Lie algebras Shlomo Sternberg April 23, 2004 2 Contents 1 The Campbell Baker Hausdorff Formula 7 1.1 The problem. 7 1.2 The geometric version of the CBH formula. 8 1.3 The Maurer-Cartan equations. 11 1.4 Proof of CBH from Maurer-Cartan. 14 1.5 The differential of the exponential and its inverse. 15 1.6 The averaging method. 16 1.7 The Euler MacLaurin Formula. 18 1.8 The universal enveloping algebra. 19 1.8.1 Tensor product of vector spaces. 20 1.8.2 The tensor product of two algebras. 21 1.8.3 The tensor algebra of a vector space. 21 1.8.4 Construction of the universal enveloping algebra. 22 1.8.5 Extension of a Lie algebra homomorphism to its universal enveloping algebra. 22 1.8.6 Universal enveloping algebra of a direct sum. 22 1.8.7 Bialgebra structure. 23 1.9 The Poincar´e-Birkhoff-Witt Theorem. 24 1.10 Primitives. 28 1.11 Free Lie algebras . 29 1.11.1 Magmas and free magmas on a set . 29 1.11.2 The Free Lie Algebra LX ................... 30 1.11.3 The free associative algebra Ass(X). 31 1.12 Algebraic proof of CBH and explicit formulas. 32 1.12.1 Abstract version of CBH and its algebraic proof. 32 1.12.2 Explicit formula for CBH. 32 2 sl(2) and its Representations. 35 2.1 Low dimensional Lie algebras. 35 2.2 sl(2) and its irreducible representations. 36 2.3 The Casimir element. 39 2.4 sl(2) is simple. 40 2.5 Complete reducibility. 41 2.6 The Weyl group. 42 3 4 CONTENTS 3 The classical simple algebras. 45 3.1 Graded simplicity. 45 3.2 sl(n +1)................................ 47 3.3 The orthogonal algebras. 48 3.4 The symplectic algebras. 50 3.5 The root structures. 52 3.5.1 An = sl(n + 1). 52 3.5.2 Cn = sp(2n), n ≥ 2. 53 3.5.3 Dn = o(2n), n ≥ 3. 54 3.5.4 Bn = o(2n + 1) n ≥ 2. 55 3.5.5 Diagrammatic presentation. 56 3.6 Low dimensional coincidences. 56 3.7 Extended diagrams. 58 4 Engel-Lie-Cartan-Weyl 61 4.1 Engel’s theorem . 61 4.2 Solvable Lie algebras. 63 4.3 Linear algebra . 64 4.4 Cartan’s criterion. 66 4.5 Radical. 67 4.6 The Killing form. 67 4.7 Complete reducibility. 69 5 Conjugacy of Cartan subalgebras. 73 5.1 Derivations. 74 5.2 Cartan subalgebras. 76 5.3 Solvable case. 77 5.4 Toral subalgebras and Cartan subalgebras. 79 5.5 Roots. 81 5.6 Bases. 85 5.7 Weyl chambers. 87 5.8 Length. 88 5.9 Conjugacy of Borel subalgebras . 89 6 The simple finite dimensional algebras. 93 6.1 Simple Lie algebras and irreducible root systems. 94 6.2 The maximal root and the minimal root. 95 6.3 Graphs. 97 6.4 Perron-Frobenius. 98 6.5 Classification of the irreducible ∆. 104 6.6 Classification of the irreducible root systems. 105 6.7 The classification of the possible simple Lie algebras. 109 CONTENTS 5 7 Cyclic highest weight modules. 113 7.1 Verma modules. 114 7.2 When is dim Irr(λ) < ∞? . 115 7.3 The value of the Casimir. 117 7.4 The Weyl character formula. 121 7.5 The Weyl dimension formula. 125 7.6 The Kostant multiplicity formula. 126 7.7 Steinberg’s formula. 127 7.8 The Freudenthal - de Vries formula. 128 7.9 Fundamental representations. 131 7.10 Equal rank subgroups. 133 8 Serre’s theorem. 137 8.1 The Serre relations. 137 8.2 The first five relations. 138 8.3 Proof of Serre’s theorem. 142 8.4 The existence of the exceptional root systems. 144 9 Clifford algebras and spin representations. 147 9.1 Definition and basic properties . 147 9.1.1 Definition. 147 9.1.2 Gradation. 148 9.1.3 ∧p as a C(p) module. 148 9.1.4 Chevalley’s linear identification of C(p) with ∧p. 148 9.1.5 The canonical antiautomorphism. 149 9.1.6 Commutator by an element of p. 150 9.1.7 Commutator by an element of ∧2p. 151 9.2 Orthogonal action of a Lie algebra. 