4. 2 Review : Multliueor algebra
°
Suppe>
e Vs , . .
.
Vr are (red) finite dim
.
Vector Spaces .
For a vector
sp .
W we wrihe L (Vs ,
. .
,
Vr ;
W ) for
the Vector
Space r - Linear mops thx . _
× Vr → W .
Def.4.li Vn , .
.
.
Vr Vectra
sp .
⑦The tensor product of Vs ,
.
.
, Vr is
V
,
.
. . ⑦ Vr : LIVE
. .
. ! ;
R ) .
② Er tun ,
. .
, vr) Ehe . . x Vr We wirke
4 ⑦ . . ⑦ Vr E ↳ ⑦ - . ① Vr
for the 40 . . ⑦
vr : ( Js , .
.
.br/i-;T=IJilu;) .
Note that the
uuop :
Vax . .
xvr → 40 - - ⑦ Vr
( 4 ,
_ .
Nr ) N 4 ④ _ .
④ Vr
is u r -
linear
mop ,
i. e .
① ELLY , . .
.
KiKa . .
Kr) .
Some properties of the tensor products
-
•
Univ For
any
r -
linear
mop
f :
Vsx . . ×
Vr → W
into a weder
sp .
W 7 ! liuew
wop Ä :
40 ⑦
Ur - W
s.hr .
f- Äo ④ .
v
,
× .
.
+ v. f- W
I ⑦
.
.
¥
(ffn nur
)
40 -
①
Vr
Fly
a . .
)!
In
particulier ,
ftsfdefinesanisomopuisnl.LY,
Nr
; W)
④" "
%EE.Y.ii.%ecg%ae.ec?r-.wB-lMo*vriW) .
•
Association :
. Uh ① 4) ⑦ ↳ ↳ ⑦
Vz ⑦ ↳
(naturel
isomorphem ) .
.
(Y ⑦ Vz) ⑦
↳ =
Y Vs ⑦
↳ ⑦ ↳
•
Basis : If { ei
,}, ±,
is a boris of V
; ,
i -
1.
.
.
,
r .
wie n
;
=
diwlvi ) .
then { (en
,
- -
⑦
erisr) }
sei ,
⇐
ni , sei Er
is a how for Vs ⑦ .
.
④ Vr dim
IV, . . ⑦ 4) =
!!dimlvi ) .
• 7 Canonical isauopn.sn : .
VÄ VI = (Vs ⑦ 4)
*
In ⑦ Jz F)
(4041141%2)
•
Via Vz =
44 , K )
In ⑦ Vz 1- Ihn hin ) vz
) .
. If fi : V
,
→
Wi are linear
mops i - 1
,
. .
,
r
,
then
by the Univ .
property 7 ! linear
cuop f,
. .
④ fr srl .
⑦
↳ x . . x
Vr - 4×0 . .
① Vr
6mm Ks :
[ fix . xfr
lfn ⑦ .
.
④ fr
W ,
x .
. × Wr ¥ W
, ① .
.
① Wr
Det Suppose V is a vector
sp .
and Wide
Lr IV , IR) : - L ( Vx . xv;
R ) =
V
*
① . .
⑦ V *
- _
-
r
r
① A r -
linea
more we Lrlv, IR ) is called alter
uating , if
W (von, i . .
Nrw)
) =
Sign lo) W ( m , . .
, vr )
✓
4 ,
- Nr
EU and RE Sr : =
{ r : { 1
,
- r
} 791,
er
}
bijediens } .
(⇐ w vannes it one
iusers an danach twice ) Notation:
We wie iv. =L:* MIR) ELRLYIR ) for
the sub
Space of r - linear
alternative mops
.
We have a natural project ieee : AH : UHR) → % ( KR),
Called alternativ
, give by :
Alt lw) k .
. .
Nr ) : =
¥ fjrguldwlu.cn.
. .
in).
Note that
, if we War LYR) ,
than Alt Lw ) = w .
¥ It follow, that ,
it r > d.mlv) ,
then Nv*
= 0
by r linear
.ly and since altern mops
Vennes if an dauert
geh
insehed twice .
