Mathematical Physics Seminar Notes Lecture 4 Global Analysis and Lie Theory

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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_at_nus.edu.sg Tel [phone]
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Nilpotent Transformations
Lemma. If
is a finite dim. complex vector space
is C-linear with eigenvalues
and
and
then
is nilpotent-some power0.
Proof.
and A, B induce
so if
then
such that
hence
with
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Minimal Polynomial, Bezout Identity, Jordan Form
Corollary. The minimal degree polynomial P such
that P(A)V 0 equals
Corollary. A solution of the Bezout Identity
gives
where
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
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Jordan-Chevalley Decomposition
Theorem. There exist polynomials p and q so that
is semisimple (diagonalizable),
is nilpotent-some power equals 0,
and
Proof. Use Chinese Remainder Theorem to compute
Clearly
and
is nilpotent on
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Ad-Semisimple
Theorem. If X is semisimple then Ad X is
semisimple
Proof. Decompose
then decompose
so that
in a corresponding manner and observe that
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Ad-Nilpotent
Theorem. If X is nilpotent then Ad X is nilpotent
Proof. Express
and observe that if
then for
therefore
vanishes.
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Derivations
Definition. A derivation of a Lie algebra
is a linear
that satisfies the identity
The set of derivations
forms a Lie subalgebra
of
(under the commutator product).
Lemma.
by the Jacobi identity.
These inner derivations form an ideal since
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Derivations
Theorem. All derivations of a semi-simple Lie
algebra
are inner derivations.
Proof. Since
is semisimple
is 1-1 and
and
are ideals and by Cartan is solvable since
Therefore a 0 since a is also semisimple.
If
then for all
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