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 )) = ( 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 + bYX - Tr ( XY ) I /3 inda nake matrix na ainihi kuma a da b gamsar da a + b = 3 ab = 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.Max-Albert Knus, Alexander Merkurjev, Markus Rost, Jean-Pierre Tignol (1998) "Composition and Triality", chapter 8 in The Book of Involutions, pp 451–511, Colloquium Publications v 44, American Mathematical Society 

 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



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 ) :



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 The term "para-quaternions" is sometimes applied to unrelated algebras. da para-octonion algebras. 

Para-Hurwitz algebra ya gamsar 



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



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

 Duba Kuma 



 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 .

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