- Also
,
if r =
dimlv) ,
then NV is 1- dein .
:
Fix u basis B =
{ es ,
. .
, er} of V
,
then de) :
KVTIR
( the detern .
of r neuer ) is on element
of NX #
.
Ihr WE NW *
,
then wlv, .
. .
µ ) =
detslv, .
.
.
. 4) when,
_ .
, er).
=) NV
*
= IR
detß Notation
ÄV : = ⑦ NV'
wihku convention
r
? 0
µV : = IR 1^4 : = V*
It is o finite dim
.
Vector
Space and
any linear
uvp f : ✓→ W
iuauce, a linear
luop f- *
: NW *
→ NV*
give by
few K .
. .
, vr ) = w (fln ),
.
.
, flur) ) .
Which exheuds to a Linear
uop f
'
: AW '
→
Ava .
Note tud ( g. f)
'
=
f :S for g
: WAZ hier
weg .
Def.4.6-we.NU ,
ze
ASV '
.
Then their
Wedge product
way E Arts wa is
giveu by :
Wir LY ,
. .
, uns ) = Alt lw
g) ↳ . . .
, v. s
)
¥.
!!!
" "
!" inkl:
%!
By liuoarhg , we war * had two to TV :
{
w
; ^
Eh;
: =
¥,
Wi 14; wie g.
EÄV'
.
Prop.4.7-thevechorspoceNVF-qfrveguippedw.hn
is an ( ossociohiue ,
uuitid )
gneded-lowuhati-edgebray.ie.
:
① ( wa
g) ne =
wnlqse) wie,
es iv.
② 1 EIR = IYV'
Sdi >
fies W two tv.
.
③ NV '
n AV '
EN "
V '
( groelend algebra ) .
④ we NV '
, qe Astra :
way =L-
1)
"
ysw
(
groelend wenn Latina ) .
More one ,
for
any
linear
map f : VTW , tue luop f-
*
: AW '
→ Aux
is a
grooud algebra morgen su : f- ( ung) =
für ff
fahr wa c NV '
.
f-
"
1 =
1 .
Prop
① If we , . .
, wr EV →
and 4 , . .
,
v
,
EV ,
then
war . .
nur 14, . .
, m ) = det Hits)) i.+ r
In
particulier , Ws ,
. .
, w ,
are
tiuuoly Inder .
⇐ Was .
1W,
# 0 .
② If 9h ,
. .
, du} is a basis of V
"
,
then
{ hier . .
dir : 1 Ein < izc . .
< ir En }
is u basis of N V '
.
4.3in
M mfd .
£
q
um
* Fix × EM Cousin
IMOP.to/ItFTQTiM ① .
. ⑦IM =
= LLTÄM ,
. .
, M.IM ,
. .
.IM ; IR) .
¥ )
- -
q
und elende
by (TM )
②
P
# M ) 9 the di>
joint Knien
• E ¥ ) over dt x EM .
.
We have a natural project on IT : TM Pox TM ② 9
→ M .
It vdeuits kw Structure
of a smookveaar made over M
( in duced from the Uecker hundle sruc . an TM and TM) .
Def
① A (smooh) Htt an M is a ( smooth ) seelen
of IT : TMQP ④HM )⑦ 9
→
M .
② We wie TP
,
(m) for the near
space of IF ) -
Tensors
an M
, www. is also a modul der (M
,
IR) .
If d E
Tf In) , y c- Tsr (M )
,
then 0/0×4 defiued by
( 10 ① y ) k) : =
Hk) ① YK) tx c- M .
is a
! ) tensor
on
M lxnlolx.vn ) 0×04.
Louvoeitiee
of sweden
maps )
Support
( V. n) is a dort
,
than { # ⑦ - ⑦
¥+0 du"
⑦ . .
#diese
ETF Iv)
form a basis of T
-
1k ) =
IM P
① II M€9
.
Kx EU -
House
my sediouo of IT can be written es :
4) $1 ¥?÷!
; #* *
# ⑦ du " * -
reden:
U
IST
for real- ualued for ¥! ¥,
an
0 .
Smooth non
of §
is eguiv.to smooth f 0¥ :) her
ayaort .
