Minimal deformations of the commutative algebra and the linear group GL(n)

We consider the relations of generalized commutativity in the algebra of formal series $M_q (x^i )$, which conserve a tensor $I_q$-grading and depend on parameters $q(i,k)$ . We choose the $I_q$-preserving version of differential calculus on $M_q$ . A new construction of the symmetrized tensor product for $M_q$-type algebras and the corresponding definition of minimally deformed linear group $QGL(n)$ and Lie algebra $qgl(n)$ are proposed. We study the connection of $QGL(n)$ and $qgl(n)$ with the special matrix algebra $\mbox{Mat} (n,Q)$ containing matrices with noncommutative elements. A definition of the deformed determinant in the algebra $\mbox{Mat} (n,Q)$ is given. The exponential parametrization in the algebra $\mbox{Mat} (n,Q)$ is considered on the basis of Campbell-Hausdorf formula.