PI-algebras with slow codimension growth.
Let \(A\) be an associative non-unitary algebra over a field of characteristic zero. The multilinear polynomial identities satisfied by \(A\) determine all the polynomial identities satisfied by \(A\). Denote by \(c_n(A)\), \(n=1,2,\dots\), the codimension of the subspace of multilinear identities of \(A\) in the \(n\)-variable multilinear component of the free associative algebra. It is known that for a PI-algebra \(A\) either the codimension sequence grows exponentially (typical case) or it is polynomially bounded. In the present paper the algebras with linearly bounded codimension sequence are classified up to PI-equivalence. The possible linearly bounded codimension sequences are determined. The authors present a list of five explicitly given finite dimensional algebras such that \(A\) has linearly bounded codimension sequence if and only if none of the given algebras belongs to the variety generated by \(A\). Linear growth of the codimensions turns out to be closely related to the property that the limit of \(l_n(A)\) exists and is bounded by \(2\), where \(l_n(A)\) counts the number of \(S_n\)-irreducibles in the \(n\)-th cocharacter of \(A\).
scientific article; zbMATH DE number 2136979
scientific article