On infinite dimensional algebras with regular gradings

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 $A_g{A}_h\subseteq A_{g+h}$ for every $g$, $h\in G$. Such a $G$-grading is called regular whenever for every $n$-tuple $(g_1,\dots ,g_n)\in G^n$ 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 $a_g{a}_h={\hat{I}}^2(g,h)a_h{a}_g$ for some ${\hat{I}}^2(g,h)\in K^{\ast }$. Here ${\hat{I}}^2(g,h)$ depends only on the choice of $g$ and $h$ but not on the elements $a_g$ and $a_h$. It is immediate that ${\hat{I}}^2$ is a bicharacter on $G$. The regular decomposition above is minimal if for every $g\in G$ with ${\hat{I}}^2(g,h)={\hat{I}}^2(g,k)$ 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 graded subalgebras of a ${\mathbb{Z}}_2$-graded regular algebra having a minimal regular decomposition.