DOCSLIB.ORG

Lecture 18 — Classification of Dynkin Diagrams 1 Examples of Dynkin

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 18 — Classification of Dynkin Diagrams 1 Examples of Dynkin 18.745 Introduction to Lie Algebras October 21, 2010 Lecture 18 | Classification of Dynkin Diagrams Prof. Victor Kac Scribe: Benjamin Iriarte 1 Examples of Dynkin diagrams. In the examples that follow, we will compute the Cartan matrices for the indecomposable root systems that we have encountered earlier. We record these as Dynkin diagrams, summarized in Figure 1. Later in the lecture, we will prove that these are actually the Dynkin diagrams of all possible indecomposable root systems. We also compute extended Dynkin diagrams specifically for the purposes of this proof. In the following examples, the rank of the root system is always denoted by r, and we get simple roots α1; : : : ; αr. The largest root, used in the extended Dynkin diagrams, is denoted by θ. Example 1.1. A (r ≥ 2): This corresponds to sl ( ). We have (" ;" ) = δ and ∆ = r r+1 F i j ij slr+1(F) Lr+1 f"i − "j j i; j 2 f1; : : : ; r + 1g and i 6= jg ⊂ V , where V is the subspace of i=1 R"i on which the sum of coordinates (in the basis f"i j i 2 f1; : : : ; r + 1gg) is zero. ∗ Take f 2 V given by f("1) = r + 1; f("2) = r; : : : ; f("r+1) = 1. Then, ∆+ = f"i − "j j i < jg, Π = f"i − "i+1 j i 2 f1; : : : ; rgg and θ = "1 − "r+1. For the Cartan matrix, recall that: 0 2 −1 0 ··· 0 1 B . C B −1 2 −1 . C B C B .. C A = B 0 −1 2 . 0 C : B C B . .. .. C @ . −1 A 0 ··· 0 −1 2 Hence, we obtain the Dynkin diagrams in Figure 1a. Example 1.2. Br(r ≥ 3): This corresponds to so2r+1(F). We have ∆so2r+1( ) = {±"i ± "j; ±"i j Lr F i; j 2 f1; : : : ; rg and i 6= jg ⊂ V = i=1 R"i: Take f given by f("1) = r; : : : ; f("r) = 1. Then, ∆+ = {±"i ± "j;"i j i; j 2 f1; : : : ; rg and i 6= jg, Π = f"i − "i+1 j i 2 f1; : : : ; r − 1gg t f"rg and θ = "1 + "2. 1 Figure 1: Dynkin diagrams and extended Dynkin diagrams of all the indecomposable root systems, from top to bottom, these correspond to: Ar;Br;Cr;Dr;E8;E7;E6;F4;G2: 2 The Cartan matrix is then: 0 2 −1 0 ··· 0 1 B . C B −1 2 −1 . C B C B .. C A = B 0 −1 2 . 0 C : B C B . .. .. C @ . −1 A 0 ··· 0 −2 2 Hence, we get the Dynkin diagrams in Figure 1b. Example 1.3. Cr(r ≥ 1): This corresponds to sp2r(F). We have ∆sp ( ) = {±"i ± "j j i; j 2 Lr 2r F f1; : : : ; rgg ⊂ V = i=1 R"i. We take the same f as in the previous case. Then, Π = f"i − "i+1 j i 2 f1; : : : ; r − 1gg t f2"rg and θ = 2"1. This gives the Cartan matrix: 0 2 −1 0 ··· 0 1 B . C B −1 2 −1 . C B C B .. C A = B 0 −1 2 . 0 C : B C B . .. .. C @ . −2 A 0 ··· 0 −1 2 The corresponding Dynkin diagrams are shown in Figure 1c. Example 1.4. D (r ≥ 4): This corresponds to so ( ). We have ∆ = {±" ± " j i; j 2 r 2r F so2r(F) i j ∗ f1; : : : ; rg and i 6= jg. Define f 2 V by f("1) = r − 1; : : : ; f("r) = 0. Then ∆ = f"i ± "j j i < jg, Π = f"1 − "2;:::;"r−1 − "r;"r−1 + "rg and θ = "1 + "2. The Cartan matrix is: 0 2 −1 0 :::::: 0 1 B . C B −1 2 −1 . C B C B .. .. C B 0 −1 . 0 0 C A = B C : B . .. C B . 2 −1 −1 C B C B . C @ . 