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(MATH514)5016c8 - final514.pdf
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FINAL EXAM OF MATH514
DUE BY 11/12
In what follows, g denotes a .nite dimensional complex Lie algebra.
1. Let g. . g be an ideal. Assume that g/g. is nilpotent and ad(x) |gis nilpotent
for any x g. Prove that g is nilpotent.
2. If[g, g] is semisimple, prove that the radical of g equals the center of g.
3.
Let w0 be the longest element in a Weyl group W of a root system. Prove that .w0 permutes the simple roots.

4.
Let g be semisimple. Prove that any nonzero homomorphism between Verma modules of g is an inclusion.

5.
Let g be semisimple. Prove that for any d N, the number of nonisomorphic g-modules of dimension d is .nite.

6.
Decompose the tensor product of a 3-dimensional and a 6-dimensional irre-ducible representations of sl3 into irreducible components (there are two essentially di.erent cases).

7.
(1) Let g. . g be a semisimple ideal. Prove that there exists an ideal h of g such that g = g. h.


(2) Let g. . g be an ideal such that g/g. is semisimple. Prove that there exists a subalgebra h of g such that g = g. h.
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