Weak polynomial identities and their applications

Abstract

Let R be an associative algebra over a field K generated by a vector subspace V. The polynomial f(x 1, . . . , xn ) of the free associative algebra K〈x 1, x 2, . . .〉 is a weak polynomial identity for the pair (R, V) if it vanishes in R when evaluated on V. We survey results on weak polynomial identities and on their applications to polynomial identities and central polynomials of associative and close to them nonassociative algebras and on the finite basis problem. We also present results on weak polynomial identities of degree three.

Polynomial identities in Novikov algebras

“…Considering the Novikov operad as a suboperad of the operad CD allows us to view the operad CD as an infinitesimal bimodule over the operad Nov. Considering sub-bimodules of this infinitesimal bimodule is very natural; it is somewhat similar to considering weak identities [14], but is more restrictive, since we only allow substitutions of Novikov elements and multiplying by a Novikov element using a Novikov product (only the first constraint would appear for a weak identity). In this context, we prove the following result resembling Theorem 1.…”

Weak polynomial identities of small degree for the Weyl algebra

“…Weak polynomial identities were also considered in [6,14,15,16,18], etc. More details on weak polynomial identities can be found in a recent survey by Drensky [5].…”

“…By Theorem 5.10, f is equivalent to a linear combination of completely reduced bracket-monomials of multidegree ∆. Assume ∆ = 1 5 . Using Example 5.4 we can see that f (x 1 , .…”