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Representation Theory of Finite Dimensional Lie Algebras

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Representation Theory of Finite Dimensional Lie Algebras Representation Theory of Finite dimensional Lie algebras July 21, 2015 2 Contents 1 Introduction to Lie Algebras 5 1.1 Basic Definitions . .5 1.2 Definition of the classical simple Lie algebras . 10 1.3 Nilpotent and solvable Lie algebras . 11 1.4 g-modules, basic definitions . 14 1.5 Testing solvability and semisimplicity . 17 1.6 Jordan Decomposition and Proof of Cartan's Criterion . 20 1.6.1 Properties . 22 1.7 Theorems of Levi and Malcev . 23 1.7.1 Weyl's complete irreducibility theorem. 26 1.7.2 Classification of irreducible finite dimensional sl(2; C) modules . 31 1.8 Universal enveloping algebras . 33 2 Representations of Lie algebras 45 2.1 Constructing new representations . 45 2.1.1 Pull-back and restriction . 45 2.1.2 Induction . 46 2.2 Verma Modules . 49 2.3 Abstract Jordan Decomposition . 52 3 Structure Theory of Semisimple Complex Lie Algebras 57 3.1 Root Space Decomposition . 58 3.2 Root Systems . 64 3.2.1 Changing scalar . 69 3.2.2 Bases of root system . 75 3.2.3 Weyl Chambers . 77 3 4 CONTENTS 3.2.4 Subsets of roots . 80 3.2.5 Classification of a parabolic subset over a fixed R+(B) . 82 3.3 Borel and Parabolic subalgebras of a complex semi simple Lie algebra. 82 4 Highest Weight Theory 85 4.1 Construction of highest weight modules . 91 4.2 Character formula . 96 4.3 Category O ............................ 97 4.4 Cartan matrices and Dynkin diagrams . 100 4.4.1 Classification of irreducible, reduced root systems/ Dynkin Diagrams . 105 Literature Humphreys: • Introduction to Lie algebras and Representation Theory • Complex Reflection Groups • Representations of semi simple Lie Algebras • Knapp: Lie groups: beyond an introduction • V.S. Varadarajan: Lie Groups, Lie Algebras and their Representations Chapter 1 Introduction to Lie Algebras (Lecture 1) 1.1 Basic Definitions Definition 1.1.1. A Lie algebra is a vector space g over some field k, together with a bilinear map [−; −]: g × g ! g such that: [x; x] = 08x 2 g(anti-symmetry) (1.1) [x; [y; z]] + [y; [z; x]] + [z; [x; y]](Jacobi identity) (1.2) Remarks. • [−; −]is called a Lie bracket. •8 x; y 2 g; [x; y] = −[y; x](because 0 = [x + y; x + y] = [x; x] + [x; y] + [y; x] + [y; y]) • Often: Say a k-Lie algebra if we have a Lie algebra over k. Examples. V a k-vector space, [v; w] = 08v; w 2 V defines a k-Lie algebra. Any associative k-algebra is naturally a k-Lie algebra by [a; b] := ab − ba8a; b 2 A . Excercise: check 1.2. 5 6 CHAPTER 1. INTRODUCTION TO LIE ALGEBRAS In particular: The vector space gl(n; k) = fA 2 Mn×n(k)g is a lie algebra via [A; B] = AB − BA (usual commutator of matrices). More generally: gl(V) = fρ :V ! Vk − linearg; V a k-vector space, is a lie algebra via [f; g] = f ◦ g − g ◦ f 8f; g 2 gl(V). These are the general linear Lie algebras. Let g1; g2 be Lie algebras, then g1 ⊕ g2 is a Lie algebra via [(x; y); (x0; y0)] = [[x; x0]; [y; y0]](take the Lie algebra component wise) . The Lie algebra g1 ⊕ g2 is called the direct sum of g1 and g2. Definition 1.1.2. Given g1; g2 k-Lie algebras, a morphism f : g1 ! g2 of k-Lie algebras is a k-linear map such that f([x; y]) = [f(x); f(y)]. Remarks. • id : g ! g is a Lie algebra homomorphism. • f : g1 ! g2; g : g2 ! g3 Lie algebra homomorphisms, then g ◦ f : g1 ! g2 is a Lie algebra homomorphism. (because g ◦ f([x; y]) = g([f(x); f(y)]) = [g ◦ f(x); g ◦ f(y)]): Hence: k-Lie algebras with Lie algebra homomorphisms form a cate- gory. Example 1.1.3. Let g be a Lie algebra. ad :g ! gl(g) x 7! ad(x) where ad(x)(y) = [x; y] 8x; y 2 g is a lie algebra homomorphism. It is called the adjoint representation. (because: linear clear, since [−; −] is bilinear, and [ad(x); ad(y)](z) = ad(x) ad(y)(z) − ad(y) ad(x)(z) = [x; [y; z]] − [y; [x; z]] Jacobi=−id [[x; y]; z] = ad([x; y])(z):) 1.1. BASIC DEFINITIONS 7 Definition 1.1.4. Let g be a k-Lie algebra, l ⊂ g a vector subspace. • l is called a sub-lie algebra if [x; y] 2 l for all x; y 2 l. • l is called an ideal if [x; y] 2 l for all x 2 l, y 2 g.