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(ECON514)7a1584 - Midterm514.pdf
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MIDTERM
DUE BY 30/10 IN CLASS
1. Let R2n be a 2n-dimensional space with coordinates pi,qi, i =1, ,n. Let A be the space of C functions on R2n . For f, g A, de.ne the Poisson bracket
.f .g .f .g
{f, g} = . .
.pi .qi .qi .pi
i
Prove that A with { , } is a Lie algebra.
2. Let charF .
= 2. Prove that the Killing form on sl(n, F ) is nondegenerate if and only if char(F ) . n.
3.
Let V = C2 be the tautological representation of sl2(C). For m, n N, write Sm(V ) . Sn(V ) as a direct sum of irreducible sl2(C)-modules. Justify your answer.

4.
Let L = ni=1Li be a direct sum of simple Lie algebras. Prove that any ideal of L is iJ Li for some J .{1, ,n}.

5.
Let L be a simple Lie algebra over C. Show that the tensor product of two nontrivial .nite dimensional L-modules is not irreducible. (Hint: HomL(V, W ) is isomorphic to the subspace of V . . W annihilated by L).


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