Limit T-subalgebras in free associative algebras.
Let \(F\langle X\rangle\) be the free unitary associative algebra over a field \(F\) on a free generating set \(X\). A unitary subalgebra \(R\) of \(F\langle X\rangle\) is called a T-subalgebra if \(R\) is closed under all endomorphisms of \(F\langle X\rangle\). A typical example of a T-subalgebra is the set of all central polynomials and polynomial identities of a PI-algebra \(A\).NEWLINENEWLINE T-subalgebras which are not finitely generated as T-subalgebras play a key role in the construction of T-ideals in positive characteristic \(p\) which are not finitely based. By a result of Shchigolev, in characteristic 0 all T-subalgebras are finitely generated. The T-subalgebra \(R\) in \(F\langle X\rangle\) is limit if it is not finitely generated and every larger T-subalgebra is finitely generated. Every not finitely generated T-subalgebra \(R\) is contained in a limit one. Hence the limit T-subalgebras are on the border between finitely generated and not finitely generated T-subalgebras. Up till now, no explicit examples of limit T-subalgebras were known.NEWLINENEWLINE In the paper under review the authors construct such examples over an arbitrary infinite field of positive characteristic \(p>2\) and when \(|X|\geq 4\). The first main result is the following. If \(q^{(k)}=x_1^{p^k-1}[x_1,x_2]x_2^{p^k-1}\), then the T-subalgebra \(\Gamma\) of \(F\langle x_1,x_2,\ldots\rangle\) generated by \(q^{(k)}\), \(k\geq 0\), and the polynomials \(x_1[x_2,x_3,x_4]\) and \(x_1[x_2,x_3][x_4,x_5]\) is a limit T-subalgebra. The second result states that for \(n\geq 4\) the intersection \(\Gamma_n=\Gamma\cap F\langle x_1,\ldots,x_n\rangle\) is a limit T-subalgebra of \(F\langle x_1,\ldots,x_n\rangle\). -- The authors conjecture that \(\Gamma\) and \(\Gamma_n\) are the only limit T-subalgebras in \(F\langle x_1,x_2,\ldots\rangle\) and \(F\langle x_1,\ldots,x_n\rangle\), respectively.
scientific article; zbMATH DE number 6336083
scientific article