0 −1 2 0 A 0 ::: 0 −1 0 2 Hence, we have the Dynkin diagrams in Figure 1d. 1 Example 1.5. E8: We have ∆E8 = {±"i±"j j i; j 2 f1;:::; 8g and i 6= jgtf 2 (±"1 ± "2 ± · · · ± "8)g even number of + signs L8 with V = i=1 R"i. Let f("1) = 32; f("2) = 6; f("3) = 5; : : : ; f("7) = 1; f("8) = 0. 1 Then, ∆+ = f"i ± "j j i < jg t f 2 ("1 ± "2 ± · · · ± "8)g. Also, Π = f"2 − "3;"3 − "4;:::"7 − even number of + signs 1 ~ "8; 2 ("1 − "2 − · · · − "7 + "8)g and θ = "1 + "2. We get the diagrams E8 and E8 in Figure 1e. 3 Exercise 1.1. E6: We have: ∆E6 =fα 2 ∆E8 j α1 + ··· + α6 = 0 and α7 + α8 = 0g =f"i − "j j i; j 2 f1;:::; 6g and i 6= jg t {±("7 − "8)g 1 1 t f (±" ± " · · · ± " ) ± (" − " )g: 2 1 2 6 2 7 8 3 + signs in the first 6 terms Pick f("1) = 32; f("2) = 7; f("3) = 6; : : : ; f("7) = 2; f("8) = 1. We can then check that Π = 1 f"2 − "3;:::;"5 − "6;"7 − "8; 2 ("1 − "2 − "3 − "4 + "5 + "6 − "7 + "8)g, in particular, "1 − "2 = 1 1 2 ("1 − "2 − "3 − "4 + "5 + "6 − "7 + "8) + 2 ("1 − "2 + "3 + "4 − "5 − "6 + "7 − "8). Also, θ = "1 − "6 for obvious reasons. We obtain the Dynkin diagrams in Figure 1g. E7 : Also, ∆E7 = fα 2 ∆E8 j α1 + α2 + ··· + α8 = 0g, so: ∆E7 =f"i − "j j i; j 2 f1;:::; 8g and i 6= jg 1 t f (±"1 ± "2 · · · ± "8)g: 2 4 + signs Again, pick f("1) = 32; f("2) = 7; f("3) = 6; : : : ; f("7) = 2; f("8) = 1. Then, Π = f"2 − "3;"3 − 1 "4;:::;"7 − "8; 2 ("1 − "2 − "3 − "4 − "5 + "6 + "7 + "8)g and θ = "1 − "8 for obvious reasons. We obtain the Dynkin diagrams in Figure 1f. 1 Exercise 1.2. F4: We have ∆F4 = {±"i; ±("i − "j); 2 (±"1 ± "2 ± "3 ± "4) j i; j 2 f1;:::; 4g and i 6= 1 jg. Pick f("1) = 30; f("2) = 3; f("3) = 2; f("4) = 1. Then, Π = f"2 − "3;"3 − "4;"4; 2 ("1 − "2 − "3 − "4)g and so θ = "1 + "2. This produces the Dynkin diagrams of Figure 1h. G2: Finally, ∆G2 = {±("1 − "2); ±("1 − "3); ±("2 − "3); ±(2"1 − "2 − "3); ±(2"2 − "1 − "3); ±(2"3 − "1 − "2)g. Pick f("1) = 2; f("2) = 1; f("3) = 4. Then, Π = f"1 − "2; −2"1 + "2 + "3g, and so θ = 2"3 − "1 − "2. We obtain the Dynkin diagrams of Figure 1i. 2 Classification of Dynkin Diagrams. Theorem 2.1. A complete non-redundant list of connected Dynkin diagrams is the following: D(Ar)(r ≥ 2), D(Br)(r ≥ 3), D(Cr)(r ≥ 1), D(Dr)(r ≥ 4), D(E6), D(E7), D(E8), D(F4) and D(G2). (See Figure 1 for these Dynkin diagrams and their extended versions.) Remark 2.2. Note that D(A1) = D(B1) = D(C1), D(B2) = D(C2) and D(D3) = D(A3). Proof. We prove that there are no other Dynkin diagrams. To do this, we find all connected graphs with connections of the 4 types depicted in Figure 2, such that the matrix of any subgraph has a positive determinant, in particular any subgraph must be a Dynkin diagram. Equivalently, we require that the determinant of all principal minors of the corresponding Cartan matrix are positive. In particular, our graphs contain no extended Dynkin diagrams as induced subgraphs, since these have determinant 0. 