Okubo algebra

A cikin algebra, **alkubra na Okubo** ko **algebra na algebra algebra** ne mai girma 8 ba tare da haɗin gwiwa ba kamar wanda Susumu Okubo yayi nazari. Okubo algebras sune algebras abun da ke ciki, algebras masu sassauƙa ( *A* ( *BA* ))&nbsp;=&nbsp;( *AB* ) *A* ), Ƙarya algebras da ake yarda da su, da haɗin gwiwa na iko, amma ba ma'amala bane, ba madadin algebras ba, kuma ba su da ƙimar ainihi.

Misalin Okubo shine algebra na 3-by-3 trace -zero matrices hadaddun, tare da samfurin *X* da *Y da* aka bayar ta *aXY*&nbsp;+&nbsp;*bYX*&nbsp;-&nbsp;Tr ( *XY* ) *I* /3 inda *nake* matrix na ainihi kuma *a* da *b* gamsar da *a*&nbsp;+&nbsp;*b*&nbsp;=&nbsp;3 *ab*&nbsp;=&nbsp;1. A Hermitian abubuwa samar da wani 8-girma real ba associative division aljabara . Irin wannan ginin yana aiki ga kowane alubabbar madaidaiciyar madaidaiciya akan filin da ke ɗauke da tushen haɗin kai na asali. Alkubra na Okubo algebra ne da aka gina ta wannan hanyar daga abubuwan da aka gano-sifili na algebra mai sauƙi -3 na tsakiya akan filin.

## Gina Para-Hurwitz algebra
Abubuwan algebras da ba a haɗa su ba ana kiran su **Hurwitz algebras.** Idan ƙasa filin K ne filin na real lambobi da kuma N ne tabbatacce-tabbataccen, sa'an nan A aka kira wani Euclidean Hurwitz aljabara .

### Scalar samfurin
Idan K yana da halayyar da ba ta yi daidai da 2 ba, to siffar bilinear (a, b ) =

### Juyin Halitta a Hurwitz algebras
Tsammani A yana da haɗin kai mai yawa, ayyana ba da izini da masu aikin ninka dama da hagu ta

: \displaystyle{\bar a=-a +2(a,1)1,\,\,\, L(a)b = ab,\,\,\, R(a)b=ba.}

Babu shakka ne involution da kuma tserar da quadratic form. Bayanin kan layi yana kuma jaddada gaskiyar cewa rikitarwa da haɗaɗɗun haɗe -haɗe sune lamurran sa. Waɗannan masu aiki suna da kaddarorin masu zuwa:

* Ƙaddamarwa shine antiautomorphism, watau a b = b a
* a mai = N (a) 1 = wani mai
* L ( a ) = L ( a )*, R ( a ) = R ( a )*, inda * nuna mai haɗin gwiwa dangane da tsari ( , )
* Re(a b ) = Re ( b a ) inda Re x = ( x + x )/2 = ( x, 1)
* Re((a b ) c ) = Re ( a ( b c ))
* L ( a 2 ) = L ( a ) 2, R ( a 2 ) = R ( a ) 2, don haka A shine madadin algebra

An tabbatar da waɗannan kaddarorin suna farawa daga sigar asalin asalin (a b, a ) = ( a, a ) ( b, b ) :

: \displaystyle{2(a,b)(c,d)=(ac,bd) + (ad,bc).}

Saitin b = 1 ko d = 1 yana samar da L ( a ) = L ( a )* da R ( c ) = R ( c )* . Don haka Re(a b ) = ( a b, 1) = ( a, b ) = ( b a, 1) = Re ( b a ) . Hakazalika (a b, c ) = ( a b, c ) = ( b, a c ) = (1, b ( a c )) = (1, ( b a ) c ) = ( b a, c ) . Don haka Re(a b ) c = (( a b ) c, 1) = ( a b, c ) = ( a, c b ) = ( a ( b c ), 1) = Re ( a ( b c )) . Ta hanyar asalin asalin N ( a ) ( c, d ) = ( a c, a d ) = ( a a c, d ) haka L ( a ) L ( a ) = N ( a ) . Aiwatar da 1 wannan yana ba a a = N ( a ) . Sauya a ta a bada wani ainihin. Sauya dabara don a a L ( a ) L ( a ) = L ( a a ) yana ba L ( a ) 2 = L ( a 2 ) .

### Para-Hurwitz algebra
Wani aikin ∗ ana iya bayyana shi a cikin Hurwitz algebra kamar

: x ∗ y = x y

Algebra (A, ∗) algebra ne wanda ba a saba da shi ba, wanda aka sani da **para-Hurwitz algebra.**  A cikin girma 4 da 8 waɗannan **para-quaternion**  da **para-octonion** algebras. 

Para-Hurwitz algebra ya gamsar 

:  (x * y ) * x = x * (y * x) = \langle x|x \rangle y \ . 

Conversely, wani aljabara tare da wani da ba degenerate fasali bilinear form gamsarwa wannan lissafi shi ne ko dai a para-Hurwitz aljabara ko wani takwas girma na karya-octonion aljabara.  Hakazalika, algebra mai sassauci yana gamsarwa

:  \langle xy | xy \rangle = \langle x|x \rangle \langle y|y \rangle \ 

ko dai aljihun Hurwitz ne, al-para-Hurwitz algebra ko algebra mai girman girma takwas.

## Duba Kuma

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** Susumu Okubo & J. Marshall Osborn (1981) "Algebras with nondegenerate associative symmetric bilinear forms permitting composition", Communications in Algebra 9(12): 1233–61,  and 9(20): 2015–73 .

## Manazarta