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Lie Algebras and Triple Systems Alma Mater Studiorum · Universita` di Bologna Scuola di Scienze Corso di Laurea Magistrale in Matematica Lie algebras and triple systems Tesi di Laurea in Algebra Relatore: Presentata da: Chiar.ma Prof.ssa Antonio Ricciardo Nicoletta Cantarini III Sessione Anno accademico 2013/2014 A chi non c'`epi`u... Introduction Jordan algebras first appeared in a 1933 paper by P. Jordan on the foun- dations of quantum mechanics. The classification of simple finite-dimensional Jordan algebras over an algebraically closed field of characteristic different from two was obtained by Albert, [1], in 1947 but a much easier proof of this clas- sification was given in the 60's, thanks to the discovery of the Tits-Kantor- Koecher (TKK) construction, [10][5][7]. This is based on the observation that if g = g−1 ⊕ g0 ⊕ g1 is a Lie algebra with a short Z-grading and f lies in g1, then the formula a • b = [[a; f]; b] defines a structure of a Jordan algebra on g−1. This leads to a bijective corre- spondence between simple unital Jordan algebras and simple Lie algebras with an sl2-triple ff; h; eg whose semisimple element h, with eigenvalues 0; −1; 1; defines a short grading of g. Over the years the TKK construction has revealed more and more relevant, due to its many generalizations. The first natural generalization is to Jordan triple systems, whose algebraic study was initiated by K. Meyberg in 1969. A Jordan triple system is a 3-algebra whose product f · ; · ; · g satisfies the following identities: fx; y; zg = fz; y; xg fu; v; fx; y; zgg = ffu; v; xg; y; zg − fx; fv; u; yg; zg + [x; y; [u; v; z]] Another natural generalization is to superalgebras: using the TKK construc- tion V. Kac, [4], obtained in 1977 the classification of simple finite-dimensional Jordan superalgebras over a field of characteristic zero, from the classification of simple finite-dimensional Lie superalgebras. More recently, the same ideas were generalized by N. Cantarini and V. Kac, [2], in order to establish the equi- valence of the category of unital linearly compact Jordan superalgebras and the category of linearly compact Lie superalgebras with a short subalgebra. This 3 4 equivalence lead to the classification of infinite-dimensional linearly compact simple Jordan superalgebras. At the same time, J. Palmkvist, [8][9], studied how to extend the TKK construction to the so-called Kantor triple systems. These are a class of triple systems including Jordan triple systems. In this case a Z-graded Lie algebra of length 5, g = g−2 + g−1 + g0 + g1 + g2, is associated to a Kantor triple system. This construction is undoubtedly more complicated, both from a conceptual and a technical point of view. It is worth mentioning that in the latest years triple systems have found several applications to different branches of physics, in particular to 3-dimensional supersymmetric gauge theories. For this reason the physicists community has shown great interest in these algebraic structures. The thesis is divided into three chapters. In the first chapter the preliminary material on Jordan and Z-graded Lie algebras is presented. The second chapter is dedicated to the Tits-Kantor-Koecher construction which is described in all details. In the third chapter the generalization of the TKK construction to triple systems is given. Also in this case, all details are provided. In Chapter 3, some examples are given, namely, the TKK construction is described in the case of g = sl2, sl4 and sp4 (with short gradings induced by sl2-triples). Introduzione Le algebre di Jordan fanno la loro prima apparizione nel 1933 in un artico- lo di P. Jordan sui fondamenti della meccanica quantistica. La classificazione delle algebre di Jordan semplici finito dimensionali su un campo algebricamente chiuso di caratteristica diversa da due viene ottenuta da Albert, [1], nel 1947 ma una dimostrazione meno complicata di questa classificazione viene data solo negli anni sessanta grazie alla scoperta della costruzione di Tits-Kantor-Koecher (TKK), [10][5][7]. Essa si basa sull'osservazione che se g = g−1 ⊕ g0 ⊕ g1 `eun'algebra di Lie con una Z-graduazione corta ed f appartiene a g1, allora il prodotto a • b = [[a; f]; b] definisce una struttura di algebra di Jordan su g−1. Ne deriva una corrispon- denza biunivoca tra algebre di Jordan semplici con unit`ae algebre di Lie con una sl2-tripla ff; h; eg il cui elemento semisemplice h, con autovalori 0; −1; 1; definisce una Z-graduazione corta su g. Nel corso degli anni la costruzione TKK si `erivelata sempre pi`uimportante, grazie alle sue molteplici generalizzazioni. Una prima naturale generalizzazione `eai Jordan triple systems, il cui studio viene cominciato da K. Meyberg nel 