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Berkeley MATH 261A - Problems

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Math 261A: Lie Groups, Fall 2008Problems, Set 41. Classify the 3-dimensional Lie algebras g over an algebraically closed field k of char-acteristic zero by showing that if g is not a direct product of smaller Lie algebras, theneither(i) g∼=sl(2, k),(ii) g is isomorphic to the nilpotent Heisenberg Lie algebra h with basis X, Y , Z suchthat Z is central and [X, Y ] = Z, or(iii) g is isomorphic to a solvable algebra s which is the semidirect product of the abelianalgebra k2by an invertible derivation. In particular s has basis X, Y , Z such that [Y, Z] = 0,and ad X acts on kY + kZ by an invertible matrix, which, up to change of basis in kY + kZand rescaling X, can be taken to be either [1 10 1], or [λ 00 1] for some nonzero λ ∈ k.10. Following problems 28–35 in Knapp, Chapter I, classify the 3-dimensional Lie algebrasover k when char(k) = 0 but k is not necessarily algebraically closed.2. (a) Show that the Heisenberg Lie algebra h in Problem 1 has the property that Z actsnilpotently in every finite-dimensional module, and as zero in every simple finite-dimensionalmodule.(b) Construct a simple infinite-dimensional h-module in which Z acts as a non-zero scalar.[Hint: take X and Y to be the operators d/dt and t on k[t].]3. Construct a simple 2-dimensional module for the Heisenberg algebra h over any fieldk of characteristic 2. In particular, if k = k, this gives a counterexample to Lie’s theorem innon-zero characteristic.4. Let g be a finite-dimensional Lie algebra over k.(a) Show that the intersection n of the kernels of all finite-dimensional simple g-modulescan be characterized as the largest ideal of g which acts nilpotently in every finite-dimensionalg-module. It is called the nilradical of g.(b) Show that the nilradical of g is contained in g0∩ rad(g).(c) Let h ⊆ g be a subalgebra and V a g-module. Given a linear functional λ : h →k, define the associated weight space to be Vλ= {v ∈ V : Hv = λ(H)v for all H ∈ h}.Assuming char(k) = 0, adapt the proof of Lie’s theorem to show that if h is an ideal and Vis finite-dimensional, then Vλis a g-submodule of V .(d) Show that if char(k) = 0 then the nilradical of g is equal to g0∩ rad g. [Hint:assume without loss of generality that k = k and obtain from Lie’s theorem that any finite-dimensional simple g-module V has a non-zero weight space for some weight λ on g0∩ rad g.Then use (c) to deduce that λ = 0 if V is simple.]5. Let g be a finite-dimensional Lie algebra over k, char(k) = 0. Prove that the thelargest nilpotent ideal of g is equal to the set of elements of r = rad g which act nilpotentlyin the adjoint action on g, or equivalently on r. In particular, it is equal to the largestnilpotent ideal of r.6. Prove that the Lie algebra sl(2, k) of 2 × 2 matrices with trace zero is simple, over afield k of any characteristic 6= 2. In characteristic 2, show that it is nilpotent.7. In this exercise, we’ll deduce from the standard functorial properties of Ext groupsand their associated long exact sequences that Ext1(N, M) bijectively classifies extensions0 → M → V → N → 0 up to isomorphism, for modules over any associative ring with unity.(a) Let F be a free module with a surjective homomorphism onto N, so we have an exactsequence 0 → K → F → N → 0. Use the long exact sequence to produce an isomorphismof Ext1(N, M) with the cokernel of Hom(F, M) → Hom(K, M).(b) Given φ ∈ Hom(K, M), construct V as the quotient of F ⊕ M by the graph of −φ(note that this graph is a submodule of K ⊕ M ⊆ F ⊕ M).(c) Use the functoriality of Ext and the long exact sequences to show that the character-istic class in Ext1(N, M) of the extension constructed in (b) is the element represented bythe chosen φ, and conversely, that if φ represents the characteristic class of a given extension,then the extension constructed in (b) is isomorphic to the given one.8. Calculate Exti(k, k) for all i for the trivial representation k of sl(2, k), where char(k) =0. Conclude that the theorem that Exti(M, N) = 0 for i = 1, 2 and all finite-dimensionalrepresentations M, N of a semi-simple Lie algebra g does not extend to i > 2.9. Let g be a finite-dimensional Lie algebra. Show that Ext1(k, k) can be canonicallyidentified with the dual space of g/g0, and therefore also with the set of 1-dimensional g-modules, up to isomorphism.10. Let g be a finite-dimensional Lie algebra. Show that Ext1(k, g) can be canonicallyidentified with the quotient Der(g)/ Inn(g), where Der(g) is the space of derivations of g,and Inn(g) is the subspace of inner derivations, that is, those of the form d(x) = [y, x] forsome y ∈ g. Show that this also classifies Lie algebra extensionsbg containing g as an idealof codimension 1.11. Let g be a finite-dimensional Lie algebra. Show that there is a canonical isomorphismExt1(g, k)∼=Ext2(k, k)⊕S2((g/g0)∗) where S2denotes the second symmetric power. The firstterm classifies those g-module extentions 0 → k →bg → g → 0 that are (one-dimensional,central) Lie algebra extensions.Addendum: This problem turned out to be harder than I thought, and I’m not even surethat it’s true.Let’s assume the ground field has char(k) 6= 2, so we can distinguish between symmetricand antisymmetric forms.The weaker result that there is a canonical injection Ext2(k, k)⊕S2((g/g0)∗) ,→ Ext1(g, k)can be proven by representing a 1-cocyle as a bilinear form on g and considering the caseswhere the form is antisymmetric or symmetric.For the stronger result, note that the identity ([x, z], z) = 0 holds for the symmetriza-tion of the form representing a 1-cocycle. Then ([x, y], z) + (y, [x, z]]) = ([x, y + z], y +z) − ([x, y], y) − ([x, z], z) = 0, so the symmetrized form is invariant. Among the invariantsymmetric forms are those whose radical contains g0. These represent 1-cocycles. But somefurther argument is needed to show that no other invariant form can arise by symmetrizinga 1-cocyle.12. Let g be a finite-dimensional Lie algebra over k, char(k) = 0. The Malcev-Harish-Chandra theorem says that all Levi subalgebras s ⊆ g are conjugate under the action of thegroup exp ad n, where n is the largest nilpotent ideal of g (note that n acts nilpotently on g,so the power series expression for exp ad X reduces to a finite sum when X ∈ n).(a) Show that the reduction we used to prove Levi’s theorem by induction in the casewhere the radical r = rad g is not a minimal ideal also works for the


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