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This Concept Map, created with IHMC CmapTools, has information related to: Quaternionic planes of octonions.cmap, SPACE OF QUATERNIONIC PLANES OF OCTONION SPACE CONTAINING PREFERRED COMPLEX PLANE IS CP_2 4. Identify the choices E_1,E_2_,E_3 which differ by SU(2) rota- tion of E_1,E_2 and by U(1) rotation of E_3, SPACE OF QUATERNIONIC PLANES OF OCTONION SPACE CONTAINING PREFERRED COMPLEX PLANE IS CP_2 2. Important point: a) complexied octo- nions can be used in tangent space so that 16-D space is in question. b) This does not mean complexification of M^4xCP_2 or of M^ as I thought for some time c) Is necessary to avoid problems cau- sed by different signa- tures of M^8 and real octonion space., Decompose the tangent space unit vectors of complexified octonions O_c to 1,e_7 spanning pre- ferred complex plane E^2 plus E_1=e_1+ie_4,E_2= e_2+ ie_5, and E_3= e_3+ ie_6 and their complex con- jugates transforming as SU(3) triplet (3) and its antitriplet (3bar) meaning that all 6-planes orthogonal to E_2 correspond to these choices and are parametrized by SU(3) leaving e_7 and the spaces spanned by the triplets invariant, CP_2=SU(3)/SU2)xU(1) parametrizes the choices and has therefore number theoretic meaning raising the hope that standard model quantum numbers could be understood number-theoretic- ally, they define the same quaternionic plane so that CP_2=SU(3)/SU2)xU(1) parametrizes the choices and has therefore number theoretic meaning, Select quaternionic 4-plane containing preferred E^2 by selecting E_1,E_2 giving rise to comple- xified quaternionic plane spanned by 1,e_7,E_1,E_2, Select quaternionic 4-plane containing preferred E^2 by selecting E_2 by fixing e_7, E_1,E_2 giving rise to comple- xified quaternionic plane spanned by 1,e_7,E_1,E_2 since multiplication of E_1 and E_2 by e_7 does not lead out from the com- plexified 4-plane so that complexi- fied quaternionic plane is in question, SPACE OF QUATERNIONIC PLANES OF OCTONION SPACE CONTAINING PREFERRED COMPLEX PLANE IS CP_2 5. a) The space of qua- ternionic planes decomposes locally to S^7xCP_2: S^7 parametrizes the choices of preferred E^2. b) this is essential for M^8-H duality since it makes possib- le to fix CP_2 point characterizing tangent lane uniquely, Identify the choices E_1,E_2_,E_3 which differ by SU(2) rota- tion of E_1,E_2 and by U(1) rotation of E_3 since they define the same quaternionic plane, SPACE OF QUATERNIONIC PLANES OF OCTONION SPACE CONTAINING PREFERRED COMPLEX PLANE IS CP_2 3. Select quaternionic 4-plane containing preferred E^2, SPACE OF QUATERNIONIC PLANES OF OCTONION SPACE CONTAINING PREFERRED COMPLEX PLANE IS CP_2 1. Decompose the tangent space unit vectors of complexified octonions O_c to 1,e_7 spanning pre- ferred complex plane E^2 plus E_1=e_1+ie_4,E_2= e_2+ ie_5, and E_3= e_3+ ie_6 and their complex con- jugates transforming as SU(3) triplet (3) and its antitriplet (3bar)