1969. Un Jordan triple system `euna 3-algebra il cui prodotto f · ; · ; · g soddisfa le seguenti relazioni: fx; y; zg = fz; y; xg fu; v; fx; y; zgg = ffu; v; xg; y; zg − fx; fv; u; yg; zg + [x; y; [u; v; z]] Un'altra generalizzazione `ealle superalgebre: usando la costruzione TKK V. Kac, [4], ottiene nel 1977 la classificazione delle superalgebre di Jordan semplici finito dimensionali su un campo di caratteristica zero, tramite la classificazione delle superalgebre di Lie semplici di dimensione finita. Di recente, generaliz- zando la stessa idea N. Cantarini e V. Kac, [2], dimostrano l'equivalenza tra la 5 6 categoria delle superalgebre di Jordan unitarie linearmente compatte e la catego- ria delle superalgebre di Lie linearmente compatte con una Z-graduazione corta. Grazie a questa equivalenza viene ottenuta la classificazione delle superalgebre di Jordan semplici linearmente compatte infinito dimensionali. Contemporaneamente, J. Palmkvist, [8][9], estende la costruzione TKK ai Kantor triple systems. Questi ultimi costituiscono una classe di triple systems contenente i Jordan triple systems. In questo caso viene associata ad un Kantor triple system un'algebra di Lie Z-graduata di lunghezza 5, g = g−2 + g−1 + g0 + g1 + g2. Nel caso dei Kantor triple systems la costruzione si rivela senza dubbio pi`ucomplicata, sia concettualmente che tecnicamente. Vale la pena di sottolineare che negli ultimi anni i triple systems hanno trovato numerose applicazioni a branche diverse della fisica, in particolare alle teorie di gauge tridimensionali supersimmetriche. Per questo motivo la comunit`a fisica ha rivolto un grande interesse a queste stutture algebriche. La tesi si divide in tre capitoli. Nel primo vengono introdotti definizioni ed esempi di algebre di Jordan e di algebre di Lie Z-graduate. Il secondo capitolo `ededicato alla costruzione di Tits-Kantor-Koecher descritta in ogni dettaglio. Nel terzo capitolo la costruzione TKK viene estesa ai triple systems. Anche in questo caso vengono forniti tutti i dettagli. Inoltre, nel terzo capitolo vengono trattati gli esempi g = sl2, sl4 e sp4 (con Z-graduazione corta indotta da una sl2-tripla). Contents Lie algebras and triple systems1 Introduction3 Introduzione5 Contents7 1 Preliminary notions9 1.1 Jordan algebras and Lie algebras..................9 2 The Tits-Kantor-Koecher construction 13 2.1 A construction of shortly graded Lie algebras........... 13 2.1.1 Ideals of L........................... 15 2.2 The Tits-Kantor-Koecher construction............... 17 3 Lie algebras and triple systems 23 3.1 Triple systems............................. 23 3.1.1 Universal Lie algebras.................... 26 3.2 The Meyberg Theorem........................ 28 3.3 Lie algebras and Jordan triple systems............... 36 3.4 Lie algebras and Kantor triple systems............... 38 Bibliography 53 7 8 CONTENTS Chapter 1 Preliminary notions on algebras 1.1 Jordan algebras and Lie algebras In what follows we will denote by F the base field. We will always assume char F = 0. Definition 1.1 (Algebra). An algebra (A; · ) is an F-vector space A with a product, i.e., a bilinear map · : A × A ! A: We will say that A is associative if the product satisfies the following relation (x · y) · z = x · (y · z) (1.1) Definition 1.2 (Jordan algebra). A Jordan algebra is an algebra (A; · ) whose product satisfies the following axioms: x · y = y · x (commutativity) (1.2) (x2 · y) · x − x2 · (y · x) = 0 (Jordan identity) Definition 1.3 (Lie algebra). A Lie algebra is an algebra (A; [ ; ] ) whose product satisfies the following axioms: [x; y] = −[y:x](anticommutativity) (1.3) [x; [y; z]] + [y; [z; x]] + [z; [x; y]] = 0 (Jacobi identity) Example 1.1.1. Let (A; · ) be an associative algebra. Then: 9 10 1. Preliminary notions + 1 a) A = (A; •), where x • y = 2 (x · y + y · x), is a Jordan algebra. Indeed • is commutative and it satisfies the Jordan identity since · is associative: 1 1 1 (x2 • y) • x − x2 • (y • x) = ( (x2 · y + y · x2) · x + x · (x2 · y + y · x2))− 2 2 2 1 1 1 − (x2 · (x · y + y · x) + (x · y + y · x) · x2) = 2 2 2 1 = (x2 · y · x + y · x3 + x3 · y + x · y · x2)− 4 1 − (x3 · y + x2 · y · x + x · y · x2 + y · x3) = 0 4 − 1 b) A = (A; [ ; ]) where [x; y] = 2 (x · y − y · x) is a Lie algebra. The product [ ; ] is anti-commutative and satisfies the Jacobi identity. Indeed: 1 1 1 [x; [y; z]] + [y; [z; x]] + [z; [x; y]] = (x · (y · z − z · y) − (y · z − z · y) · x)+ 2 2 2 1 1 1 1 1 1 + (y· (z·x−x·z)− (z·x−x·z)·y)+ (z· (x·y−y·x)− (x·y−y·x)·z) = 2 2 2 2 2 2 1 = (x · y · z − x · z · y − y · z · x + z · y · x)+ 4 1 + (y · z · x − y · x · z − z · x · y + x · z · y)+ 4 1 + (z · x · y − z · y · x − x · y · z + y · x · z) = 0 4 Definition 1.4 (gl(V )).
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