On infinite dimensional algebras with regular gradings
Lucio Centrone, Plamen Koshlukov, Kauê Pereira
TL;DR
This work analyzes infinite-dimensional algebras graded by $\mathbb{Z}_2$ with regular, minimal decompositions. It proves that any such algebra $A$ contains a graded copy of the Grassmann algebra $E$, constructed via direct limits of $n$-regular subalgebras $F_n$ isomorphic to Grassmann algebras $E_n$. The authors then characterize the graded identities generated by $A$, showing they coincide with those of $E$ precisely when the associated finite-dimensional commutative $\mathbb{Z}_2$-graded algebra $C$ is regular and contains a graded simple summand $K\oplus cK$. This leads to a description of the graded variety generated by $E$ and a structural classification of finitely generated graded subalgebras of $A$ as direct limits of Type I–IV constructions, illuminating how PI identities and graded structures interact in the regular $\mathbb{Z}_2$-graded setting.
Abstract
Let $G$ be a finite abelian group and let $K$ be an algebraically closed field of characteristic 0. We consider associative unital algebras $A$ over $K$ graded by $G$, that is $A=\oplus_{g\in G} A_g$, where the vector subspaces $A_g$ satisfy $A_gA_h\subseteq A_{g+h}$ for every $g$, $h\in G$. Such a $G$-grading is called regular whenever for every $n$-tuple $(g_1,\ldots,g_n)\in G^n$ there exist homogeneous elements $a_i\in A_{g_i}$ such that $a_1\cdots a_n\ne 0$ in $A$; furthermore, for every $g$, $h\in G$ and every $a_g\in A_g$, $a_h\in A_h$ one has $a_ga_h=β(g,h)a_ha_g$ for some $β(g,h)\in K^*$. Here $β(g,h)$ depends only on the choice of $g$ and $h$ but not on the elements $a_g$ and $a_h$. It is immediate that $β$ is a bicharacter on $G$. The regular decomposition above is minimal if for every $g\in G$ with $β(g,h)=β(g,k)$ one has $h=k$. In this paper we prove that if $G=\mathbb{Z}_2$ then every $G$-graded regular algebra whose regular decomposition is minimal, contains a copy of the infinite dimensional Grassmann algebra. By applying this result we are able to describe the generating algebras of the variety of $\mathbb{Z}_2$-graded algebras defined by the Grassmann algebra. Furthermore we describe the finitely generated subalgebras of a $\mathbb{Z}_2$-graded regular algebra having a minimal regular decomposition.