**On parametrization of the finite transformations of the unitary group
SU(4)**

**Abstract:**

The Dirac matrix basis is taken to construct the theory of the unitary
group *SU(4)*.
Parametrization of finit matrices *G* of the complex linear group *GL(4,C)*
in terms of four
complex vector-parameters *G=G(k,m,n,l)* is proposed and investigated.
Additional restrictions
separating some sub-groups of *GL(4,C)* are given explicitly. In the given
parametrization, the problem
of inverting any matrix *G* is solved. Expression for determinant of *G* is
found: det *G = F(k,m,n,l*).
Unitarity conditions in the theory on *SU(4)* group on the base of complex
vector parametrization are investigated.
Unitarity conditions have been formulated in the form of non-linear
cubic algebraic equations for 16 complex-valued
pavariables. Several simplest types of solutions have been constructed:
1-parametric Abelian sub-group; three
2-parametric sub-groups ; one 4-parametric unitary sub-group.
Curvilinear coordinates to cover these sub-groups
have been found.

Joint work with A.A. Bogush and N.G. Tokarevskaya (B.I. Stepanov Institute of Physics, Minsk, Belarus).