
quadratic jordan beejaganitam

ganita saastramlo, quadratic jordan algibras anedi jordan algibras yokka saadhaaraneekarana. leanier jordan beejaganitam yokka quadratic praatinidhyam yokka praathamika gurtimpulu ekapaksha lakshanam unna kshetrampai quadratic jordan beejaganitam nirvachinchadaaniki suutraalugaa upayoginchabadataayi. parimita-dimensional saadhaarana quadratic jordan algibras yokka ekareeti vivarana undhi, idhi lakshanam nundi swatantramgaa untundi. gunakaala rangamlo 2 vilomam ayithe, quadratic jordan algibras siddhaantam leanier jordan algibras siddhaantaaniki taggutundi.
nirvachanam.
oka chaturasraakaara jordan beejaganitamlo oka pratyaekamaina moolakam 1 thoo oka kshetram "k" paie vector space "a" untundi mariyu "a" yokka chaturasraakaara patam "a", "a", "q" ("a") yokka "k"-endomorphisms loki untundi, idhi paristhitulanu santruptiparustundihi
inca, eee lakshanaalu scalarla yokka edainaa podigimpu kinda unchukoovaali.
amsaalu.
oka moolakam "a" vilomam ayithe vilomam avutundi mariyu dheeniki vilomam unnavi unnaayi mariyu atuvanti "b" pratyekamainadi mariyu "b" anedi "a" kii vilomam anie memu chebutamu. jordan vibhajana beejaganitam anedi prati sunna kaani moolakam vilomam cheyagaladi.
nirmaanam.
b" anedi "a" yokka upasthanamgaa undanivvandi. b" looni anni "b" laku "q" ("b") chitram "b" loe unte "b" nuu chaturasraakaara ideal ledhaa antargata idealga nirvachinchandi; "a" looni anni "a" laku "b" nuu prati "q" ("a") dwara mapp chesinatlayithe "b" nuu bahya idealga nirvachinchandi. a" yokka ideal anedi oka subspace, idhi antargata mariyu bahya ideal rendoo. oka chaturasraakaara jordan beejaganitam chinnapati aadarsaalu laenitlayitae adhi chaalaa sulabham.
icchina "b" kosam, "q" ("b") yokka chitram oka antargata ideal dheenini "b" paie pradhana antargata ideal anie pilustaamu.
a" yokka centroid γ anedi endomorphisms "t" thoo kuudina mugimpu "k" ("a") yokka upasamiti, idhi "q" thoo "commute" chestundi, antey andarikee "a"
oka saadhaarana beejaganitam yokka kendrakam oka kshetram daani kendrakam kevalam "k" ayithe "a" kendramgaa untundi.
udaaharanalu.
anubandha beejaganitam nundi quadratic jordan beejaganitam.
a" anedi gunakaaram × thoo "k" paie ekasambandha anubandha beejaganitam ayithe, appudu "q" anae chaturasraakaara pataanni "a" nundi mugimpu "k" ("a") dwara "q" ("a") dwara nirvachinchavacchuh "b" "a" "×" b "" × "a". idhi "a" paie quadratic jordan beejaganitam nirmaanaanni nirvachistundi. atuvanti beejaganitam yokka upa beejaganitam yokka samaruupata unte, oka chaturasraakaara jordan beejaganitam pratyekamainadi, lekapothe asaadhaaranamainadi.
chaturasraakaara roopam nundi chaturasraakaara jordan beejaganitam.
a" anedi "q" anae chaturasraakaara roopam mariyu anubandha samaroopa dvimukha roopam "q" ("x", "y") = "q" ("x" + "y")-"q" ("x")-"q" ("y") thoo "k" paie vector space gaa undanivvandi. e" anedi "a" yokka "bays point", antey "q" ("e") = 1 thoo oka moolakam. oka sarala kriyaatmaka "t" ("y") = "q" ("y", "e") mariyu oka "pratibimbam" y "≤ =" t "(" y ")" e "-" y "nirvachinchandi. prati "x" kii manam "q" ("x") nii nirvachistaamu
appudu "q" anedi "a" paie quadratic jordan beejaganitam nirvachistundi.
leanier jordan beejaganitam nundi quadratic jordan beejaganitam.
a" anedi 2kii samanam kaani lakshanam gala "k" feeled meeda ekjordan beejaganitam avandi. "a" loe "a" kosam, "l" anedi anubandha enveloping beejaganitamlo edama gunakaara pataanni suuchistumdi.
