TUTORIAL 2
①
SLlu.IR) =
{AEGLlu.IR) : det A -
1
} EGLLu.IR) EMUHR)
⑨ open
f : Ella .IR) → IR f- det -
1
is a map
, regulierend f- '
lo) = Sllu, IR) .
By Prop .
3.1
, Tidslln , )
tut.
of Mulk ) .
= Ker (Tat ) =
kerltiddet ) .
Tadeln .IR) MULIR
) CEMUUR)
UI c- (f) =
ldttccurveiuGLK.pl.
TIdsllu.IR) .
-
.
clo) - Id <
'
lo) - C
(Kadett #
t.ge/-lldttc)-detlldttC)=detles,...,en)ttldetl4.ez. .
. .eu)
\
+ . . .
t
denen,
ich;)C
;
=
( ei =
ttucdunnof C
+ fern>
of Wagner oder
int .
dFdIt.doetlldttcI_dgfto@e.tLG , e. . .
een
! - -
t
.
t del (es , .
.
, G)
=
41 t . . t
Cnn = tr ( C) .
detlldt.tl/=1tt-trlc)thiguooderbeusiu--3T,dSLlu,lR) =
{ CEMNUR ) : trk ) = 0
} .
k! ( Tadet ) weg
Tadet # C - trlc ) .
Alternative ly , you mag orgue
like this :
It C
nepper triauguuouudrix :
C =
'
i
-
en!
"
)
¥!!
hatte
) =
! !!
"
^
.
÷"
) =
=
¥) #Itten ) .
.
.
(tttcua ) =
Cent - -
t
Gu
=
trlc) .
Any CE MUIR ) ts
triangulierte over Q
,
iue . 7 AEGLIU, G)
s.tn .
A. C. A-
^
is
triangle .
der lldtt ACA-1 )
c-
G)*
endet ( ldttc ) =
#Her (Alldtt c) A- 1) = HLACÄ)
=
trk) .
⑤ Fix A E SLIU.IR) :
Wenzel to work this
üeuj
#
IB) = A- '
B A ( ABA
-1
=
cauj .
)
(
Ohja : SLlu.IR ) →
SLlu.IR/f#ceuja:MullR)-MullR) is a Linear
map and
Jo also Smooth .
Tidcoujn :
T.dsllu.IR) → TDSLIUIIR)
Tiacaej +
(c) = AYCA by liueaoohy of ¥) .
② Ad : Sun.IR ) → GLITdsllu.IR) ) E- Hautfalte.IR)
,
Ä '→ Idealist
T.dslk.IR).
Shutout :
µ : Multi ) × MNLIR ) - Hulk ) tuultpi .
isch .
=) AH JA halb) =
MIA, B )
Antti
↳ (B) =
µ (B. A)
↳ im swwok .
Also
,
inversen is smooth i : A - A1 :
F :
GLIu.IR/xGLlu,tR)-GLlu.lR)xGLlu,lR)
FIA ,
B) = Itt ,
AB ) is smooth .
Tat ( c ,
D) = ( C
,
Ct D) is a linear Dauerheim .
=) ↳
udly eraud lld .
Id ) ,
F odunts a seuoohieuuse
FBy
definition ,
EIH ,
lol ) =
( A ,
ILA ) ) = (A , A)
and Vence i is Smooth .
Also ,
¢ CEMULIR ) ,
clt ) = ldttc Curve a
GLK.IR/clt).clt)-1=ld.Differeutieiggives:0--dd-I+=ft).cHT1=c'(o ) .
do)-1
+ do) - d
FLEHT!
-
¥1 ¥
"
=
-
c.
6)
"
c
'
6) el
!
-
1
= - eko ) .
- -
F
-
-
Heine
,
Ad :
SLlu.IR) → GLLTIdshlu.IR) )
A n
Talonja ( Cm AYCA )
Learjet =
JA- i ft
Tee Ad :
Tasche.IR ) →
Td GUT,ask.IR))
= Hau IIDSLKIRISLK.IR/)
Hatd) (c) (D) =
d)Edlen) (D) =
# TDI)
= - c
'
lo) Dclo) t do)
-
'
De
'
)
=
Id
-
Id
E)
= CD
+ DC =
- IC ,
D] .
Kanutin
of wahres .
Reine Tanken Taking can
) : B - A.BA
-
1
( mal A- IBA )
leid, fler H ) then just [ C. D] .
(Tdsuu.IR ) ,
T ,
] ) is the hie
dgenraafklu.IR) .
[
,
] :
T.dshlu.IR/xTdSLlu.tR)-TaSLln.lR)bilihew,skew-symuaneund [C.
EDI] tED.EE ,
Jacobi kaaiitg .
TEE ,
EC, D)=D
Siuiherly ,
(Td G- Llu , IR) =
MNHR) ,
[ ,
] ) is the hie
wg .
of
G-Llu ,
IR ) und
simihdg for my
uuetix
graep.ie.
ok hie Sub
group of GLK.IR) .
② IR
"
equipped with on inner product of Signature ( me ) :
< x. y) =
!?" bi
In
Yi
* II.es In.
=
! .
.
.
.
:) -
IR
"
wir an inner producer of Sign . (1,3 ) =
=
Minkowski Space -
Olpiq) :
=
{
AEGLluilRIiLAxiAy7-LxiyYVxytIRkG@LAx.Ay) =
# ftp.eay-xt/ttIp.gAy 4)
(*) = +
'
Ip , ey ttxiyER"
⇐ At Ing A =
Ing ⇐
At Ima
=
Imp A- 1
f : GLK.IR ) → Munz )
⇐ Irie At Ing
- A -1
.
Ah A- Ip ,
At Ing
f-^
( O ) =
Olp, g) .
-
f (A)
t
=
ftp.e At ) E
-
Ing
=
A
Ing At Ipy
= FIA ) .
f-
: GLln.IR) → MI"
LIR) E Mu LIR ) .
Regula,
hey of f : AE 01mg ) ,
SE MY ( IR )
Then see ( =
ELEA .
If (C ) =
ftp.gcttctp.g At =
-
.
.
Ein
4) =
tz ( A
Ip.pk Ine A)
t
t
Stine AIne
At )"
i A
räst =
± Is a s
) = S .
Ins
- 01mg ) is a submfd .
ef NUHR ) of dim .
nz Utz
=
Zu?
-
n
?
-
h
=
nln -1 )
→
⑤ A ,
B E Ol
.mg
)
(A- B)t.Ip.ee/AB)t=ABIBtAt--Ip,g .
\
=) A BE
O (ng ) .
"
" 9
A-
"
Ine # It =
Ipp #
Ip.jp#t)-1=Ip.q.A-1E0lpcp) Ist
=) 01mg ) E GLIu.IR ) is a sub
group of GLIu.IR) .
und uudnx ueultplicote • Ohne) × Olp, g) -104,9)
B Sueooh as msn.cieeefasmootmop.MN/lP)xMulIR)
→
MAUR) .
Haue ,
Olma ) is a hie
group .
⑦ ② Tdolp, a) =
Ker (Tat )
,
{ CEMUUR ) :
Inqctt CIp.JO
⇐ > Lte Ip,
ftp.gIR "
011,3 ) ar 013,1) is
called Lorentz group -