  In this paper we postulate an algebraic model to explain how the symmetry of
three lepton species plays its role in the Lorentz extension. Inspired by the
two-to-one mapping between the group SL (2, C) and the Lorentz group, we design
a mapping between SL (3, C) group, which displays the family symmetry, and a
generalized Lorentz group. Following the conventional method, we apply the
mapping results to Dirac equation to discuss its transformation invariance, and
it turns out that only when the vertex matrix is extended to the combination
can the Dirac-equation-form be reserved. At the same time we find that the
Lorentz group has to be extended with an additional generator . The generalized
vertex matrix is helpful in understanding the axial-like form of weak
interaction and the neutrino oscillations.
