On the Additive Structure and Asymptotics of Codimensions c_n in the Algebra F⁽⁵⁾

In this paper, we investigate the additive structure of the algebra F⁽⁵⁾, i.e., a relatively free, associative, countably-generated algebra with the identity [x₁, . . . , x₅] = 0 over an infinite field of characteristic ≠2, 3. We study the space of proper multilinear polynomials in this algebra and means of basis construction in one of its basic subspaces. As an additional result, we obtain estimations of codimensions c_n = dimP_n/P_n∩ T⁽⁵⁾, where P_n is the space of multilinear polynomials of degree n in F⁽⁵⁾ and T⁽⁵⁾ is the T-ideal generated by the long commutator [x₁, . . . , x₅].