4 Figure 2: The four Dynkin diagram connection types, corresponding to the four types of 2 × 2 Cartan matrix minors. Figure 3: Some diagrams for the simply laced case. Part 1. We first classify all simply-laced Dynkin diagrams D(A), i.e. diagrams using only or − connections (which correspond to a symmetric Cartan matrix A). Such a diagram contains no cycles, since otherwise it contains D(A~r). It has simple edges and it may or may not contain branching vertices. If there are no branching vertices, we get D(Ar). If there are two branching vertices, the diagram contains D(D~r), which is not possible. If there is precisely one branching vertex, it has at most 3 branches, since D(D~4) is the 4-star in Figure 3a. Therefore, it remains to consider the case when our graph D(A) is of the form Tp;q;r, p ≥ q ≥ r ≥ 2, depicted in Figure 3b. If r = 3, then Tp;q;r contains D(E~6) = T3;3;3, which is impossible, so r = 2. If q = 2, we have type D. Now, assume q ≥ 3. If q > 3, then D(A) contains D(E~7) = T4;4;2, which is impossible, so we are left with the case Tp;3;2 with p ≥ 3. The case p = 3 is E6, the case p = 4 is E7 and the case p = 5 is E8. The case p = 6 is E~8, so Tp;3;2 with p > 5 is impossible. Part 2. We now classify all non-simply-laced diagrams D(A), i.e. those containing double- or triple-edge connections (corresponding to a non-symmetric Cartan matrix A). This can be done by computing many large determinants, but we would rather argue using the following result: Exercise 2.1. Let A be an r × r matrix, and let B (resp. C) be the submatrices of A obtained by removing the first row and column (resp.
Load more

Recommended publications
  • Lie Algebras and Representation Theory Andreasˇcap
    Lie Algebras and Representation Theory Fall Term 2016/17 Andreas Capˇ Institut fur¨ Mathematik, Universitat¨ Wien, Nordbergstr. 15, 1090 Wien E-mail address: [email protected] Contents Preface v Chapter 1. Background 1 Group actions and group representations 1 Passing to the Lie algebra 5 A primer on the Lie group { Lie algebra correspondence 8 Chapter 2. General theory of Lie algebras 13 Basic classes of Lie algebras 13 Representations and the Killing Form 21 Some basic results on semisimple Lie algebras 29 Chapter 3. Structure theory of complex semisimple Lie algebras 35 Cartan subalgebras 35 The root system of a complex semisimple Lie algebra 40 The classification of root systems and complex simple Lie algebras 54 Chapter 4. Representation theory of complex semisimple Lie algebras 59 The theorem of the highest weight 59 Some multilinear algebra 63 Existence of irreducible representations 67 The universal enveloping algebra and Verma modules 72 Chapter 5. Tools for dealing with finite dimensional representations 79 Decomposing representations 79 Formulae for multiplicities, characters, and dimensions 83 Young symmetrizers and Weyl's construction 88 Bibliography 93 Index 95 iii Preface The aim of this course is to develop the basic general theory of Lie algebras to give a first insight into the basics of the structure theory and representation theory of semisimple Lie algebras. A problem one meets right in the beginning of such a course is to motivate the notion of a Lie algebra and to indicate the importance of representation theory. The simplest possible approach would be to require that students have the necessary background from differential geometry, present the correspondence between Lie groups and Lie algebras, and then move to the study of Lie algebras, which are easier to understand than the Lie groups themselves.