153 9.2.1 Expression for ν in terms of dual bases. 153 9.2.2 The adjoint action of a reductive Lie algebra. 153 9.3 The spin representations. 154 9.3.1 The even dimensional case. 155 9.3.2 The odd dimensional case. 158 9.3.3 Spin ad and Vρ. 159 10 The Kostant Dirac operator 163 10.1 Antisymmetric trilinear forms. 163 10.2 Jacobi and Clifford. 164 10.3 Orthogonal extension of a Lie algebra. 165 2 10.4 The value of [v + ν(Casr)]0. 167 10.5 Kostant’s Dirac Operator. 169 10.6 Eigenvalues of the Dirac operator. 172 10.7 The geometric index theorem. 178 10.7.1 The index of equivariant Fredholm maps. 178 10.7.2 Induced representations and Bott’s theorem. 179 10.7.3 Landweber’s index theorem. 180 6 CONTENTS 11 The center of U(g). 183 11.1 The Harish-Chandra isomorphism. 183 11.1.1 Statement . 184 11.1.2 Example of sl(2). 184 W 11.1.3 Using Verma modules to prove that γH : Z(g) → U(h) . 185 11.1.4 Outline of proof of bijectivity. 186 11.1.5 Restriction from S(g∗)g to S(h∗)W . 187 11.1.6 From S(g)g to S(h)W . 188 11.1.7 Completion of the proof. 188 11.2 Chevalley’s theorem. 189 11.2.1 Transcendence degrees. 189 11.2.2 Symmetric polynomials. 190 11.2.3 Fixed fields. 192 11.2.4 Invariants of finite groups. 193 11.2.5 The Hilbert basis theorem. 195 11.2.6 Proof of Chevalley’s theorem. 196 Chapter 1 The Campbell Baker Hausdorff Formula 1.1 The problem. Recall the power series: 1 1 1 1 exp X = 1 + X + X2 + X3 + ··· , log(1 + X) = X − X2 + X3 + ··· . 2 3! 2 3 We want to study these series in a ring where convergence makes sense; for ex- ample in the ring of n×n matrices. The exponential series converges everywhere, and the series for the logarithm converges in a small enough neighborhood of the origin. Of course, log(exp X) = X; exp(log(1 + X)) = 1 + X where these series converge, or as formal power series. In particular, if A and B are two elements which are close enough to 0 we can study the convergent series log[(exp A)(exp B)] which will yield an element C such that exp C = (exp A)(exp B). The problem is that A and B need not commute. For example, if we retain only the linear and constant terms in the series we find log[(1 + A + ··· )(1 + B + ··· )] = log(1 + A + B + ··· ) = A + B + ··· . On the other hand, if we go out to terms second order, the non-commutativity begins to enter: 1 1 log[(1 + A + A2 + ··· )(1 + B + B2 + ··· )] = 2 2 7 8 CHAPTER 1. THE CAMPBELL BAKER HAUSDORFF FORMULA 1 1 1 A + B + A2 + AB + B2 − (A + B + ··· )2 2 2 2 1 = A + B + [A, B] + ··· 2 where [A, B] := AB − BA (1.1) is the commutator of A and B, also known as the Lie bracket of A and B. Collecting the terms of degree three we get, after some computation, 1 1 1 A2B + AB2 + B2A + BA2 − 2ABA − 2BAB] = [A, [A, B]]+ [B, [B, A]]. 12 12 12 This suggests that the series for log[(exp A)(exp B)] can be expressed entirely in terms of successive Lie brackets of A and B. This is so, and is the content of the Campbell-Baker-Hausdorff formula. One of the important consequences of the mere existence of this formula is the following. Suppose that g is the Lie algebra of a Lie group G. Then the local structure of G near the identity, i.e. the rule for the product of two elements of G sufficiently closed to the identity is determined by its Lie algebra g. Indeed, the exponential map is locally a diffeomorphism from a neighborhood of the origin in g onto a neighborhood W of the identity, and if U ⊂ W is a (possibly smaller) neighborhood of the identity such that U · U ⊂ W , the the product of a = exp ξ and b = exp η, with a ∈.