for Lie olds group : the to going basics Friday Fish - - PowerPoint PPT Presentation

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for Lie olds group : the to going basics Friday Fish - - PowerPoint PPT Presentation

Dec 16, 2022 407 likes •647 views

classes Characteristic for Lie olds group : the to going basics Friday Fish 2410712020 , - fixed M Motivagcio - s Mppalbdle classes :P ,VBc4BE HMM ) Characteristic Chern ESM classes Cpk v. B : - fixed M

slide-1
SLIDE 1

Characteristic classes

for

Lie

group

  • lds
:

going

to

the basics Friday

Fish , 2410712020
slide-2
SLIDE 2 ①

Motivagcio

M

  • fixed
Characteristic classes :P
  • s Mppalbdle

,VB→c4BE

HMM) → Chern classes : ESM Cpk
  • v. B
slide-3
SLIDE 3 ②

Motivagcio

M

  • fixed
→ Chern classes :

EFM

Cpk v. Bms CKCEIEHTM) Classifying space B Chern
  • Well
theory f ' Qu

EE

Quinn Choose 7 :

Rei

  • wjyxjei
→ curvature I :NTM→EndfE )

I

I

A- dwttfw ,w ] M
  • t
  • >
B n i i i

detziztrttf-EG.CH

'

Cite

) f , CEHFB ) Crye)EH" (M )
slide-4
SLIDE 4 ③

Motivation

: Characteristic classes for EE Rep CA) :

( Marius Rui )

→ cry E) EH "
  • YA ) :
Image VE : Hay (g) → HIA )
slide-5
SLIDE 5 ④

Motivation

: Characteristic classes for EE Rep CA) :

( Marius Rui )

→ Construction

:P

: TIAKTCEI → TIE ) Local frame e
  • f

Etu

: The agitated . the TIA ) we := Iwite C' IAI ④ gln → Car . . . (Ew) :=trlw÷weIEH" " '(A) travel
  • trlwf )= des
V

LIREEs

trave )
  • tray ) # de X ud
X Replace we by

weft

I ? ? ? )
slide-6
SLIDE 6

⑤

Definition :

§

,

.EE/2epal5l:49:ExTEy

⇐

G÷µ§¥E

)

rt

÷ . Defa :

HIGH

' HIGLIED HCG )
slide-7
SLIDE 7

⑥

Definition

at co chains

GIGUE)

Defa : Hakan ' HCGLIED HCG ) Locally :

E=MxEn→GLlEt56ln=Gln( G)

, roof
  • id
E :E×→Eyh As It tt M → * C' CGen ) tallest Ccg )
slide-8
SLIDE 8 ⑦ → G- Lie group : Cohomology
  • f
BG and R Gh : Wto , k )
  • s

Cl

B.G ) → Computation
  • f
C
  • Coger ,
um ) RGhN C- ( Gln )

(

Polar decomposition
  • f
matrices ) ur local formulas for ch . classes .
slide-9
SLIDE 9

⑧

Cohomology
  • f
BG A G Ch . classes P : Gppaecrbdle " µ, ¥ IF

!

f*HtBG ) Model for HTBG) : Bott
  • Shulman
complex
  • E. G
  • simplicial
:

riieoai

  • IIE
as → n'

Eat

  • →

Ep*G=G

" Epa .ci 't ' Td T T r 't EOG)

Edyta

) -4 IYER )
  • →
Td Td Td ICE

.GS#CEnG1Er4EaG )

  • t
slide-10
SLIDE 10 ⑨ Cohomology
  • f
BG Model for HTBG) : Bolt
  • Shulman
complex
  • E. G
  • simplicial :

tIEoGhry

as →

ryiaa

, . . , EpG=G' " AG t Td T T BPG = GP I 't EOG)

Edyta

) -8 I' l Eeg )
  • →
→ xeg : ix. ↳ Td Td Td ICE .GS#CEnG1Ed4EaG)--iix:r9-tEpGHtt-tEpG)Lx:tttEpf)-ir9-fEpG ) → DIE . G)
  • f
  • dog
  • algebra
U HINE . G) basic)= HIBG ) → ICE . G) basic
  • Keri
. A Kerk
slide-11
SLIDE 11

⑧

Chern
  • Well
theory 9- Lie algebra us Well algebra : WP

'f=SPg*④NP¥y*

piezo → g
  • deg
  • algebra
: Ix
  • contraction
  • nto
; trivial
  • n Soy
' L×=adI , Xeg F- Lela )

g*¥r'