mariyu "a" yokka "k"-endomorphismnu nirvachinchandi, dheenini quadratic representation anie pilustaaru,
appudu "q" anedi quadratic jordan beejaganitamnu nirvachistundi.
quadratic jordan beejaganitam leanier jordan beejaganitam dwara nirvachinchabadindi.
chaturasra gurtimpulanu viloma muulakaanni upayoginchina garishta kocharnu anusarinchi r ledhaa c paie parimita-dimensional jordan beejaganitamlo niroopinchavacchu. avi ekasambandha anubandha beejaganitam (oka "pratyeka" jordan beejaganitam) dwara nirvachinchabadina jordan beejaganitam loe kooda niroopinchadam sulabham, endukante aaa sandarbhamlo "q" ("a") "b" = "aba". avi 2kii samanam kaani lakshanam gala kshetrampai edainaa jordan beejaganitamlo chellubaatu avtayi. idhi jacobson chetha oohinchabadindi mariyu niroopinchabadindih muudavadaaniloo saralamaina moodu chararaasulaloo bahupadi gurtimpu edainaa pratyeka jordan beejaganitamlo chellubaatu ayithe, adhi anni jordan algibralalo untundi. praathamika rujuvulo, mccrimmon mariyu meyberg kaaranamgaa, 2kii samanam kaani lakshanaala kshetrampai jordan algibralaku ivvabadutundi.
coacher yokka rujuvu.
koecher vaadanalu vaastava ledhaa sanklishta sankhyalapai parimita-dimensional jordan algibralaku vartistaayi.
praathamika gurtimpu i.
a" looni moolakam r ["a"] ledhaa c ["a"] loe invertible ayithe daanini "invertible" antaaru. b" anedi vilomaanni suuchisthee, "a" yokka shakti anubandhata anedi "l" ("a") mariyu "l" ("b") prayaanaanni chooputundi.
vaastavaaniki "q" ("a") invertible ayithe matrame "a" invertible avutundi. aaa sandarbhamlo
nijaaniki "q" ("a") vilomamgaa unte adhi r ["a"] nuu thana meeda taanu mosukelutundi. marovaipu "q" ("a") 1 = "a" 2, kabatti
jordan gurtimpu
a" nuu "a" + "tc" thoo bharti cheyadam dwara mariyu "t" yokka gunakam teesukovadam dwara dhruveekarinchavachhu. dheenini "sii" digubadiki vartinche operatorga tirigi vraayandi
eee dhruveekruta jordan gurtimpulo "b" = "a" −1 teesukovadam valla phalitam labhistundi.
a" nuu daani vilomamtho bharti chesthe, "l" ("a") mariyu "l" ("a"-1) vilomamgaa unte sambandham anusaristundi. lekapothe idhi "a" + ε1 kii ε ekapakshamgaa chinnadigaa mariyu anduvalla kooda parimitiloo untundi.
a" loe "c" mariyu "f" ("a") kosam "a" loe oka function, chivara "a" loe viluvalatho, lett
"d" "c" "f" "(" "a" ") anedi" "f" "(" "a" "+" "tc" ") yokka" "t" "= 0 vadda utpannam avutundi". appudu
ikkada "q" ("a", "b") anedi "q" yokka dhruveekarana ayithe"
l" ("a") anedi "l" ("a" −1) thoo prayaanistundi kabatti
anduvalla
tadwara
d" "c" nuu "l" ("a" −1) "q" ("a") = "l" ("a") kuu vartimpajeyadam mariyu "b" = "c" −1 paie panicheyadam valla digubadi labhistundi ".
marovaipu "l" ("q" ("a") "b") anedi oka bahiranga dattamaina samiti meeda vilomamgaa untundi, ikkada "q" ("a") "b" kooda vilomamgaa undaali.
paie vyakteekaranalooni chararashi "bii" loe utpannam "dii" sii "nuu teesukovadam valla istundi
idhi dattamaina viloma muulakaala samuuhaaniki praathamika gurtimpunu istundi, kabatti idhi saadhaaranamgaa konasagimpu dwara anusaristundi. a" mariyu "b" vilomamgaa unte mariyu "q" ("c") vilomaaniki suutraanni isthe "c" = "q" ("a") "b" vilomamgaa untundani praathamika gurtimpu suuchistumdi. dheenini "sii" kii vartimpajeyadam valla puurti saadhaaraneekaranalo viloma gurtimpunu istundi.
