomorphic to † In 0 U(n,n)= M GL(2n, C) M In,nM = In,n , where In,n I . { ∈ | } ≡ 0 − n The group of complex symplectic matrices, Sp(n,C) is defined as the set of 2n 2n complex matrices,1 × Sp(n, C) M GL(2n, C) M TJM = J , (1) ≡{ ∈ | } where the matrix J is defined in block form by, 0 In J , (2) ≡ In 0 − with 0 denoting the n n zero matrix and In the n n identity matrix. Sp(n,C) is a complex × × 2 simple Lie group. Moreover, the determinant of any complex symplectic matrix is 1. If the transpose is replaced by hermitian conjugate in eq. (1), then one obtains the group of conjugate symplectic matrices, G = M GL(2n, C) M †JM = J , (3) { ∈ | } following the terminology or Refs. [1, 2]. Indeed, the group of conjugate symplectic matrices is a Lie group that is neither semi-simple nor complex.
    [Show full text]
  • Special Unitary Group - Wikipedia
    Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Special unitary group In mathematics, the special unitary group of degree n, denoted SU( n), is the Lie group of n×n unitary matrices with determinant 1. (More general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.) The group operation is matrix multiplication. The special unitary group is a subgroup of the unitary group U( n), consisting of all n×n unitary matrices. As a compact classical group, U( n) is the group that preserves the standard inner product on Cn.[nb 1] It is itself a subgroup of the general linear group, SU( n) ⊂ U( n) ⊂ GL( n, C). The SU( n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in quantum chromodynamics.[1] The simplest case, SU(1) , is the trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+ I, − I}. [nb 2] SU(2) is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations. Contents Properties Lie algebra Fundamental representation Adjoint representation The group SU(2) Diffeomorphism with S 3 Isomorphism with unit quaternions Lie Algebra The group SU(3) Topology Representation theory Lie algebra Lie algebra structure Generalized special unitary group Example Important subgroups See also 1 of 10 2/22/2018, 8:54 PM Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Remarks Notes References Properties The special unitary group SU( n) is a real Lie group (though not a complex Lie group).
    [Show full text]