Notes for Math 261A Lie groups and Lie algebras June 28, 2006 Contents Contents ......................... 1 How these notes came to be ................. 3 Lecture 1 ........................ 4 Lecture 2 ........................ 6 Tangent Lie algebras to Lie groups, 6 Lecture 3 ........................ 9 Lecture 4 ........................ 12 Lecture 5 ........................ 16 Simply Connected Lie Groups, 16 Lecture 6 - Hopf Algebras .................. 21 The universal enveloping algebra, 24 Lecture 7 ........................ 26 Universality of Ug, 26 Gradation in Ug, 27 Filtered spaces and algebras, 28 Lecture 8 - PBW theorem, Deformations of Associative Algebras . 31 Deformations of associative algebras, 32 Formal deformations of associative algebras, 34 Formal deformations of Lie algebras, 35 Lecture 9 ........................ 36 Lie algebra cohomology, 37 Lecture 10 ........................ 41 Lie algebra cohomology, 41 H2(g, g) and Deformations of Lie algebras, 42 Lecture 11 - Engel’s Theorem and Lie’s Theorem ......... 46 The radical, 51 Lecture 12 - Cartan Criterion, Whitehead and Weyl Theorems . 52 Invariant forms and the Killing form, 52 Lecture 13 - The root system of a semi-simple Lie algebra ..... 59 Irreducible finite dimensional representations of sl(2), 63 Lecture 14 - More on Root Systems .............. 65 Abstract Root systems, 66 The Weyl group, 68 Lecture 15 - Dynkin diagrams, Classification of root systems . 71 Isomorphisms of small dimension, 76 1 2 Lecture 16 - Serre’s Theorem................. 78 Lecture 17 - Constructions of Exceptional simple Lie Algebras . 82 Lecture 18 - Representations ................. 87 Representations, 87 Highest weights, 89 Lecture 19 - The Weyl character formula ............ 93 The character formula, 94 Lecture 20 ........................100 Compact Groups, 102 Lecture 21 ........................105 What does a general Lie group look like?, 105 Comparison of Lie groups and Lie algebras, 107 Finite groups and Lie groups, 108 Algebraic groups (over R) and Lie groups, 109 Lecture 22 - Clifford algebras and Spin groups ..........111 Clifford Algebras and Spin Groups, 112 Lecture 23 ........................117 Clifford Groups, 118 Lecture 24 ........................123 Spin representations of Spin and Pin groups, 123 Triality, 126 More about Orthogonal groups, 127 Lecture 25 - E8 ......................129 Lecture 26 ........................134 Construction of E8, 136 Lecture 27 ........................140 Construction of the Lie group of E8, 143 Real forms, 144 Lecture 28 ........................146 Working with simple Lie groups, 148 Every possible simple Lie group, 150 Lecture 29 ........................152 Lecture 30 - Irreducible unitary representations of SL2(R) . 159 Finite dimensional representations, 159 The Casimir operator, 162 Lecture 31 - Unitary representations of SL2(R) .. .. .. .165 Background about infinite-dimensional representations, 165 The group SL2(R), 166 Solutions to (some) Exercises.................171 References ........................176 Index ..........................178 3 How these notes came to be Among the Berkeley professors, there was once Alan Knutson, who would teach Math 261. But it happened that professor Knutson was on sabbati- cal elsewhere (I don’t know where for sure), and eventually went there (or somewhere else) for good. During this turbulent time, Maths 261AB were cancelled two years in a row. The last of these four semesters (Spring 2006), some graduate students gathered together and asked Nicolai Reshetikhin to teach them Lie theory in a giant reading course. When the dust settled, there were two other professors willing to help in the instruction of Math 261A, Vera Serganova and Richard Borcherds. Thus Tag Team 261A was born. After a few lectures, professor Reshetikhin suggested that the students write up the lecture notes for the benefit of future generations. The first four lectures were produced entirely by the “editors”. The remaining lectures were LATEXed by Anton Geraschenko in class and then edited by the people in the following table. The columns are sorted by lecturer. Nicolai Reshetikhin Vera Serganova Richard Borcherds 1 Anton Geraschenko 11 Sevak Mkrtchyan 21 Hanh Duc Do 2 Anton Geraschenko 12 Jonah Blasiak 22 An Huang 3 Nathan George 13 Hannes Thiel 23 Santiago Canez 4 Hans Christianson 14 Anton Geraschenko 24 Lilit Martirosyan 5 Emily Peters 15 Lilit Martirosyan 25 Emily Peters 6 Sevak Mkrtchyan 16 Santiago Canez 26 Santiago Canez 7 Lilit Martirosyan 17 Katie Liesinger 27 Martin Vito-Cruz 8 David Cimasoni 18 Aaron McMillan 28 Martin Vito-Cruz 9 Emily Peters 19 