FIRST GENERATION FERMIONS: In the D4-D5-E6 model, the first generation of spinor fermions is represented by the octonions O, the second by O+O , and the third by O+O+O. The global structure of the D4-D5-E6 model with 8-dimensional spacetime and first generation fermions is given by a O+ Ov E6 = F4 + ( O+* b O- ) Ov* O-* (-a-b) where a O+ Ov ( O+* b O- ) = J3(O)o Ov* O-* (-a-b) Here, the O+ and O- in J3(O)o represent the O first generation fermion particles and antiparticles. Using the octonion basis { 1, e1, e2, e3, e4, e5, e6, e7 } with quaternionic subalgebra basis { 1, e1, e2, e6 }, the representation is: Octonion Fermion Particle Basis Element 1 e-neutrino e1 red up quark e2 green up quark e6 blue up quark e4 electron e3 red down quark e5 green down quark e7 blue down quark The Ov represents 8-dimensional spacetime. The Hermitian Symmetric Space representing the first generation fermions is the coset space E6 / (Spin(10) x U(1)) It has 78 - 45 - 1 = 32 real dimensions, or 16 complex dimensions, and is the Rosenfeld projective plane (CxO)P2. Since the complex numbers C have only one algebraically independent imaginary i, (CxO)P2 corresponds to a plane of one independent octonion for each of the two dimensions of the plane. (CxO)P2 corresponds to a bounded complex domain of type EIII, whose Shilov boundary is 16-dimensional, being two copies of RP1 x S7, or two copies of the octonions, one for first generation fermion particles and one for first generation fermion antiparticles.
SECOND GENERATION FERMIONS: To represent the O+O second generation of fermions, you need a structure that generalizes the equation for the first generation by having two copies of the fermion octonions. The simplest such generalization is a O+ Ov 0 O+ 0 ?E? = F4 + ( O+* b O- ) + ( O+* 0 O- ) Ov* O-* (-a-b) 0 O-* 0 This proposal fails because the O+ and O- in ?E? are not embedded in J3(O)o and therefore do not transform like the first generation O+ and O- that are embedded in J3(O)o. Therefore, make the second simplest generalization: a O+ Ov a O+ Ov ??E?? = F4 + ( O+* b O- ) + ( O+* b O- ) Ov* O-* (-a-b) Ov* O-* (-a-b) The ??E?? proposal also fails, because the algebraic structure of the two copies of J3(O)o is incomplete. To complete the algebraic structure, a third copy of J3(O)o must be added, and all three copies must be related algebraically like the imaginary quaternions { i, j, k }. This can be done by tensoring J3(O)o with the imaginary quaternions S3 = SU(2) = Spin(3) = Sp(1). Since the order of the octonions in O+O should be irrelevant (for example, the octonion pair { ei, 1 } should represent the same fermion as the octonion pair { 1, ei }, the structure must include the derivation algebra of the automorphism group of the quaternions, SU(2). The resulting structure is the 133-dimensional exceptional Lie algebra E7: a O+ Ov E7 = F4 + SU(2) + ( S3 x O+* b O- ) Ov* O-* (-a-b) Therefore E7 is the global structure algebra of the second generation fermions. The Hermitian Symmetric Space representing the secopnd generation fermions is the coset space E7 / (Spin(12) x SU(2)) It has 133 - 66 - 3 = 64 real dimensions, and is the Rosenfeld projective plane (QxO)P2. Since the quaternions Q have two algebraically independent imaginaries i and j, (QxO)P2 corresponds to a plane of two independent octonions for each of the two dimensions of the plane.
THIRD GENERATION FERMIONS: For the third generation of fermions, note that three algebraically independent copies of J3(O)o generate seven copies, corresponding to the imaginary octonions { 1, e1, e2, e3, e4, e5, e6, e7 }; that the imaginary octonions can be represented by S7; and that the derivation algebra of the automorphism group of the octonions is G2. Therefore, the 248-dimensional exceptional Lie algebra E8 a O+ Ov E8 = F4 + G2 + ( S7 x O+* b O- ) Ov* O-* (-a-b) is the global structure algebra of the third generation fermions. The Hermitian Symmetric Space representing the secopnd generation fermions is the coset space E8 / Spin(16) It has 248 - 120 = 128 real dimensions, and is the Rosenfeld projective plane (OxO)P2. Since the octonions have three algebraically independent imaginaries i, j, and E, (OxO)P2 corresponds to a plane of three independent octonions for each of the two dimensions of the plane.
Since there are only three Lie algebras in the series E6, E7, E8, there are only three generations of fermions. These E6-E7-E8 structures are based on the Freudenthal-Tits Magic Square.
The Lie groups E6, E7, and E8 are described in the book Lectures on Exceptional Lie Groups by J. F. Adams, published posthumously by Un. of Chicago Press in 1996, edited by Zafer Mahmud and Mamoru Mimura.
Rosenfeld planes are described in two books: Einstein Manifolds, Arthur L. Besse (Springer 1987), particularly tables at pages 313 and 316; Geometry of Lie Groups, Boris Rosenfeld (Kluwer 1997).
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