exo-ple ee issaks equialnar ho smc2khovanov/ma2_fall/files/lect_1.pdf · 2020. 9. 21. · mkovaov -...
TRANSCRIPT
MKovaov -
St 9, 220
S G ana m nahan GxG G( ulei caton)
unit ela t G , '8*33 eG, asso a Ehagh
MAC Crs
de Ta aM iek X. Ta s mndnes Sy (x .a
easie do udston h.
Exo-ple ee vs. issaks equialnaR
ho SmC2
C2- -2a cos Smp mas a9 anu >60
Sn smmties n-anass st
R S ies (innsh lua maps) veakn spa
2)GLn (1R)
Dha&al n D. 2a aks -smes f lau
S)On)- Sxm es nScaR
4) S
Exed: )R«I- an kjest
CAR C
anSunS. OnRp
us exenciu as ma
P Pprir)
Fie umk fuo
tra rape hen g& is a prods ma PCies.
Pna Aan a ch easin unndan on ass a aki
os >C,,*a2 .
-27 Ma ma have te dpns asic dw eas
IN-l,, 2,. . natl umas +
Qo&or amns IR a wnlnt, Colax mmlens
A+ B, A B M(IR) nrn mahias nAmadias
De A R-(R,4, ) isa ct Rum tho bina
sach dlat
(1 R,t) is en a ar groP ,
. a don + tmns R
(2) Onado aSSoave. Sh ab Lnsttas a b
(3) (ndnackon axioms RAn t a. . ht ons iht &shiuhue laws ld
(atb)c = act bc cla+b) Ca tcb.
(1) atlh=bta , ab) tC =at(btc) o ia6h (R,+)
a dihe iasy - a aie nvan af a +a =atO <a
ael-a) =(-a) +a =0
(a) (ab) c= a(be). Usuay 4iu t ptha (2)
a eR Aa af- a
Calld. wgs We ms st aaS onsi 1, aSSu
enhk kr +
Basic Praies O 0+0 O. a = (O+0>a >
O.a O.arOa Oa OarOa aaliar saup sdan t
bth=bho
Oaso
) Oa= aO = O a eR uliacadon 0 s0
-a ab a(-b)
- (a b)= -a
3 a swh's q uabon ?
ta us Precadlanca oven -/t
Ob=la l-a) )b = ab +(-adb ,
ans ob =0
Jasthihubi O= a-a)
a 8 hue vensp
O abt (-a)b > (-a) - - ab
a, a-b) - a Can
Ya,beR 2 -a)b = al-b)--al (-a) C-5)= ab
oed
(-()l Pro u. -2a = (-a)t(a)
na at at.. +a 3aat ata 0.aO
na - (na) = -(atat ta -2a -a -a. aa 2-a =ata
na rodtun ua to Ha
(eR ItleR aee 2eR, t. ER de aeR
s
dskagush hala,
-IER - =(-)n =(-t}+-t)T t-)ER nRade'.
MCe map dmes
ouo is pro duct R La mP
i, tive 0t ta (ifDo)
teud suwes a) t(a)t. t (a) tnco
N , , IR, , 2adan maduus 4-
M. IR)
s C
Mu ador in N, 2,, R, C ism u ta e
in Ma (R AB BA Ar an madnias , usua (n>)
R R is calkd
No
Des2 COm da ve T alh ba Ya,beR
oa Com nta tive ab#ba Ror Soma a, eR.
Po posi.) Z,, R, C Comnda te u a
2) MaCIR) Is a nam nunba hue , Br n>l,
Gtnnalis isd hude lawsowelr
prduîs Lmta unluss
s onudahie
intach
Ca tb)Crd) = lar5) c + (ar) d = ac r-lbctadtb&
Biu ehon ca
CatT. +0a)lh,* bt. +5) = 4, ra,h,t. to,bat ahyt@ab. 1 Q.6
(o)E;)- 20b Cannts
(o+6)latb) = aa+ abt ba + ob a tab+ba +R
aR a-a a o=a.a a..
e O o a a.. a saSsoda e
ell-dafwd
Cab)"= ababah.. ah lors, sm
R s o tae Can
(ab) atab ba tb a* 2abtb (a+b(a'rab + be tl2) (a«b) = ata+aba+ab + ba-babt bar
Ris snei 3
a3ab+ 3a b
Po R Ris Gmad and,b eR un Peo - bino i ofhcdt
Cab) ()a*"
l(n-u) Inskaun inducde ihsn
R (A)= (")+ Ca-') Pasca tia
n=l-2.3.. . so tns
Cann avid R =Z, r isa au
2
eaheam is an 3 3
Inde a malus 4 64
Slo to S
(ab)-abab. .ab = a^b"
R R
Co-b)=la-l.) Ca-)= as-a- oa + = a- 2 ab+L?
Tn e sids mdulo n -6-
Ha n 20 an onSi o an equvana laXa ton on
ab na -b
CL a, a ba an b~c
Evalena Sases Rapuga es o, 1,.. n-i OtnZ, taz n-tn Z
a dihon ans us pli a don 1 2 Jescends do a ell-dak naed a9 n ond MuOApkd a on on e w na Sostes
(atn2)+(btn 2) = atben 7Z
ell-des, (2/n, +) isa r-p on elu n.
2)lata 7)(bta z) = abtn Z
needdo ad psdatues
alanu Ao ss a'=a * kn : b +en
aen2=atn2, b'tnZ = btn2 . a a'bZ = abtnZ
a'b=la tn ) (b . en) zab + (al rkl+ke)n. Ge a trdue (Z/%, t, . ) usiluus ml n
s don Ro Cen e chack u Tc aKom3 Zisa
xa~k n-6 Zls = o, 1, 2, 31,s] 3SE2 (nl 6) Ou ds r+k 3+SP2 3-S IS3 (~l c)
3 C 3 in 6. 23 =0
Zeo y 10 in R en a Oa a eR
a =a aom) O-a =0 (eve
.a -O aO a=O Ris a si lawt R o}
so nonphism sA mnpe - isan isononpism R
s a bijechan sets
Ra ttuns
atb )= Jla)t (6) arU a, R
ab) = 1la) (&) ¥a,b eR
muln R ult n S.
Canso urit ipe
r R Exench aAn isomopisn leR do le$
L (o) =O 1r inS.
SR He nve od ,also on iso msrpsn
Copsin isoess S an ophaS m
Gentna Sahabon Ar isanopis jekh is ^ijrkon A srds
Soepis m eds (have sa candina G)
so pis ven spaas (hae so dari)
snws oS.
-8 Gnven a ,a sulun SeR ic a su el sudL Ot is an alon snant R
(2 S is sud undn nulhi a dio n So mes oni on (3)
drop aJ (î uully
ExamouS tan Tts on
7 IR CC
) sulus
L R
2) RC M (R) ala eR tor ()
aaeR
to Z. t e { JDL.-4
So oa.
Su, m EZ
n no 4iq
n
all a Dr mas
Exen u 2 [) is 5u PCm tory
20Pow Can
n=pP*? 2) Wan i 24)-zLt)? 3) Does hae sdmE hgoas foe ue ualy le
,2, Z[4),a?