1. Mathematical Physics Seminar Notes Lecture 4 Global Analysis and Lie Theory Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email matwml@nus.edu.sg Tel (65) 6874-2749 1

2. Nilpotent Transformations 2 Lemma. If and is C-linear with eigenvalues is a finite dim. complex vector space then and is nilpotent-some power=0. Proof. and A, B induce so ifthen such thathencewith

3. Minimal Polynomial, Bezout Identity, Jordan Form 3 Corollary. There exists a bases for V for which A has the upper Jordan Canonical Form, the exponents in P are minimal sizes of eigenblocks, and P divides CP Corollary. The minimal degree polynomial P such Corollary. A solution of the Bezout Identity that P(A)V = 0 equals giveswhere

4. Jordan-Chevalley Decomposition 4 Theorem. There exist polynomials p and q so that and Proof. Use Chinese Remainder Theorem to compute is semisimple (diagonalizable), is nilpotent-some power equals 0, Clearly and is nilpotent on

5. Ad-Semisimple 5 Theorem. If X is semisimple then Ad X is semisimple then decompose Proof. Decompose so that in a corresponding manner and observe that

6. Ad-Nilpotent 6 Theorem. If X is nilpotent then Ad X is nilpotent therefore Proof. Express and observe that ifthen for vanishes.

7. Derivations 7 Definition. A derivation of a Lie algebrais a linear of(under the commutator product). The set of derivations that satisfies the identity forms a Lie subalgebra Lemma. These inner derivations form an ideal since by the Jacobi identity.

8. Derivations 8 Theorem. All derivations of a semi-simple Lie algebra are inner derivations. Proof. Since If is 1-1 and is semisimple and are ideals and by Cartan is solvable since then for all Therefore a = 0 since a is also semisimple.