Constants of formal derivatives of non-associative algebras, Taylor expansions and applications

Abstract

We study unitary multigraded non-associative algebras R generated by an ordered set X over a field K of characteristic 0 such that the mappings d_k: x_l->delta_{kl}, x_k,x_l in X, can be extended to derivations of R. The class of these algebras is quite large and includes free associative and Jordan algebras, absolutely free (non-associative) algebras, relatively free algebras in varieties of algebras, universal enveloping algebras of multigraded Lie algebras, etc. There are Taylor-like formulas for R: Each element of R can be uniquely presented as a sum of elements of the form (...(r_0 x_{j_1})...x_{j_{n-1}}) x_{j_n}, where r_0 is a constant and j_1 <= ...<= j_n. We present methods for the description of the algebra of constants, including an approach via representation theory of the general linear or symmetric groups. As an application of the Taylor expansion for non-associative algebras, we consider the solutions of ordinary linear differential equations with constant coefficients from the base field.