Parametrization of $4\times 4$-matrices $G$ of the complex linear group $GL(4,C)$ in terms of four complex 4-vector parameters $(k,m,n,l)$ is investigated. Additional restrictions separating some subgroups of $GL(4,C)$ are given explicitly. In the given parametrization, the problem of inverting any $4\times 4$ matrix $G$ is solved. Expression for determinant of any matrix $G$ is found: $\det G = F(k,m,n,l)$. Unitarity conditions $G^{ +} = G^{ -1}$ have been formulated in the form of non-linear cubic algebraic equations including complex conjugation. Several simplest solutions of these unitarity equations have been found: three 2-parametric subgroups $G_{ 1}$, $G_{ 2}$, $G_{ 3}$ - each of subgroups consists of two commuting Abelian unitary groups; 4-parametric unitary subgroup consisting of a product of a 3-parametric group isomorphic SU(2) and 1-parametric Abelian group. The Dirac basis of generators $\Lambda_{ k}$, being of Gell-Mann type, substantially differs from the basis $\lambda_{ i}$ used in the literature on SU(4) group, formulas relating them are found - they permit to separate SU(3) subgroup in SU(4). Special way to list 15 Dirac generators of $GL(4,C)$ can be used $\{ \Lambda_k\} = \{ \alpha_i\oplus\beta_j\oplus(\alpha_iV\beta_j = { \boldsymbol K} \oplus { \boldsymbol L}\oplus{ \boldsymbol M})\}$, which permit to factorize SU(4) transformations according to $S = e^{ i\vec{ a}\vec{ \alpha}} e^{ i\vec{ b}\vec\beta}} e^{ i{ \boldsymbol k}{ \boldsymbol K}} e^{ i{ \boldsymbol l}{ \boldsymbol L}} e^{ i\boldsymbol m}{ \boldsymbol M}}$, where two first factors commute with each other and are isomorphic to SU(2) group, the three last ones are 3-parametric groups, each of them consisting of three Abelian commuting unitary subgroups.
Categories: math-ph math.MP