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Lie Algebras Lie Algebras Brooks Roberts University of Idaho Course notes from 2018{2019 Contents 1 Basic concepts1 1.1 References...............................1 1.2 Motivation..............................1 1.3 The definition.............................1 1.4 Some important examples......................3 1.5 The adjoint homomorphism.....................5 2 Solvable and nilpotent Lie algebras7 2.1 Solvability...............................7 2.2 Nilpotency............................... 12 3 The theorems of Engel and Lie 15 3.1 The theorems............................. 15 3.2 Weight spaces............................. 15 3.3 Proof of Engel's Theorem...................... 17 3.4 Proof of Lie's Theorem........................ 20 4 Some representation theory 23 4.1 Representations............................ 23 4.2 Basic results.............................. 24 4.3 Representations of sl(2)....................... 25 5 Cartan's criteria 33 5.1 The Jordan-Chevalley decomposition................ 33 5.2 Cartan's first criterion: solvability................. 34 5.3 Cartan's second criterion: semi-simplicity............. 40 5.4 Simple Lie algebras.......................... 42 5.5 Jordan decomposition........................ 45 6 Weyl's theorem 51 6.1 The Casmir operator......................... 51 6.2 Proof of Weyl's theorem....................... 55 6.3 An application to the Jordan decomposition............ 58 iii iv CONTENTS 7 The root space decomposition 63 7.1 An associated inner product space................. 81 8 Root systems 83 8.1 The definition............................. 83 8.2 Root systems from Lie algebras................... 84 8.3 Basic theory of root systems..................... 85 8.4 Bases................................. 92 8.5 Weyl chambers............................ 98 8.6 More facts about roots........................ 101 8.7 The Weyl group............................ 104 8.8 Irreducible root systems....................... 114 9 Cartan matrices and Dynkin diagrams 123 9.1 Isomorphisms and automorphisms................. 123 9.2 The Cartan matrix.......................... 124 9.3 Dynkin diagrams........................... 129 9.4 Admissible systems.......................... 130 9.5 Possible Dynkin diagrams...................... 142 10 The classical Lie algebras 145 10.1 Definitions............................... 145 10.2 A criterion for semi-simplicity.................... 147 10.3 A criterion for simplicity....................... 148 10.4 A criterion for Cartan subalgebras................. 151 10.5 The Killing form........................... 153 10.6 Some useful facts........................... 155 10.7 The Lie algebra sl(` + 1)...................... 155 10.8 The Lie algebra so(2` + 1)..................... 161 10.9 The Lie algebra sp(2`)....................... 174 10.10The Lie algebra so(2`)........................ 182 11 Representation theory 193 11.1 Weight spaces again......................... 193 11.2 Borel subalgebras........................... 195 11.3 Maximal vectors........................... 196 11.4 The Poincar´e-Birkhoff-WittTheorem................ 201 Chapter 1 Basic concepts 1.1 References The main reference for this course is the book Introduction to Lie Algebras, by Karin Erdmann and Mark J. Wildon; this is reference [4]. Another important reference is the book [6], Introduction to Lie Algebras and Representation The- ory, by James E. Humphreys. The best references for Lie theory are the three volumes [1], Lie Groups and Lie Algebras, Chapters 1-3,[2], Lie Groups and Lie Algebras, Chapters 4-6, and [3], Lie Groups and Lie Algebras, Chapters 7-9, all by Nicolas Bourbaki. 1.2 Motivation Briefly, Lie algebras have to do with the algebra of derivatives in settings where there is a lot of symmetry. As a consequence, Lie algebras appear in various parts of advanced mathematics. The nexus of these applications is the theory of symmetric spaces. Symmetric spaces are rich objects whose theory has com- ponents from geometry, analysis, algebra, and number theory. With these short remarks in mind, in this course we will begin without any more motivation, and start with the definition of a Lie algebra. For now, rather than be con- cerned about advanced applications, the student should instead exercise critical thinking as basic concepts are introduced. 