Mathematical Physics Seminar Notes Lecture 3 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.sgTel (65) 6874-2749

1

Commuting Vector Fields

2

Theorem. If

such that

exist local coordinates RM:,...,1 kuuVect(M)V,...,V k1 then there

iuiV iff 0]V,[V ji

Proof. p 471 J. Lee Introduction to Smooth Manifolds

Corollary. If G is a Lie group with Lie algebra

then

andk1 V,...,V spans an abelian subalgebra

0)exp(G 00 is an abelian subgroup of G

and Rui iijj j s,s))Vs(exp(

of

Lie Algebras and Lie Groups

3

Lemma (standard homotopy result) Every connected Lie group is the quotient of a unique simply connected Lie group (obtained as its universal covering space) with a discrete central subgroup. Lie groups are locally isomorphic iff they have the same s.c. covering groups

Theorem (Lie). There is a 1-to-1 correspondence between Lie algebras and s. c. Lie groups.

Theorem (Frobenius) There is a 1-to-1 correspondence between Lie algebras and (not necessarily closed) Lie subgroups – e.g. subgroup R of the two-dim torus

Lemma A closed subgroup of a Lie group is a L. g.

Adjoint Representation

4

Definition

Definition )(End : ad

)GL(G:Ad uG, g,gugAd(g)(u) -1

vu,,]vu,[ad(u)vTheorem For

p(su))vad(u)Ad(ex)vAd(exp(su)dsd

Rs ,vu,

su))v(exp(Ad)vexp(ad(su)

Ideals and Normal Subgroups

5

Definition A Lie subalgebra

Definition A Lie subgroup

0 is an ideal if

00 v,u,v][u,

0-1 GhG,gh,ghg

Theorem There is a 1-to-1 correspondence between normal connected Lie subroups of a Lie group and ideals of its Lie algebra

GG0 is normal if

Killing Form

6

Definition.

)vu,()vu,()(Aut KK

ad(u)ad(v)Tracev)K(u, C:K

Theorem

Corollary

0v)(w)adu,K(v)u,(w)K(ad

Nilpotent and Solvable Algebras

7

Definition Lie algebra 0

0 nilpotent, solvable if

],[],,[ nn1nn01n

Theorem (Lie) A subalgebra of GL(V) is solvable iff its elements are simultaneously triangulable

terminates

Theorem (Engel) A Lie algebra is nilpotent iff ad(u) is nilpotent (some power = 0) for every element u

Simple and Semisimple Algebras

8

Definition Lie algebra is simple, semisimple if it has no ideals, abelian ideals other that itself and {0}

Theorems (Cartan) A Lie algebra

0 is solvable iff10 v,u0,v)K(u,

Proof D. Sattinger and O. Weaver, Lie Groups and Algebras with App. to Physics, Geom. &Mech.

Theorem The sum of any two solvable ideals is a solvable ideal, hence every algebra has a unique maximal solvable ideal – called its radicalTheorem (Levi) Every Lie algebra is the semidirect sum of its radical and a semisimple subalgebra

semisimple iff K is nondegenerate, so SSLA = + SI

Examples

9

Euclidean Motion Groups

Heisenberg Groups

Solvable RSO(2) 2s

Poincare Group

Radical SS R3)SO(m 2s s

Nilpotent RR s2 d

RadicalSS RLorentz s4

s

Affine Groups

RadicalSS R2)GL(d sd

s d

s2 R)R(ZRad SL(d),SS

Solvable R)1GL( s

Cartan’s Classification of Complex Semisimpil LA

10

Classical

1)2n(2n0XT

XnDso(2n)

1)n(2n0JXJT

XnCsp(2n)

1)2n(2n0XT

XnB1)so(2n

12

n0XTr 1nAsl(n)

DimensionConstraint

Exceptional

24876 G,F,E,E ,E

Lagrange’s Equations in Action

Lagrangian L := T – U in Action

Principle of Least Action: for

2

1

dt LSt

t

0S2

1

t

t qL

qL dtqq

q

L

q

L

dt

d

Lagrange Equations

described as a section of T(T(M)), ie in Vect(T(M))

0)(q)(q21

tt

11

Geodesics

If V = 0 then L = T defines a Riemannian manifold M with metric tensor g

jiij qqqg )()qT(q, 21

Lagrange’s equations

describe trajectories that minimize squared magnitude of velocity, and hence minimize length and have constant speed, therefore they are geodesics 12

nmmni,jij qqqg where the components of the Christoffel symbol

i

mn

m

in

n

immni, qqq

2

ggg

Hamilton’s Equations

Hamiltonian R(M)T:H * defined by the Legendre Transformation

Lagrange’s equations are equivalent to Hamilton’s

),( maxp)H(q, qqLqp,q satisfies

dqq

Ldpqdq

q

Hdp

p

HHd

p

Hp

p

Hq

, and UTH 13

Symplectic Structure

The Liouville 1-form

(M)T*induces the symplectic structure on

The Hamiltonian vector field v is

))((dqp *1ii MT

given by the nondegenerate 2-form

iiii dqdpdqpd )(dH),v(

hence the Lie derivative satisfies

0)v,v(, vdHHLv0,...,0,0 d

vvv LLL14

Poincare’s Recurrence Theorem

If

(M)TO *is a Hamiltonian flow then for every open set

and

R t(M),T (M)T:F **t

Op0t

and OpF tn )(

)()()( 32 OFOFOF ttt

there exists

0n such that

Proof. Consider the (infinite) union

Since the volume (induced by the symplectic form) of each set is positive and equal, they can not be disjoint, and the conclusion follows. 15

The Kirillov Form on Co-Adjoint Orbits

Theorem. If is a Lie group with Lie algebra

of

G G:))((AdM * gg

Ad(g)v,v,(g)Ad*

then the orbit* under the coadjoint antirepresentation

Proof. Tangents u, v to M at p are represented by curves in M, hence by curves in G through 1 that

*ΓΓ,vG,g

define elements t(v)t(u), so the 2-form

t(v)][t(u),p,v)(u, is symplectic.

admits a symplectic structure.

16

Weyl-Chevalley Normal Form

Theorem. If is a complex semisimple Lie algebra

then

0with Cartan subalgebra

)(ad semisimple (diagonalizable) for all

are roots and

oΓ *

vhvcvc ,,)(],[

where

17

(maximal abelian with

)0

],[

N],[where 0N unless is a root.