Algebras with bounded multiplicities of the central cocharacters

Let $A$ be an associative PI-algebra over a field of characteristic zero. The central polynomials of $A$ give rise to a sequence $\{{\chi }_n^z(A){\}}_{n\ge 1}$ of $S_n$-characters, called its central cocharacters. A natural problem is to understand which algebraic properties of $A$ are imposed by bounds on the multiplicities occurring in the decompositions of these cocharacters into irreducible characters. We prove that these multiplicities are bounded by one if and only if there exist scalars $(\alpha ,\beta )\mathrm{\ne }(0,0)$ such that $\alpha [x_1,x_2]x_1+\beta x_1[x_1,x_2]$ takes values in the center of $A$. This provides a central analogue of the Ananin-Kemer theorem for ordinary cocharacters. We also characterize the algebras whose central cocharacter multiplicities are uniformly bounded.