Maximal \(T\)-spaces of a free associative algebra. (Q645265)

Let \(k\) be any field and let \(X\) be a nonempty set. A T-space is a subspace of the free (unitary or nonunitary) associative algebra \(k\langle X\rangle\) which is closed under the endomorphisms of the algebra (i.e., under substitution of the variables with arbitrary elements of the free algebra). Every T-space is contained in a maximal one. In the paper under review the authors investigate the maximal T-spaces. They reduce the study to problems for the polynomial algebra in the commuting variables \(X\) and then to the polynomial algebra in one variable only. Considering algebras without 1, over an infinite field \(k\) the only maximal T-space is generated by \(x_1x_2\). When the field \(k\) is finite with \(q\) elements, the algebra \(k\langle X\rangle\) has infinitely many maximal T-spaces. Examples are the maximal T-spaces containing the polynomial \(x+x^{q^{2^n}}\) if \(\text{char}(k)>2\) and \(x+x^q,x^{q^n+1}\) if \(\text{char}(k)=2\). For the free unitary algebra the answer is similar but the T-spaces are generated by different elements. For example, over an infinite field the only maximal T-space of \(k\langle X\rangle\) is generated by the commutator \([x_1,x_2]\).