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 ExtensionofaLiealgebrahomomorphismtoitsuniversal 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 L . . . . . . . . . . . . . . . . . . . 30 X 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 A =sl(n+1). . . . . . . . . . . . . . . . . . . . . . . . . 52 n 3.5.2 C =sp(2n),n≥2. . . . . . . . . . . . . . . . . . . . . . 53 n 3.5.3 D =o(2n), n≥3. . . . . . . . . . . . . . . . . . . . . . 54 n 3.5.4 B =o(2n+1) n≥2. . . . . . . . . . . . . . . . . . . . . 55 n 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 dimIrr(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 V . . . . . . . . . . . . . . . . . . . . . . . . . 159 ρ 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 10.4 The value of [v2+ν(Cas )] . . . . . . . . . . . . . . . . . . . . . 167 r 0 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 11.1.3 Using Verma modules to prove that γ :Z(g)→U(h)W. 185 H 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 expX =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- ampleintheringofn×nmatrices. Theexponentialseriesconvergeseverywhere, and the series for the logarithm converges in a small enough neighborhood of the origin. Of course, log(expX)=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[(expA)(expB)] which will yield an element C such that expC =(expA)(expB). 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 (cid:0)A2B+AB2+B2A+BA2−2ABA−2BAB](cid:1)= 1 [A,[A,B]]+ 1 [B,[B,A]]. 12 12 12 This suggests that the series for log[(expA)(expB)] 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 thefollowing. SupposethatgistheLiealgebraofaLiegroupG. Thenthelocal 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=expξ and b=expη, with a∈U and b∈U is then completely expressed in terms of successive Lie brackets of ξ and η. We will give two proofs of this important theorem. One will be geometric - theexplicitformulafortheseriesforlog[(expA)(expB)]willinvolveintegration, andsomakessenseovertherealorcomplexnumbers. Wewillderivetheformula from the “Maurer-Cartan equations” which we will explain in the course of our discussion. Oursecondversionwillbemorealgebraic. Itwillinvolvesuchideas astheuniversalenvelopingalgebra,comultiplicationandthePoincar´e-Birkhoff- Witt theorem. In both proofs, many of the key ideas are at least as important as the theorem itself. 1.2 The geometric version of the CBH formula. Tostatethisformulaweintroducesomenotation. LetadAdenotetheoperation of bracketing on the left by A, so adA(B):=[A,B]. Define the function ψ by zlogz ψ(z)= z−1 which is defined as a convergent power series around the point z =1 so log(1+u) u u2 u u2 ψ(1+u)=(1+u) =(1+u)(1− + +···)=1+ − +··· . u 2 3 2 6 1.2. THE GEOMETRIC VERSION OF THE CBH FORMULA. 9 In fact, we will also take this as a definition of the formal power series for ψ in terms of u. The Campbell-Baker-Hausdorff formula says that Z 1 log((expA)(expB))=A+ ψ((expad A)(exptad B))Bdt. (1.2) 0 Remarks. 1. The formula says that we are to substitute u=(expad A)(exptad B)−1 intothedefinitionofψ,applythisoperatortotheelementB andthenintegrate. In carrying out this computation we can ignore all terms in the expansion of ψ in terms of ad A and ad B where a factor of ad B occurs on the right, since (ad B)B = 0. For example, to obtain the expansion through terms of degree three in the Campbell-Baker-Hausdorff formula, we need only retain quadratic and lower order terms in u, and so 1 t2 u = ad A+ (ad A)2+tad B+ (ad B)2+··· 2 2 u2 = (ad A)2+t(ad B)(ad A)+··· Z 1(cid:18) u u2(cid:19) 1 1 1 1+ − dt = 1+ ad A+ (ad A)2− (ad B)(ad A)+··· , 2 6 2 12 12 0 where the dots indicate either higher order terms or terms with ad B occurring on the right. So up through degree three (1.2) gives 1 1 1 log(expA)(expB)=A+B+ [A,B]+ [A,[A,B]]− [B,[A,B]]+··· 2 12 12 agreeing with our preceding computation. 2. The meaning of the exponential function on the left hand side of the Campbell-Baker-Hausdorff formula differs from its meaning on the right. On therighthandside,exponentiationtakesplaceinthealgebraofendomorphisms of the ring in question. In fact, we will want to make a fundamental reinter- pretation of the formula. We want to think of A,B, etc. as elements of a Lie algebra, g. Then the exponentiations on the right hand side of (1.2) are still takingplaceinEnd(g). Ontheotherhand,ifgistheLiealgebraofaLiegroup G, then there is an exponential map: exp: g→G, and this is what is meant by the exponentials on the left of (1.2). This exponential map is a diffeomorphism on some neighborhood of the origin in g, and its inverse, log, is defined in some neighborhood of the identity in G. This is the meaning we will attach to the logarithm occurring on the left in (1.2). 3. The most crucial consequence of the Campbell-Baker-Hausdorff formula is that it shows that the local structure of the Lie group G (the multiplication law for elements near the identity) is completely determined by its Lie algebra. 4. For example, we see from the right hand side of (1.2) that group multi- plication and group inverse are analytic if we use exponential coordinates. 10 CHAPTER 1. THE CAMPBELL BAKER HAUSDORFF FORMULA 5. Consider the function τ defined by w τ(w):= . (1.3) 1−e−w This is a familiar function from analysis, as it enters into the Euler-Maclaurin formula, see below. (It is the exponential generating function of (−1)kb where k the b are the Bernoulli numbers.) Then k ψ(z)=τ(logz). 6. The formula is named after three mathematicians, Campbell, Baker, and Hausdorff. But this is a misnomer. Substantially earlier than the works of any of these three, there appeared a paper by Friedrich Schur, “Neue Begruendung der Theorie der endlichen Transformationsgruppen,” Mathematische Annalen 35 (1890), 161-197. Schur writes down, as convergent power series, the com- position law for a Lie group in terms of ”canonical coordinates”, i.e., in terms of linear coordinates on the Lie algebra. He writes down recursive relations for the coefficients, obtaining a version of the formulas we will give below. I am indebted to Prof. Schmid for this reference. Our strategy for the proof of (1.2) will be to prove a differential version of it: d log((expA)(exptB))=ψ((expad A)(expt ad B))B. (1.4) dt Since log(expA(exptB)) = A when t = 0, integrating (1.4) from 0 to 1 will prove (1.2). Let us define Γ=Γ(t)=Γ(t,A,B) by Γ=log((expA)(exptB)). (1.5) Then expΓ=expAexptB and so d d expΓ(t) = expA exptB dt dt = expA(exptB)B = (expΓ(t))B so d (exp−Γ(t)) expΓ(t) = B. dt We will prove (1.4) by finding a general expression for d exp(−C(t)) exp(C(t)) dt where C =C(t) is a curve in the Lie algebra, g, see (1.11) below.