Triality and Dual Equivalence Between Dirac Field and Topologically Massive Gauge Field

It is proved that there exists a vector representation of Dirac's spinor field and in one sense it is equivalent to biquaternion (i.e. complexified quaternion) representation. This can beconsidered as a generalization of Cartan's idea of triality toDirac's spinors. In the vector representation the first-orderDirac Lagrangian is dual-equivalent to the two-order Lagrangian oftopologically massive gauge field. The potential field which corresponds to the Dirac field is obtained by using master (or parent) action approach. The novel gauge field is self-dual andcontains both anti-symmetric Lee and symmetric Jordan structure.