    [Show full text]
  • Note on the Decomposition of Semisimple Lie Algebras
    Note on the Decomposition of Semisimple Lie Algebras Thomas B. Mieling (Dated: January 5, 2018) Presupposing two criteria by Cartan, it is shown that every semisimple Lie algebra of finite dimension over C is a direct sum of simple Lie algebras. Definition 1 (Simple Lie Algebra). A Lie algebra g is Proposition 2. Let g be a finite dimensional semisimple called simple if g is not abelian and if g itself and f0g are Lie algebra over C and a ⊆ g an ideal. Then g = a ⊕ a? the only ideals in g. where the orthogonal complement is defined with respect to the Killing form K. Furthermore, the restriction of Definition 2 (Semisimple Lie Algebra). A Lie algebra g the Killing form to either a or a? is non-degenerate. is called semisimple if the only abelian ideal in g is f0g. Proof. a? is an ideal since for x 2 a?; y 2 a and z 2 g Definition 3 (Derived Series). Let g be a Lie Algebra. it holds that K([x; z; ]; y) = K(x; [z; y]) = 0 since a is an The derived series is the sequence of subalgebras defined ideal. Thus [a; g] ⊆ a. recursively by D0g := g and Dn+1g := [Dng;Dng]. Since both a and a? are ideals, so is their intersection Definition 4 (Solvable Lie Algebra). A Lie algebra g i = a \ a?. We show that i = f0g. Let x; yi. Then is called solvable if its derived series terminates in the clearly K(x; y) = 0 for x 2 i and y = D1i, so i is solv- trivial subalgebra, i.e.
    [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]
  • Lecture 5: Semisimple Lie Algebras Over C
    LECTURE 5: SEMISIMPLE LIE ALGEBRAS OVER C IVAN LOSEV Introduction In this lecture I will explain the classification of finite dimensional semisimple Lie alge- bras over C. Semisimple Lie algebras are defined similarly to semisimple finite dimensional associative algebras but are far more interesting and rich. The classification reduces to that of simple Lie algebras (i.e., Lie algebras with non-zero bracket and no proper ideals). The classification (initially due to Cartan and Killing) is basically in three steps. 1) Using the structure theory of simple Lie algebras, produce a combinatorial datum, the root system. 2) Study root systems combinatorially arriving at equivalent data (Cartan matrix/ Dynkin diagram). 3) Given a Cartan matrix, produce a simple Lie algebra by generators and relations. In this lecture, we will cover the first two steps. The third step will be carried in Lecture 6. 1. Semisimple Lie algebras Our base field is C (we could use an arbitrary algebraically closed field of characteristic 0). 1.1. Criteria for semisimplicity. We are going to define the notion of a semisimple Lie algebra and give some criteria for semisimplicity. This turns out to be very similar to the case of semisimple associative algebras (although the proofs are much harder). Let g be a finite dimensional Lie algebra. Definition 1.1. We say that g is simple, if g has no proper ideals and dim g > 1 (so we exclude the one-dimensional abelian Lie algebra). We say that g is semisimple if it is the direct sum of simple algebras. Any semisimple algebra g is the Lie algebra of an algebraic group, we can take the au- tomorphism group Aut(g).
    [Show full text]
  • Holomorphic Diffeomorphisms of Complex Semisimple Lie Groups
    HOLOMORPHIC DIFFEOMORPHISMS OF COMPLEX SEMISIMPLE LIE GROUPS ARPAD TOTH AND DROR VAROLIN 1. Introduction In this paper we study the group of holomorphic diffeomorphisms of a complex semisimple Lie group. These holomorphic diffeomorphism groups are infinite di- mensional. To get additional information on these groups, we consider, for instance, the following basic problem: Given a complex semisimple Lie group G, a holomor- phic vector field X on G, and a compact subset K ⊂⊂ G, the flow of X, when restricted to K, is defined up to some nonzero time, and gives a biholomorphic map from K onto its image. When can one approximate this map uniformly on K by a global holomorphic diffeomorphism of G? One condition on a complex manifold which guarantees a positive solution of this problem, and which is possibly equiva- lent to it, is the so called density property, to be defined shortly. The main result of this paper is the following Theorem Every complex semisimple Lie group has the density property. Let M be a complex manifold and XO(M) the Lie algebra of holomorphic vector fields on M. Recall [V1] that a complex manifold is said to have the density property if the Lie subalgebra of XO(M) generated by its complete vector fields is a dense subalgebra. See section 2 for the definition of completeness. Since XO(M) is extremely large when M is Stein, the density property is particularly nontrivial in this case. One of the two main tools underlying the proof of our theorem is the general notion of shears and overshears, introduced in [V3].