They are called the Loud coordiwde
express ia of $ Wir .
to Win) Reine
We how
locdly way
henkers and vie
par titans
of
un
.ly also
globally .
G- iveu § E
TI ( M) we Lea landet a
map ;
also deuohed
by d)
9iua.by:2 ~
§ : TLTM ) × . . x TLTM) t TITM ) x . .
XTHM)
→ Olm .IR ) ¥)
| Lw: . .
ii. u . . . . 4) * E
"
*).
By ueustrucliau
,
this
more is
CYR) linear
in eaa
entry and
by H ) 011W ! . .
w! ! .
. .
,
sg) is iadeediu ETM
!since loudly on the domain!) a
Chat it is gib by
{ d ¥,
wi .
. w
! ) .
.
sir which is a
pm of a
Product
of Imad fcls .
Reine A Sea end of IT is Sunde ⇐ 0/41.
.
, w
! } , . .
, <
g)
In for sund 1- keins wi audveded Leeds
sj .
( ol ¥ =
olluldü:-. ,
dän
, ¥.
. . .
, ¥.
) i
Ro
Rennert :
Special loses .
.
10 E TI ( M ) = T (TM ) is a 1- form and we
know
dnedy 01h ) : M → IR is for SE TCTM ) .
•
OIET! IM ) =
T ( TM ) is a veda held :
§ (w) = w (y) : M → IR wihiu is a for Olt TM).
Prop.tl#)defiuesaliueorisauorpwsn between
Trevi and the
www.cewi-kdiiiii
"
: ¥: %;)
y
IM AR) multi
Linear
mops .
Proof We alneody know that d E
TI (M ) gics rise to
an deinen IÜWP and that 4) is linear fand in
jeder ) .
Carver sdy , lel 01 : THM ) x - x TITM) → Cola, IR) he
LM
,
IR) linear
.
Then we love to show that
0/4) (wik) .
.
. . sgk) ) : =
011W? . . . 4) (x) .
for 1- form, wi and reger fields sj just Mepm
on the Value s
of the w
'
s und z
'
s at × .
In this Core
,
+ AM×
is a ( Y ) . heuser .
It is sufticieul to show that if ay of the 1- fans arueao
finds r venus at ×
,
so does
011W? Ise ) k) .
Support first r raines ident
udlg on
open heights. U
of x
and letfecdM.IR) sie .
f) M¥1 and fk) - O (
we!!),
Then fr =
r and my Calm , IR) liuaerhy,
we
" " "
) '
hae
j.fr,
.
.
g) 4) =
fk) dlwi.ir, . .
so) k) = 0
This in
plies
that for a how ( Ku ) wir XEO ,
¢ Lw! . .
, so) ) u
( in
polinnen utx ) , ja ,
ldlpeuds un tue neshidihs of the w 's and s
'
s
to U We
love
aha, .
. .
.si/u--EdiI::u:-u!sI...gie ↳ * ).
-
0119dnis
,
_ .
.
%) .
→ If r 4) =
0 ,
then so one the load nordisch lepaessie
at X
and heule klar ? Sg ) (e) = 0
by TG) .
Rennen Elements T! ( M) ( resp .
T
; ( M ) ) are called
p
- Times contre vibriert ( resp .
p
Times
Covenant tensors ) .
Er . (pseudo.)-
←
A tensor
go.TL IM ) is called a Riemann: an
uahäc on
My
it for
any
x EM the hihi wer fan gx : IMXIM → IR
is
symuetrc and non degenerde
-
H M is come chad ,
the
signature (ng ) of gk) does not
depeud on x and is netter referent to a the
Signature ofg .
In particulier , g
is called a Riemen nie mehr C
,
if
ghos sign . (n
,
o ) and Lorentzen if the
signdue
is In -1,1) or ( 1 , n e)
.
TM - R
"
Standard inner
gmail give , Hot)
Euch den uehic
g
=
du②dritt . - tdx"
die .
Standard Lereuttia Neuer product an IR"
Girls ( flat ) Lorentz in / Minkowski neue
gr dxt
① die + die die t et
die ④die .