commutation identity i.
paina chuupina vidhamgaa, "a" vilomam ayithe,
chararashi icchinatlugaa "a" thoo "d" "c" nii teesukovadam
a" nii "a"-1 thoo bharti cheyadam, "q" ("a") nii vartimpajeyadam mariyu praathamika gurtimpunu upayoginchadam dwara istundi.
anduvalla
bii" mariyu "sii" parasparam marpidi cheyadam valla labhistundi
marovaipu dheeni dwara nirvachinchabadindi, kabatti idhi suuchistumdi
tadwara "a" kosam invertible mariyu anduvalla anni "a" kosam konasagimpu dwara"
mccrimmon-mayberg rujuvu.
commutation identity II.
a" nuu "a" + "tc" thoo bharti cheyadam dwara mariyu "t" yokka gunakam teesukovadam dwara jordan gurtimpunu dhruveekarinchavachhu. idhi istundi
operator sangnaamaanamlo idhi suuchistumdi
a" loe dhruveekarana mallii istundi
dii" paie panichese operatorluga vrayabadi, idhi istundi
sii" nii "bii" mariyu "bii" nii "eah" thoo bharti cheyadam valla vastundi
alaage, kudi chethi vaipu "b" mariyu "c" loe samaruupamgaa unnanduna, edama vaipuna "b" mariyu "c" lanu parasparam maarchukoni, teesivestuu, commutatorlu ["l" ("b"), l ("c")] jordan beejaganitam yokka utpannaalu.
veelu.
appudu "q" ("a") jordan gurtimpu dwara "l" ("a") thoo prayaanistundi.
nirvachanaala nundi anubandha cemetric bilinear mapping ayithe, appudu mariyu
antekakunda
nijaaniki
rendava mariyu modati dhruveekruta jordan gurtimpula dwara idhi suuchistumdi
dhruveekarinchabadina samskarana
ippudu, adhi anusaristundi
kabatti "bii" sthaanamlo "abb" thoo chivari gurtimpu dwara idhi parivartana gurtimpunu suchistundihi
q" ("a") "r" ("b", "a") = "r" ("a", "b") "q" ("a") anae gurtimpunu balopetam cheyavachu.
nijaaniki "sii" kii vartistundi, modati rendu padaalu istayi
bii" mariyu "sii" maaradam taruvaata istundi
praathamika gurtimpu II.
abaddham brocket sambandhaalanu upayoginchi gurtimpu niroopinchabadutundi
nijaaniki gurtimpu yokka "sii" loe dhruveekarana istundi
dii" kii rendu vaipula vartimpajeyadam, idhi choopistundi
mukhyamgaa eee sameekaranaalu "x" = "ab" kosam untaayi. marovaipu "t" = ["l" ("a"), "l" ("b")] ayithe "d" ("z") = "tz" anedi jordan beejaganitam nundi udbhavinchinadi, tadwara
abaddhapu brocket sambandhaalu anusaristaayi endukante "r" ("a", "b") = "t" + "l" ("ab").
edama vaipuna unna abaddham brocket pratisamiteeyamgaa untundi kabatti,
phalitamgaa
nijaaniki "a" = "y", "b" = "x", "c" = "z", "d" = "x" anie sett chesi, rendu vaipula "y" paie charya teesukondi.
marovaipu
vaastavaaniki idhi "x" = "q" ("a") "b" nii amarchadam dwara anusaristundi.
anduvalla, eee sameekaranaalanu balopetam chesina parivartana gurtimputoe kalapadam,
leanier jordan beejaganitam anedi quadratic jordan beejaganitam dwara nirvachinchabadindi.
a" anedi r ledhaa c paie quadratic jordan beejaganitam anukundam. taruvaata, oka leanier jordan beejaganitam nirmaanaanni "a" thoo anubandhinchavacchu, antey "l" ("a") anedi jordan gunakaaram ayithe, appudu chaturasraakaara nirmaanam "q" ("a") = 2 "l" ("a") 2 − "l" ("a" 2) dwara ivvabadutundi.
modata suutram "q" ("a") "r" ("b", "a") = "r"
("a", "b") "q" ("a") nuu balopetam cheyavachu
nijaaniki "sii" kii vartistundi, modati rendu padaalu istayi
bii" mariyu "sii" maaradam taruvaata istundi
ippudu veelu
paina perkonna gurtimpulo "bii" nuu "eah" mariyu "eah" nuu 1 thoo bharti cheyadam valla vastundi.
