  Motivated by the group entropy theory, in this work we generalize the algebra
of real numbers (that we called G-algebra), from which we develop an associated
G-differential calculus. Thus, the algebraic structures corresponding to the
Tsallis and Kappa statistics are obtained as special cases when the Tsallis and
Kappa group classes are chosen. We employ the G-algebra to formulate a
generalized G-deformed Schr\"{o}dinger equation and we illustrate it with the
infinite potential well, where the effective mass is related with the G-algebra
structure and the $q$-deformed (standard) Schr\"{o}dinger equation results an
special case for the Tsallis (Boltzmann-Gibbs) group class. The non-uniform
zeros spacing of the G-deformed eigenfunctions is expressed in terms of the
generalized sum of the G-algebra.
