Weak polynomial identities and their applications (Q2245616)

Let \(R\) be an associative algebra over a field \(K\), and let \(V\) be a subspace of \(R\) generating it as a \(K\)-algebra. An element \(f\) of the free associative algebra is a \textit{weak polynomial identity} for the pair \((R,V)\) if \(f\) vanishes under any substitution of its variables by elements from the subspace \(V\) of \(R\). This paper is a survey on weak polynomial identities, their role in the construction of central polynomials, in finding bases of identities in associative, or Lie, Jordan, or other non-associative algebras. Open problems and conjectures are also stated in the paper.