Math 261A: Lie Groups, Fall 2008Problems, Set 31. [corrected 10/3] (a) Let S be a commutative k-algebra. Show that a linear operatorX : S → S is a derivation if and only if it kills 1 and its commutator with the operator ofmultiplication by every function is the operator of multiplication by another function.(b) Grothendieck’s inductive definition of differential operators on S goes as follows:the differential operators of order zero are the operators of multiplication by functions;the space D≤nof operators of order at most n is then defined inductively for n > 0 byD≤n= {X : [X, f] ∈ D≤n−1for all f ∈ S}. Show that the differential operators of all ordersform a filtered algebra D, and that when S is the algebra of smooth [analytic, holomorphic]functions on an open set in Rn[or Cn], D is a free left S-module with basis consisting of allmonomials in the coordinate derivations ∂/∂xi.2. Calculate all terms of degree ≤ 4 in the Baker-Campbell-Hausdorff formula.3. Let F (d) be the free Lie algebra on generators X1, . . . , Xd. It has a natural Ndgradingin which F (d)(k1,...,kd)is spanned by bracket monomials containing kioccurences of eachgenerator Xi. Use the PBW theorem to prove the generating function identityYk1(1 − tk11· · · tkdd)dim F (d)(k1,...,kd)=11 − (t1+ · · · + td).4. Words in the symbols X1, . . . , Xdform a monoid under concatentation, with identitythe empty word. Define a primitive word to be a non-empty word that is not a power of ashorter word. A primitive necklace is an equivalence class of primitive words under rotation.Use the generating function identity in Problem 3 to prove that the dimension of F (d)k1,...,kdis equal to the number of primitive necklaces in which each symbol Xiappears kitimes.5. A Lyndon word is a primitive word that is the lexicographically least representativeof its primitive necklace.(a) Prove that w is a Lyndon word if and only if w is lexicographically less than v forevery factorization w = uv such that u and v are non-empty.(b) Prove that if w = uv is a Lyndon word of length > 1 and v is the longest proper rightfactor of w which is itself a Lyndon word, then u is also a Lyndon word. This factorizationof w is called its right standard factorization.(c) To each Lyndon word w in symbols X1, . . . , Xdassociate the bracket polynomialpw= Xiif w = Xihas length 1, or, inductively, pw= [pu, pv], where w = uv is the rightstandard factorization, if w has length > 1.Prove that the elements pwform a basis of F (d).6. Prove that if q is a power of a prime, then the dimension of the subspace of totaldegree k1+ · · · + kq= n in F (q) is equal to the number of monic irreducible polynomials ofdegree n over the field with q elements.7. This problem outlines an alternative proof of the PBW theorem.(a) Let L(d) denote the Lie subalgebra of T (X1, . . . , Xd) generated by X1, . . . , Xd. With-out using the PBW theorem—in particular, without using F (d) = L(d)—show that thevalue given for dim F (d)(k1,...,kd)by the generating function in Problem 3 is a lower boundfor dim L(d)(k1,...,kd).(b) Show directly that the Lyndon monomials in Problem 5(b) span F (d).(c) Deduce from (a) and (b) that F (d) = L(d) and that the PBW theorem holds forF (d).(d) Show that the PBW theorem for a Lie algebra g implies the PBW theorem for g/a,where a is a Lie ideal, and so deduce PBW for all finitely generate Lie algebras from (c).(e) Show that the PBW theorem for arbitrarty Lie algebras reduces to the finitely gen-erated case.8. Let B(X, Y ) be the Baker-Campbell-Hausdorff series, i.e., eB(X,Y )= eXeYin non-commuting variables X, Y . Let F (X, Y ) be its linear term in Y , that is, B(X, sY ) =X + sF (X, Y ) + O(s2).(a) Show that F (X, Y ) is characterized by the identityXk,l≥0XkF (X, Y )Xl(k + l + 1)!= eXY.(b) Let λ, ρ denote the operators of left and right multiplication by X, and let f be theseries in two commuting variables such that F (X, Y ) = f(λ, ρ)(Y ). Show thatf(λ, ρ) =λ − ρ1 − eρ−λ(c) Deduce thatF (X, Y ) =ad X1 − e− ad X(Y ).9. Let G be a Lie group, g = Lie(G), 0 ∈ U0⊆ U ⊆ g and e ∈ V0⊆ V ⊆ G openneighborhoods such that exp is an isomorphism of U onto V , exp(U0) = V0, and V0V0⊆ V .Define β : U0× U0→ U by β(X, Y ) = log(exp(X) exp(Y ), where log: V → U is the inverseof exp.(a) Show that β(X, (s + t)Y ) = β(β(X, tY ), sY ) whenever all arguments are in U0.(b) Show that the series (ad X)/(1 − e− ad X), regarded as a formal power series in thecoordinates of X with coefficients in the space of linear endomorphisms of g, converges forall X in a neighborhood of 0 in g.(c) Show that on some neighborhood of 0 in g, β(X, tY ) is the solution of the initial valueproblemβ(X, 0) = Xddtβ(X, tY ) = F (β(X, tY ), Y ),where F (X, Y ) =(ad X)/(1 − e− ad X)(Y ).(d) Show that the Baker-Campbell-Hausdoff series B(X, Y ) also satisfies the identity inpart (a), as an identity of formal power series, and deduce that it is the formal power seriessolution to the IVP in part (c), when F (X, Y ) is regarded as a formal series.(e) Deduce from the above an alternative proof that B(X, Y ) is given as the sum ofa series of Lie bracket polynomials in X and Y , and that it converges to β(X, Y ) whenevaluated on a suitable neighborhood of 0 in g.(f) Use part (c) to calculate explicitly the terms of B(X, Y ) of degree 2 in Y .10. (a) Show that the Lie algebra so(3, C) is isomorphic to sl(2, C).(b) Construct a Lie group homomorphism SL(2, C) → SO(3, C) which realizes the iso-morphism of Lie algebras in part (a), and calculate its kernel.11. (a) Show that the Lie algebra so(4, C) is isomorphic to sl(2, C) × sl(2, C).(b) Construct a Lie group homomorphism SL(2, C)×SL(2, C) → SO(4, C) which realizesthe isomorphism of Lie algebras in part (a), and calculate its kernel.12. Show that every closed subgroup H of a Lie group G is a Lie subgroup, so that theinclusion H ,→ G is a closed immersion.13. Let G be a Lie group and H a closed subgroup. Show that G/H has a uniquemanifold structure such that the action of G on it is smooth [analytic, holomorphic].14. [clarified 10/22] Show that the intersection of two Lie subgroups H1, H2of a Lie groupG can be given a canonical structure of Lie subgroup so that its Lie algebra is Lie(H1) ∩Lie(H2) ⊆ Lie(G).15. Find the dimension of the closed linear group SO(p, q, R) ⊆ SL(p + q, R) consistingof elements which preserve a non-degenerate symmetric bilinear form