Killing Forms of Lie Algebras Open Access

Malagon, Audrey Lynne (2009)

Permanent URL: https://etd.library.emory.edu/concern/etds/xw42n867q?locale=en
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Abstract

Abstract
One approach to the problem of classifying Lie Algebras is to find invari-
ants. One such invariant is the Killing form. In this dissertation, I give
a formula for computing the Killing form of any semisimple isotropic Lie
algebra defined over an arbitrary field of characteristic zero, based on the
Killing form of a subalgebra containing its anisotropic kernel. I then explic-
itly compute the Killing form for several Lie algebras of exceptional type
and give a general formula for the Killing form of all Lie algebras of inner
type E6, including the anisotropic ones.

Table of Contents

Contents 1 Introduction 1 2 Background Information I 2 2.1 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Central Simple Algebras and The Brauer Group . . . . . . . . 5 2.3 Cli ord Algebras and Merkurjev's Theorem . . . . . . . . . . 6 2.4 Cohomological Invariants . . . . . . . . . . . . . . . . . . . . . 7 3 Background Information II 10 3.1 Introduction to Lie Algebras . . . . . . . . . . . . . . . . . . . 10 3.2 Root Systems and Dynkin Diagrams . . . . . . . . . . . . . . 11 3.3 Killing Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.4 Chevalley Basis and Killing Form . . . . . . . . . . . . . . . . 21 4 Isotropic Lie Algebras 25 4.1 Tits Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 Tits Indices and F-Split Tori . . . . . . . . . . . . . . . . . . . 28 5 Previous Results on Killing Forms of Lie Algebras 38 6 Killing Forms of Isotropic Lie Algebras 41 6.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.2 Calculating the Killing form on ZH(A) . . . . . . . . . . . . . 43 7 Classical Results with New Method 46 7.1 Type 7.2 Type 7.3 Real Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 50 An . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Dn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 8 Results for Lie algebras of type 9 Results for Lie algebras of type 10 Results for Lie algebras of type Bibliography 65 1E6 532E6 59E7 63 4.3 Tits Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.4 Weight Space Decomposition . . . . . . . . . . . . . . . . . . . 32

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