Anton Geraschenko 29 Anton Geraschenko 10 Qingtau Chen 20 Hanh Duc Do 30 Lilit Martirosyan 31 Sevak Mkrtchyan Richard Borcherds then edited the last third of the notes. The notes were further edited by Crystal Hoyt, Sevak Mkrtchyan, and Anton Geraschenko. Send corrections and comments to [email protected]. Lecture 1 4 Lecture 1 Definition 1.1. A Lie group is a smooth manifold G with a group structure such that the multiplication µ : G G G and inverse map ι : G G are × → → smooth maps. ◮ Exercise 1.1. If we assume only that µ is smooth, does it follow that ι is smooth? n Example 1.2. The group of invertible endomorphisms of C , GLn(C), is a Lie group. The automorphisms of determinant 1, SLn(C), is also a Lie group. Example 1.3. If B is a bilinear form on Cn, then we can consider the Lie group A GL (C) B(Av, Aw)= B(v, w) for all v, w Cn . { ∈ n | ∈ } If we take B to be the usual dot product, then we get the group On(C). T 0 1m If we let n = 2m be even and set B(v, w) = v 1m 0 w, then we get Sp (C). − 2m Example 1.4. SU SL is a real form (look in lectures 27,28, and 29 for n ⊆ n more on real forms). Example 1.5. We’d like to consider infinite matrices, but the multiplication wouldn’t make sense, so we can think of GL GL via A A 0 , then n ⊆ n+1 7→ 0 1 define GL as n GLn. That is, invertible infinite matrices which look like ∞ the identity almost everywhere. S Lie groups are hard objects to work with because they have global charac- teristics, but we’d like to know about representations of them. Fortunately, there are things called Lie algebras, which are easier to work with, and representations of Lie algebras tell us about representations of Lie groups. Definition 1.6. A Lie algebra is a vector space V equipped with a Lie bracket [ , ] : V V V , which satisfies × → 1. Skew symmetry: [a, a] = 0 for all a V , and ∈ 2. Jacobi identity: [a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0 for all a, b, c V . ∈ A Lie subalgebra of a lie algebra V is a subspace W V which is closed ⊆ under the bracket: [W, W ] W . ⊆ Example 1.7. If A is a finite dimensional associative algebra, you can set [a, b] = ab ba. If you start with A = M , the algebra of n n matri- − n × ces, then you get the Lie algebra gl . If you let A M be the algebra n ⊆ n of matrices preserving a fixed flag V V V Cn, then you get 0 ⊂ 1 ⊂ ··· k ⊆ parabolicindexparabolic subalgebras Lie sub-algebras of gln. Lecture 1 5 Example 1.8. Consider the set of vector fields on Rn, Vect(Rn) = ℓ = i ∂ { Σe (x) [ℓ1,ℓ2]= ℓ1 ℓ2 ℓ2 ℓ1 . ∂xi | ◦ − ◦ } ◮ Exercise 1.2. Check that [ℓ1,ℓ2] is a first order differential operator. Example 1.9. If A is an associative algebra, we say that ∂ : A A is a → derivation if ∂(ab) = (∂a)b + a∂b. Inner derivations are those of the form [d, ] for some d A; the others are called outer derivations. We denote the · ∈ set of derivations of A by (A), and you can verify that it is a Lie algebra. D Note that Vect(Rn) above is just (C (Rn)). D ∞ The first Hochschild cohomology, denoted H1(A, A), is the quotient (A)/ inner derivations . D { } Definition 1.10. A Lie algebra homomorphism is a linear map φ : L L → ′ that takes the bracket in L to the bracket in L′, i.e. φ([a, b]L) = [φ(a), φ(b)]L′ . A Lie algebra isomorphism is a morphism of Lie algebras that is a linear isomorphism.1 A very interesting question is to classify Lie algebras (up to isomorphism) of dimension n for a given n. For n = 2, there are only two: the trivial bracket [ , ] = 0, and [e1, e2] = e2. For n = 3, it can be done without too much trouble. Maybe n = 4 has been done, but in general, it is a very hard problem. If e is a basis for V , with [e , e ]= ck e (the ck are called the structure { i} i j ij k ij constants of V ), then the Jacobi identity is some quadratic relation on the k 3n cij, so the variety of Lie algebras is some quadratic surface in C . Given a smooth real manifold M n of dimension n, we can construct Vect(M n), the set of smooth vector fields on M n. For X Vect(M n), we ∈ can define the Lie derivative LX by (LX f)(m) = Xm f, so LX acts on n · · C∞(M ) as a derivation. ◮ Exercise 1.3. Verify that [L ,L ]= L L L L is of the form X Y X ◦ Y − Y ◦ X L for a unique Z Vect(M n). Then we put a Lie algebra structure on Z ∈ Vect(M n)= (C (M n)) by [X,Y ]= Z. D ∞ There is a theorem (Ado’s Theorem2) that any Lie algebra g is isomor- phic to a Lie subalgebra of gln, so if you understand everything about gln, you’re in pretty good shape. 1The reader may verify that this implies that the inverse is also a morphism of Lie algebras. 2Notice that if g has no center, then the adjoint representation ad : g → gl(g) is already faithful.