Star-algebras and almost polynomial growth of central polynomials

Abstract

Let A be an algebra with involution ⁎ over a field of characteristic zero. There are three numerical sequences attached to the ⁎-polynomial identities ${\mathrm{Id}}^{⁎}\left(A\right)$ satisfied by A: the sequence of codimensions $c_n^{⁎}\left(A\right)$, the sequence of central codimensions $c_n^{⁎.z}\left(A\right)$ and the sequence of proper central codimensions $c_n^{⁎,\delta }\left(A\right)$ $n=1,2,\dots$.

They give a measure of the ⁎-polynomial identities, the central ⁎-polynomials and the proper central ⁎-polynomials of the algebra A.

It has been proved ([9], [18]) that when A satisfies a non-trivial identity, the three sequences either grow exponentially or are polynomially bounded. Now, an algebra A has almost polynomial growth of the codimensions if $c_n^{⁎}\left(A\right)$ grows exponentially and for any algebra B such that ${\mathrm{Id}}^{⁎}\left(B\right)⊋{\mathrm{Id}}^{⁎}\left(A\right)$, $c_n^{⁎}\left(B\right)$ is polynomially bounded. Similarly we have the definitions of almost polynomial growth of the central codimensions $c_n^{⁎.z}\left(A\right)$ and of the proper central codimensions $c_n^{⁎,\delta }\left(A\right)$.

We aim to classify, up to ⁎-PI-equivalence, the algebras with almost polynomial growth of one of the above codimensions. This has already been done in [8] regarding the codimensions $c_n^{⁎}\left(A\right)$.

Here we classify, up to ⁎-PI-equivalence, the algebras having almost polynomial growth of the central ⁎-polynomials by exhibiting three subalgebras of upper triangular matrices. We also construct three more finite dimensional algebras giving the classification of almost polynomial growth of the proper central ⁎-polynomials.