Minimum deformations of commutative algebra and linear groupGL(n)

In the algebra of formal seriesM _q (x ⁱ ), the relations of generalized commutativity that preserve the tensorI _q grading and depend on parametersq(i, k) are considered. A norm of the differential calculus onM _q consistent with theI _q grading is chosen. A new construction of a symmetrized tensor product of algebras of the typeM _q (x ⁱ ) and a corresponding definition of the minimally deformed linear groupQGL(n) and Lie algebraqgl(n) are proposed. A study is made of the connection ofQGL(n) andqgl(n) with the special matrix algebra Mat(n, Q), which consists of matrices with noncommuting elements. The deformed determinant in the algebra Mat(n, Q) is defined. The exponential mapping in the algebra Mat(n, Q) is considered on the basis of the Campbell-Hausdorff formula.Institute of Applied Physics, Tashkent State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 3, pp. 403–417, June, 1993.