Triality Transformation And Lie Group Spin₇

The Spin group Spin(8)has three irreducible representations:R⁸,V₈⁺,V₈^-.These representations are all orientation preserving and isometric.E.Cartan had showed that there exits a kind of Triality transformation among these representations which is an automorphism of the Spin group Spin(8).In this paper,we use the method of Zhou [12],represent the Spinor space V₈⁺=C(?)₈^even·A₈(1+β₈)and V₈^-=C(?)₈^odd·A₈(1+β₈) as subspaces of Clifford algebra C(?)₈.By these representations we study the Triality transformation and give some applications.Lie group Spin₇ is the isotropy subgroup of the group SO(8)acting on the spinor A₈(1+β₈)and its subgroup G₂ is an exceptional group.By Triality transformation, we show that the group Spin₇ is isomorphic to the Spin group Spin(7).Then we show that the Grassmann manifold G(3,7)is diffeomorphic to CAY which is a submanifold of G(4,8).We also show that the Grassmann manifolds G(2,8)and G(3,8)are all the quotient spaces of Lie group Spin₇.