(denoted l C g ( () [x; y] 2 l for all x 2 g, y 2 l)). Given a Lie algebra g,I C g, then the vector space g=I becomes a Lie algebra via [x + I; y + I] = [x; y] + I. To check this is well defined: Assume x+I = x0+I; y+I = y0+I =) x0 = x+u; y0 = y+v; u; v 2 I =) [x0 + I; y0 + I] = [u + x + I; v + y + I] = [u + x; v + y] + I = [u; v] + [u; y] + [x; v] + [x; y] + I = [x; y] + I =) well defined. Proposition 1.1.5. Let f : g1 ! g2 be a Lie algebra homomorphism. Then 1 ker(f) C g1. 2 im(f) is a Lie subalgebra of g2. 3 Universal Property: If I C g1; ker(f) ⊂ I, the following diagram com- mutes: f x g1 - g2 @ @ @ @R ? x + I 2 g1=I ∼ In particular: g1= ker(f) = im(f) is an isomorphism of Lie algebras. Proof. : Standard. Remark 1.1.6. There are the usual isomorphism Theorems: a.I ; J C g; I ⊆ J, then J=I C g=I and (g=I)=(J=I) ∼= g=J 8 CHAPTER 1. INTRODUCTION TO LIE ALGEBRAS b.I ; J C g. Then I + J C g;I \ J C g, and I=I \ J ∼= I + J=J Definition 1.1.7. A Lie algebra g is called simple if g contains no ideals I C g except for I = 0 or I = g, and if [g; g] 6= 0. Def: Remark 1.1.8. [g; g] = 0 () [x; y] = 08x; y 2 g (i.e g is abelian). If g is simple, then: a. Z(g) = fx 2 g :[x; y] = 08y 2 gg b.[ g; g] = g (because: Z(g)Cg and g not abelian, hence Z(g) = 0); Z(g)Cg because 8x; y 2 g; z 2 Z(g): [[x; z]; y] = [x; [z; y]] + [z; [y; x]] [g; g] is obviously an ideal, because for x; y; z 2 g,[x; [y; z]] 2 [g; g]. [g; g] is called the derived lie algebra of g. Also, [g; g] is the smallest ideal of g such that g=[g; g] is abelian. Example 1.1.9. sl(2; k) := fA 2 Mat(2 × 2; k)j Tr(A) = 0g Is a lie subalgebra of gl(2; k), because Tr([A; B]) = Tr([A; B]) = Tr(AB − BA) = 0 8A; B 2 sl(2; k). Fact 1.1.10. sl(2; k) is simple () char(k) 6= 2. Proof. Choose a standard basis (sl2 "triple") as follows: 0 1 0 0 1 0 e = 0 0 f = 1 0 h = 0 −1 Then we have (excercise!): [h; e] = 2e [h; f] = −2f [e; f] = h These relations define the lie bracket. If char(k) = 2, then the vector subspace h spanned by h is a non-trivial ideal because [e; h] = 0; [f; h] = 0. Since [h; h] = 0, then g = sl(2; k) is not simple. 1.1. BASIC DEFINITIONS 9 Remark 1.1.11. gl(n; k) is not simple, because 1 spans a non-trivial ideal (Note Z(gl(n; k)) 6= 0). Lemma 1.1.12. ker(ad(g) ! gl(g)) = fx 2 gj ad(x) = 0g = fx 2 gj[x; y] = 08x; y 2 gg = Z(g). Proof. Just definitions. Corollary 1.1.13. Let g be a simple Lie algebra, then g is a linear Lie algebra (i.e. g is a Lie subalgebra of some Lie algebra of matrices, i.e., of gl(V) for some vector space V). Proof. Since g is simple, then Z(g) = 0, which implies ad is injective, hence g ∼= ad(g) is an isomorphism of Lie algebras, and ad(g) is a Linear lie algebra. Theorem 1.1.14. (Theorem od Ado) Every finite dimensional Lie algebra is linear. Proof. Later. Theorem 1.1.15. (Cartan-Killing Classification) Every complex finite-dimensional simple Lie algebra is isomorphic to exactly one of the following list: • Classical Lie Algebras: An sl(n + 1; C); n ≥ 1 Bn so(2n + 1; C); n ≥ 2 Cn sp(2n; C); n ≥ 3 Dn so(2n; C); n ≥ 4 • Exceptional Lie Algebras: E6 E7 E8 F4 10 CHAPTER 1. INTRODUCTION TO LIE ALGEBRAS G2 Remark 1.1.16. Every finite dimensional complex Lie algebra which is a direct sum of simple Lie algebras is called semi-simple. Remark 1.1.17. (Without further explanations) connected, compact Lie group with trivial center 1:1! s.s. complex f.d. Lie alg G 7! C ⊗R Lie(G) Where Lie(G) is the tangent space of G at the origin. −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− (Lecture 2- 6th April 2011) 1.2 Definition of the classical simple Lie al- gebras • sl(n + 1; C) = fA 2 gl(n + 1; C) : Tr(A) = 0g • sp(2n; C) = fA 2 gl(2n; C): f(Ax; y) = −f(x; Ay)8x; y 2 C2ng where f is the bilinear form given by f(x; y) := xtMy where M = 0 In .
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