mukhyamgaa
inca "a" vilomam ayithe
adevidhamgaa 'bii' vilomam ayithe
jordan utpatti dwara ivvabadutundi
tadwara
paina perkonna suutram 1 anedi oka gurtimpu anie choopistundi. a" "2 nuu" "a" "δ" a "=" "q" ("a") 1 dwara nirvachinchadam, dhruveekarinchavalasina ekaika sharatu jordan gurtimpu matrame ".
praathamika gurtimpulo
a" nii "a" + "t" thoo bharti cheyandi, "b" = 1 nii sett cheyandi mariyu rendu vaipula "t" 2 yokka gunakaalanu polchandi
rendava suutramloe "b" = 1 amarchadam istundi
anduvalla "l" ("a") anedi "l" ("a" 2) thoo prayaaninchaali.
gurtimpu marchandi.
ekakaalika sarala jordan beejaganitamlo shift gurtimpu nokki chebutundi
taruvaata, praathamika gurtimpu yokka dhruveekruta roopaalu mariyu parivartana ledhaa homotopy gurtimpu yokka pratyaksha paryavasanamga dheenini sthaapinchavachu. idhi rendu chararaasulanu matrame kaligi unna operator gurtimpu kabatti idhi mcdonald siddhaantam yokka parinaamam kooda.
ekakaalika sarala jordan beejaganitamlo "a" kosam "a" chaturasra praatinidhyam ivvabadutundi
kabatti sambandhita cemetric bilinear mapping
itara operatorlu suutram dwara ivvabadataayi
tadwara
commutation ledhaa homotopy gurtimpu
a" loe dhruveekarinchavachhu. a" nii "a" + "t" 1 thoo bharti chesi, "t" yokka gunakam teesukunte istundi.
praathamika gurtimpu
a" loe dhruveekarinchavachhu. a" nuu "a" + "t" 1 thoo bharti chesi, "t" yokka gunakaalanu teesukoni istundi ("a" mariyu "b" lanu parasparam marchukovadam)
munupati pradarsinchina rendu gurtimpulanu kalapadam valla labham labhistundi
praathamika gurtimpulo "a" nuu "a" + "t" 1 thoo bharti cheyadam mariyu "t" 2 yokka gunakam teesukovadam valana vastundi.
kudi chethi vaipu samaruupamgaa unnanduna idhi suuchistumdi
shift gurtimpunu niroopinchadaaniki eee gurtimpulanu upayoginchavachhuh
idhi gurtimputoe samanam
munupati pradarsinchina gurtimpu dwara idhi samanam
marovaipu, moodava pradarsinchabadina gurtimpu dwara brocket chesina padaalanu saralikrutam cheyavachu. idhi rendu vaipula samanamga unnayani suuchistumdi.
parimita-dimensional unital jordan algibras kosam, shift identityni utparivartanalanu upayoginchi marinta neerugaa chudavachu. a" mariyu "b" vilomamgaa undanivvandi, mariyu
 a" b "loe jordan gunakaaram ayi undandi. appudu
. antekakunda
marovaipu mariyu adevidhamgaa "a" mariyu "b" parasparam marai. anduvalla
eee vidhamgaa
kabatti shift identity "q" ("b") nuu raddu cheyadam dwara anusaristundi. saandrata vaadana anedi viloma bhavananu tolaginchadaaniki anumatistundi.
jordan jantalu.
oka leanier unital jordan beejaganitam praathamika gurtimpu, homotopy gurtimpu yokka parivartana mariyu shift identityni santruptipariche quadratic mapping "q" mariyu anubandha mapping "r" kuu daariteestundi. oka jordan jatalo rendu vector space mariyu nundi nundi rendu quadratic mappinglu untaayi. ivi ekkada nundi suutram dwara dvimukha mappinglanu nirnayistaayi. ± subscriptsnu tolagiste, ivi santrupti chendaali
praathamika gurtimpu
commutation ledhaa homotopy gurtimpu
mariyu maarpu gurtimpu
ekjordan beejaganitam "a" anedi "v" ± = "a" nuu daani chaturasraakaara nirmaana pataalu "q" mariyu "r" thoo teesukoni jordan jatanu nirvachistundi.