1.3 The definition Lie algebras are defined as follows. Throughout this chapter F be an arbitrary field. A Lie algebra over F is an F -vector space L and an F -bilinear map [·; ·]: L × L −! L that has the following two properties: 1 2 CHAPTER 1. BASIC CONCEPTS 1.[ x; x] = 0 for all x 2 L; 2.[ x; [y; z]] + [y; [z; x]] + [z; [x; y]] = 0 for all x; y; z 2 L. The map [·; ·] is called the Lie bracket of L. The second property is called the Jacobi identity. Proposition 1.3.1. Let L be a Lie algebra over F . If x; y 2 L, then [x; y] = −[y; x]. Proof. Let x; y 2 L. Then 0 = [x + y; x + y] = [x; x] + [x; y] + [y; x] + [y; y] = [x; y] + [y; x]; so that [x; y] = −[y; x]. If L1 and L2 are Lie algebras over F , then a homomorphism T : L1 ! L2 is an F -linear map that satisfies T ([x; y]) = [T (x);T (y)] for all x; y 2 L1. If L is a Lie algebra over F , then a subalgebra of L is an F -vector subspace K of L such that [x; y] 2 K for all x; y 2 K; evidently, a subalgebra is a Lie algebra over F using the same Lie bracket. If L is a Lie algebra over F , then an ideal I of L is an F -vector subspace of L such that [x; y] 2 I for all x 2 L and y 2 I; evidently, an ideal of L is also a subalgebra of A. Also, because of Proposition 1.3.1, it is not necessary to introduce the concepts of left or right ideals. If L is a Lie algebra over F , then the center of L is defined to be Z(L) = fx 2 L :[x; y] = 0 for all y 2 Lg: Clearly, the center of L is an F -subspace of L. Proposition 1.3.2. Let L be a Lie algebra over F . The center Z(L) of L is an ideal of L. Proof. Let y 2 L and x 2 Z(L). If z 2 L, then [[y; x]; z] = −[[x; y]; z] = 0: This implies that [y; x] 2 Z(L). If L is a Lie algebra over F , then we say that L is abelian if Z(L) = L, i.e., if [x; y] = 0 for all x; y 2 L. Proposition 1.3.3. Let L1 and L2 be Lie algebras over F , and let T : L1 ! L2 be a homomorphism. The kernel of T is an ideal of L1. Proof. Let y 2 ker(T ) and x 2 L1. Then T ([x; y]) = [T (x);T (y)] = [T (x); 0] = 0, so that [x; y] 2 ker(T ). 1.4. SOME IMPORTANT EXAMPLES 3 1.4 Some important examples Proposition 1.4.1. Let A be an associative F -algebra. For x; y 2 A define [x; y] = xy − yx; so that [x; y] is just the commutator of x and y. With this definition of a Lie bracket, the F -vector space A is a Lie algebra. Proof. It is easy to verify that [·; ·] is F -bilinear and that property 1 of the definition of a Lie algebra is satisfied. We need to prove that the Jacobi identity is satisfied. Let x; y; z 2 A. Then [x; [y; z]] + [y; [z; x]] + [z; [x; y]] = x(yz − zy) − (yz − zy)x + y(zx − xz) + (zx − xz)y + z(xy − yx) + (xy − yx)z = xyz − xzy − yzx + zyx + yzx − yxz − zxy + xzy + zxy − zyx − xyz + yxz = 0: This completes the proof. Note that in the last proof we indeed used that the algebra was associative. If V is an F -vector space, then the F -vector space gl(V ) of all F -linear operators from V to V is an associative algebra over F under composition, and thus defines a corresponding Lie algebra over F , also denoted by gl(V ), with Lie bracket as defined in Proposition 1.4.1. Similarly, if n is a non-negative integer, then F -vector space gl(n; F ) of all n×n matrices is an associative algebra under multiplication of matrices, and thus defines a corresponding Lie algebra, also denoted by gl(n; F ). The example gl(n; F ) shows that in general the Lie bracket is not associative, i.e., it is not in general true that [x; [y; z]] = [[x; y]; z] for all x; y; z 2 gl(n; F ). For example, if n = 2, and 1 1 x = ; y = ; z = 1 1 then 1 [x; [y; z]] = x(yz − zy) = xyz − xzy = xyz − = xyz 1 and 1 1 [[x; y]; z] = (xy − yx)z = xyz − yxz = xyz − = xyz − : 1 We describe some more important examples of Lie algebras. 4 CHAPTER 1. BASIC CONCEPTS Proposition 1.4.2. Let n be a non-negative integer, and let sl(n; F ) be the subspace of gl(n; F ) consisting of elements x such that tr(x) = 0. Then sl(n; F ) is a Lie subalgebra of gl(n; F ). Proof. It will suffice to prove that tr([x; y]) = 0 for x; y 2 sl(n; F ). Let x; y 2 sl(n; F ). Then tr([x; y]) = tr(xy − yx) = tr(xy) − tr(yx) = tr(xy) − tr(xy) = 0. The example sl(2;F ) is especially important. We have a b sl(2;F ) = f : a; b; c 2 F g: c −a An important basis for sl(2;F ) is 1 1 e = ; f = ; h = : 1 −1 We have: [e; f] = h; [e; h] = −2e; [f; h] = 2f: Proposition 1.4.3. Let n be a non-negative integer, and let S 2 gl(n; F ). Let t glS(n; F ) = fx 2 gl(n; F ): xS = −Sxg: Then glS(n; F ) is a Lie subalgebra of gl(n; F ). Proof. Let x; y 2 glS(n; F ).
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