Minimum deformations of commutative algebra and linear group $GL(n)$

In the algebra of formal series $M_q(x^i)$, the relations of generalized commutativity that preserve the tensor $I_q$ grading and depend on parameters $q(i,k)$ are considered. A norm of the differential calculus on $M_q$ consistent with the $I_q$ grading is chosen. A new construction of a symmetrized tensor product of algebras of the type $M_q(x^i)$ and a corresponding definition of the minimally deformed linear group $QGL(n)$ and Lie algebra $qgl(n)$ are proposed. A study is made of the connection of $QGL(n)$ and $qgl(n)$ with the special matrix algebra $\operatorname {Mat}(n,Q)$, which consists of matrices with noncommuting elements. The deformed determinant in the algebra $\operatorname {Mat}(n,Q)$ is defined. The exponential mapping in the algebra $\operatorname {Mat}(n,Q)$ is considered on the basis of the Campbell–Hausdorff formula.