Verbally prime T-ideals and graded division algebras

TL;DR

This paper classifies graded verbally prime T-ideals in free G-graded algebras, revealing their structure as identities of finite-dimensional G-graded division algebras over algebraically closed fields.

Contribution

It extends the classification of verbally prime T-ideals to the graded setting, distinguishing between verbally prime and strongly verbally prime cases, and characterizes the latter via division algebra structures.

Findings

Classified G-graded verbally prime T-ideals.

Identified strongly verbally prime T-ideals as identities of finite-dimensional G-graded division algebras.

Connected the structure of these ideals to the existence of roots of unity in the base field.

Abstract

Let $F$ be an algebraically closed field of characteristic zero and let $G$ be a finite group. We consider graded Verbally prime $T$-ideals in the free $G$-graded algebra. It turns out that equivalent definitions in the ordinary case (i.e. ungraded) extend to nonequivalent definitions in the graded case, namely verbally prime $G$-graded $T$-ideals and strongly verbally prime $T$-ideals. At first, following Kemer's ideas, we classify $G$-graded verbally prime $T$-ideals. The main bulk of the paper is devoted to the stronger notion. We classify $G$-graded strongly verbally prime $T$-ideals which are $T$-ideal of affine $G$-graded algebras or equivalently $G$-graded $T$-ideals that contain a Capelli polynomial. It turns out that these are precisely the $T$-ideal of $G$-graded identities of finite dimensional $G$-graded, central over $F$ (i.e. $Z(A)_{e}=F$) which admit a $G$-graded division algebra twisted form over a field $k$ which contains $F$ or equivalently over a field $k$ which contains enough roots of unity (e.g. a primitive $n$-root of unity where $n = ord(G)$).