    [Show full text]
  • Note to Users
    NOTE TO USERS This reproduction is the best copy available. The Rigidity Method and Applications Ian Stewart, Department of hIathematics McGill University, Montréal h thesis çubrnitted to the Faculty of Graduate Studies and Research in partial fulfilrnent of the requirements of the degree of MSc. @[an Stewart. 1999 National Library Bibliothèque nationale 1*1 0fC-& du Canada Acquisitions and Acquisilions et Bibliographie Services services bibliographiques 385 Wellington &eet 395. nie Weüington ôuawa ON KlA ON4 mwaON KlAW Canada Canada The author has granted a non- L'auteur a accordé une Licence non exclusive licence aiiowing the exclusive pennettant à la National Lïbrary of Canada to Brbhothèque nationale du Canada de reproduce, Ioan, distri'bute or sell reproduire, prêter, distribuer ou copies of this thesis m microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/filnl de reproduction sur papier ou sur format éIectronique. The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or otherwise de ceiie-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. Abstract The Inverse Problem ol Gnlois Thcor? is (liscuss~~l.In a spticifiç forni. the problem asks tr-hether ewry finitr gruiip occurs aç a Galois groiip over Q. An iritrinsically group theoretic property caIIed rigidity is tlcscribed which confirnis chat many simple groups are Galois groups ovrr Q.
    [Show full text]
  • Geometries, the Principle of Duality, and Algebraic Groups
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Expo. Math. 24 (2006) 195–234 www.elsevier.de/exmath Geometries, the principle of duality, and algebraic groups Skip Garibaldi∗, Michael Carr Department of Mathematics & Computer Science, Emory University, Atlanta, GA 30322, USA Received 3 June 2005; received in revised form 8 September 2005 Abstract J. Tits gave a general recipe for producing an abstract geometry from a semisimple algebraic group. This expository paper describes a uniform method for giving a concrete realization of Tits’s geometry and works through several examples. We also give a criterion for recognizing the auto- morphism of the geometry induced by an automorphism of the group. The E6 geometry is studied in depth. ᭧ 2005 Elsevier GmbH. All rights reserved. MSC 2000: Primary 22E47; secondary 20E42, 20G15 Contents 1. Tits’s geometry P ........................................................................197 2. A concrete geometry V , part I..............................................................198 3. A concrete geometry V , part II .............................................................199 4. Example: type A (projective geometry) .......................................................203 5. Strategy .................................................................................203 6. Example: type D (orthogonal geometry) ......................................................205 7. Example: type E6 .........................................................................208
    [Show full text]
  • On Dynkin Diagrams, Cartan Matrices
    Physics 220, Lecture 16 ? Reference: Georgi chapters 8-9, a bit of 20. • Continue with Dynkin diagrams and the Cartan matrix, αi · αj Aji ≡ 2 2 : αi The j-th row give the qi − pi = −pi values of the simple root αi's SU(2)i generators acting on the root αj. Again, we always have αi · µ 2 2 = qi − pi; (1) αi where pi and qi are the number of times that the weight µ can be raised by Eαi , or lowered by E−αi , respectively, before getting zero. Applied to µ = αj, we know that qi = 0, since E−αj jαii = 0, since αi − αj is not a root for i 6= j. 0 2 Again, we then have AjiAij = pp = 4 cos θij, which must equal 0,1,2, or 3; these correspond to θij = π=2, 2π=3, 3π=4, and 5π=6, respectively. The Dynkin diagram has a node for each simple root (so the number of nodes is 2 2 r =rank(G)), and nodes i and j are connected by AjiAij lines. When αi 6= αj , sometimes it's useful to darken the node for the smaller root. 2 2 • Another example: constructing the roots for C3, starting from α1 = α2 = 1, and 2 α3 = 2, i.e. the Cartan matrix 0 2 −1 0 1 @ −1 2 −1 A : 0 −2 2 Find 9 positive roots. • Classify all simple, compact Lie algebras from their Aji. Require 3 properties: (1) det A 6= 0 (since the simple roots are linearly independent); (2) Aji < 0 for i 6= j; (3) AijAji = 0,1, 2, 3.
