Description of the algebras generating the variety of trace-killers. (Q863322)

Let \(F\) be an infinite field. A polynomial \(f(x_1,\dots,x_n)\) from the free associative algebra \(F\langle X\rangle\) is a trace-killer for the \(k\times k\) matrix algebra \(M_k(F)\) if there exists a polynomial \(h(x_1,\dots,x_n,z)\in F\langle X\rangle\) such that \(f(x_1,\dots,x_n)\text{Tr}(z)=h(x_1,\dots,x_n,z)\) is a trace identity for \(M_k(F)\). Trace-killers form a T-ideal \(\Gamma^{(k)}\) in \(F\langle X\rangle\), called also the conductor ideal. In the paper under review the authors study the ideal \(\Gamma^{(3)}_n=\Gamma^{(3)}\cap F\langle x_1,\dots,x_n\rangle\). The main result describes \(\Gamma^{(3)}_n\) over a field of characteristic different from 2 and 3. The authors construct an explicitly given finite dimensional algebra \(B\) such that for each \(n\) there exists a finite dimensional local noncommutative algebra \(C\) such that \(\Gamma^{(3)}_n\) is the intersection of the T-ideals in \(F\langle x_1,\dots,x_n\rangle\) of the polynomial identities of \(M_2(F)\), \(B\) and \(C\). As a consequence, if \(f(x_1,\dots,x_n)\) is a polynomial identity for \(M_2(F)\) which is of sufficiently large degree and belongs to the fifth power of the commutator ideal of \(F\langle x_1,\dots,x_n\rangle\), then \(f(x_1,\dots,x_n)\) is a trace-killer for \(M_3(F)\). As the authors easily show, the T-ideal of the trace-killers of \(M_2(F)\) is generated by the polynomial identities \([[x_1,x_2],x_3]\) and \([x_1,x_2][x_3,x_4]\).