General notions of lie groupsI General linear groups

GL n.lk fAEMnxnH23 A invertibleGL h E At Mn.mil Ainreatibk

Glen M C Gun E is asubgroup

2 Special linear group

SL h 4 4 At Mun I def A lyOrthogonal group

01h 4 A C Mnxn 11141 ATA lyY Special orthogonal groupSolnl 4 At Manik ATA 1 detail

5 Unitary groupIn 4 At Mnxnc.CI A A lY

I 9At A IT

6 Special unitary group

5014 4 A f Un detail

fl Wi G Un E C 644,112is a subgroup of Gum N

Hint a I 1122 asreal vector span

Pnp All eigenvalues d of AE 01h has14 1

Pf Lef v f f be a non zone

eigenvector 5 t A v Av

AV Av CV V

Av Av car ar 1 7 2 iv usso 1 171

Corollary Let At 50th and d isan eigenrame of A

then 1 1 1 ane J is alsoan eigenvalue of A

pf HI Al is a real polynomial

Conjugacy class in Se i3

From Lecture 3 we know 4 1 is an

eigenvalue of A C 5013

Then there exists an eigenvector vflR3Av v

Let W Hult then IRV he 1123

and

prop A preserves w

Pf let wth

Aw Av W v o

LAW V 7 0

Aw C W

So A is an orthogonal operation onW

orthonormalg there exists basis of W K

3sit A Va Vik Ch Vsg bio SinoV ITT since sosine Alr ru.rs Yi's'd I59

I Vi V2 VI

FTand def A L so

uselet A

Sino o

Let D Vi Va vs then DTP In

If dit P I replace a by u

then det p i l

se P Al f I7hm The

conjugacy class of Soos isdetermined by trau function

Pf Every A can be conjugate to

FAPI n

O IiIf Ai L Ying IiAz y Egf

s

and to A _KAL

then 0 Oz ou 0 Oz

If 01 02 d Oz

choose IT C 014 deff 1

Sit I moi sine

no e P619 s no

Sinor and

pp

then P A P tAu

Geometrically If Av u and

Bw w

vA is an rotation

w of legneegCounterclockwise

from13positive v direction

BO

positive W direction

Let PE 5013 such thatpv w

then PAP B

More generally

7hm Let A C Seen then I Pf 50ins't l

Pf Leta is an eigenvalueoffIf 11 1 then Let VIO

AVIV and choose Wfµyt

inductionon W

If D l then there existsanotherbigenvalue d I I otherwise

Itt Atl AITf x't t't't d is not real

A y 1ft VD X Yt b V qTake span iv hey EH

Dimension of 50123 is 1

µSon I 51

UH 4 At 6 151 14 son

O l y OSino

size10

VII I 50127 I s

el din't Sphere

5013 A v w var

a

circleperpendicular

to u

I dim't circle

vxw

Prop dim 5013 2 1 3

Why important

Heather's 7hm Every symmetry gives a

conservation Law

Sol angular momentum 15components

time translation energy1123 t

span translationmomentum

I 3 Components

Sun THE A A 7

detA I

A't A I

A uta Iah A't

g a d c 5

det Lff bar lait lb El

GI xty.FI be 2 wifiX y t w f IR

then Tty't t 1W

Papi 50121 1 A Ig aka't

rt Aina A Es

Iggythx

1914612 1A A zAt Ail

conjugacy classes in Suez

Poop Given At 50123 I P f sungs t 15 Ap L with

1 1 1

Pf p Ev u

fi

anet x be Tigenvalne of Anita eigenvector uthen Av du Let Vi v thV ab and call 1142 1

Let v L E Tha

Vi Vu 0 Vc is the basisof

Gh t

A preserve c so Ah C164,1why Av Ah cu Va o

Icv Ah

CY Are 7 0

so A Vc M Va

Sina A Lv ru V ua duthen Xµ l 3 µ I

Pnp Conjugacy classes of sun isdetermined

by trace faction

dim sun 3

i L IEh Lf basis of

space ofmatriusfff.ba xx

I Ess