    [Show full text]
  • Semi-Simple Lie Algebras and Their Representations
    i Semi-Simple Lie Algebras and Their Representations Robert N. Cahn Lawrence Berkeley Laboratory University of California Berkeley, California 1984 THE BENJAMIN/CUMMINGS PUBLISHING COMPANY Advanced Book Program Menlo Park, California Reading, Massachusetts ·London Amsterdam Don Mills, Ontario Sydney · · · · ii Preface iii Preface Particle physics has been revolutionized by the development of a new “paradigm”, that of gauge theories. The SU(2) x U(1) theory of electroweak in- teractions and the color SU(3) theory of strong interactions provide the present explanation of three of the four previously distinct forces. For nearly ten years physicists have sought to unify the SU(3) x SU(2) x U(1) theory into a single group. This has led to studies of the representations of SU(5), O(10), and E6. Efforts to understand the replication of fermions in generations have prompted discussions of even larger groups. The present volume is intended to meet the need of particle physicists for a book which is accessible to non-mathematicians. The focus is on the semi-simple Lie algebras, and especially on their representations since it is they, and not just the algebras themselves, which are of greatest interest to the physicist. If the gauge theory paradigm is eventually successful in describing the fundamental particles, then some representation will encompass all those particles. The sources of this book are the classical exposition of Jacobson in his Lie Algebras and three great papers of E.B. Dynkin. A listing of the references is given in the Bibliography. In addition, at the end of each chapter, references iv Preface are given, with the authors’ names in capital letters corresponding to the listing in the bibliography.
    [Show full text]
  • Algebraic Groups of Type D4, Triality and Composition Algebras
    1 Algebraic groups of type D4, triality and composition algebras V. Chernousov1, A. Elduque2, M.-A. Knus, J.-P. Tignol3 Abstract. Conjugacy classes of outer automorphisms of order 3 of simple algebraic groups of classical type D4 are classified over arbitrary fields. There are two main types of conjugacy classes. For one type the fixed algebraic groups are simple of type G2; for the other type they are simple of type A2 when the characteristic is different from 3 and are not smooth when the characteristic is 3. A large part of the paper is dedicated to the exceptional case of characteristic 3. A key ingredient of the classification of conjugacy classes of trialitarian automorphisms is the fact that the fixed groups are automorphism groups of certain composition algebras. 2010 Mathematics Subject Classification: 20G15, 11E57, 17A75, 14L10. Keywords and Phrases: Algebraic group of type D4, triality, outer au- tomorphism of order 3, composition algebra, symmetric composition, octonions, Okubo algebra. 1Partially supported by the Canada Research Chairs Program and an NSERC research grant. 2Supported by the Spanish Ministerio de Econom´ıa y Competitividad and FEDER (MTM2010-18370-C04-02) and by the Diputaci´onGeneral de Arag´on|Fondo Social Europeo (Grupo de Investigaci´on de Algebra).´ 3Supported by the F.R.S.{FNRS (Belgium). J.-P. Tignol gratefully acknowledges the hospitality of the Zukunftskolleg of the Universit¨atKonstanz, where he was in residence as a Senior Fellow while this work was developing. 2 Chernousov, Elduque, Knus, Tignol 1. Introduction The projective linear algebraic group PGLn admits two types of conjugacy classes of outer automorphisms of order two.
    [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]
  • View This Volume's Front and Back Matter
    Functions of Several Complex Variables and Their Singularities Functions of Several Complex Variables and Their Singularities Wolfgang Ebeling Translated by Philip G. Spain Graduate Studies in Mathematics Volume 83 .•S%'3SL"?|| American Mathematical Society s^s^^v Providence, Rhode Island Editorial Board David Cox (Chair) Walter Craig N. V. Ivanov Steven G. Krantz Originally published in the German language by Friedr. Vieweg & Sohn Verlag, D-65189 Wiesbaden, Germany, as "Wolfgang Ebeling: Funktionentheorie, Differentialtopologie und Singularitaten. 1. Auflage (1st edition)". © Friedr. Vieweg & Sohn Verlag | GWV Fachverlage GmbH, Wiesbaden, 2001 Translated by Philip G. Spain 2000 Mathematics Subject Classification. Primary 32-01; Secondary 32S10, 32S55, 58K40, 58K60. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-83 Library of Congress Cataloging-in-Publication Data Ebeling, Wolfgang. [Funktionentheorie, differentialtopologie und singularitaten. English] Functions of several complex variables and their singularities / Wolfgang Ebeling ; translated by Philip Spain. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 83) Includes bibliographical references and index. ISBN 0-8218-3319-7 (alk. paper) 1. Functions of several complex variables. 2. Singularities (Mathematics) I. Title. QA331.E27 2007 515/.94—dc22 2007060745 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society.
    [Show full text]