(G) =D 'lEoG )
  • connection
left
  • inv
. MC
slide-12
SLIDE 12

①

Chern
  • Well
theory Well algebra : WP '9-=SPg*④N¥* 9- dg
  • algebra
: ix. W → W " ' , Lx :W÷W ' Xtg F- Lieu )
  • f
  • 0,1
(G) =D 'lEoG )
  • connection
Thmttexeev
  • Meinrenken )
Wog -4 site .G) g- dg . spaces $99 .nu 'S SHBG ) 9- da
  • algebras
In cohom .
slide-13
SLIDE 13 ④ Chern
  • Well
theory F- Lie LG) : DENKI

tell

Wog -9 sites ) 9- dg
  • sp
. Thmttexeev
  • Mein renken )
G
  • Compact
, Connected 9- da
  • algebras

fogginess

' I BG) Is a h . equiv . In Cohoon . Thru I S . ) G
  • connected
, KCG Max . compact subgroup with k heck ) . Then (Wg ) r
  • basic
SHE . G) r . bas . ICB .GL Is ah . equiv
slide-14
SLIDE 14 ⑤ Chern
  • Well
theory g- Leela ) : 9*-71161-7 Wg ICE . Thru ( S . ) G
  • connected
, KC

Gmax

. compact , R=Lie(k) (Walk
  • basic
Altar .basic'll

.BG/lsah.egniv

slide-15
SLIDE 15

④

Chern
  • Well
theory Thru ( S . ) G
  • connected
, KC

Gmax

. compact , R
  • heck)
1 . (Walk . basic ICE . G) r.ba . ftp.GIlsah.egniv 2 . Ifan

:%4kbasrIlEoGlr

. basic

ftp.GKCCG

) Van Est Integration I Melnrenken
  • S
. )
slide-16
SLIDE 16 ④ Description of HadGlen at cochalns Uh )cGLn Max . compact , he UhI=un=fAegHA*=
  • At
I

lighten

, . basic

""

Clan ) Polar decomposition

p=fAEgLn/A= # I

9 Ln = want top : A
  • ALI
t ALI
slide-17
SLIDE 17

④

Description of HadGlen at cochalns Polar decamp . : Uni -_fAeglnIA*=
  • At

P=fAE9lnlA= # I

9 Ln = want top : A --Az#tAtzA a. [ Ulm , UIHDCUIN ,
  • b. WIN
, PKD ,
  • C. [ D. DKUIN
Lemma : g=R④P . If b. ⇒

IN

te basic =

# PTR

  • inv
Itb & c ⇒ Hlg . HIND 'T r . inv
slide-18
SLIDE 18 ④ Description of HadGlen at cochalns g Ln = want top : It lgln.VN/=MP*)wini-inv Characteristic coeffcge-15-glil.nu :

detttI-AI-a.IG.tn

  • t
159 milk . . . .cn ] Cat AHHHH → Carlan map : r :Csight

.net/Tt-blnlmvTKgt--cleOIzq.i

, Eeg.it/tii..sAzg.i)--EEttrlAhn' ' ' Arkan) TESLA
  • I
slide-19
SLIDE 19 ⑧ Description of HadGlen at cochalns gln = win top : High

.VN/=fNDYuim.FnulN9ln)ulht-

basic T a AE
  • A
A*=A → Carlan map : r :(551ft

.net/Tt-'9lnlmvTKqI--cleOIzq.i

, Eeg.it/tii..sAzg.i)--EE7rlArni ' 'Ahearn) 8ES2q
  • I

Luzg it

  • ' IQs
. , Ip ) * = 1155¥)umm=H' " login

.hn/Hl9ln.UlnD=AlUi.Us....hzn-

it

slide-20
SLIDE 20 ④ Description of HadGlen at cochalns I

lightning

. bas

.fm/UlIClG1nliHlGlnl--Nv..Vs....lkn..lh2q

  • I
⇒ Vzq
  • I
Polar decomposition Glen
  • PUM ;
e :P P A =e×U : XEP , NEVIN

GLYNN

=p E p extent ⇐

ex

a x → t . EXUM :=et×UM , to IR
slide-21
SLIDE 21

⑧

Description of HadGlen at cochalns I

lightskin

, . bas

.FM/UIClGlnliHlGlnt--Nv..Vs....lkn..l/Nj#)umHnrU29--

' I → " 29--1 → g In =p @ win ) → Gln
  • _
PUM

Vi

= Rcolnlucmltrlss) E C' (Gln) :

IN D)

um , .mu
  • ICP )?
" atop A=e×v :

tE¥'

etxep

vdexulfiraltrpll

's

.HN#xlFfE.tH=trlxl

/ ✓ L ' I
  • extent
⇐

ex

a × → t . EXUM :=et×UM , te IR
slide-22
SLIDE 22

④ The

1st

ch . Class

vs

IE) e C' (5)

GIGUE)

Defa :

Hudak

)) ' ¥ HIGLIED HCG ) Locally : E=MxE "
  • GLIEII
Gen
  • Gink )
, E : Ex → Eyes As R Mcginty ; C' CGent taken
  • 4
Ccg )

vitEHgf-trfx.gl

YE ,

U¥EIkI=trlwkLt5

I Ag=e×9UjEagTEtg, ,xgtD C' (A) wy-wzatftfwz.at ) in P
slide-23
SLIDE 23

④

Questions : → Ch . classes for rep . Up to homotopy → Intrinsic classes for group olds I Adjoint ) → Applications