On finite dimensional regular gradings

TL;DR

This paper characterizes finite-dimensional algebras with regular gradings by finite abelian groups, describing their structure, computing graded codimension sequences, and relating regular decomposition minimality to algebraic exponent properties.

Contribution

It provides a complete structural description of algebras with regular gradings and links minimal regular decomposition to the algebra's PI-exponent.

Findings

The graded PI-exponent equals the ungraded PI-exponent for these algebras.

Regular decomposition is minimal if and only if the algebra's exponent equals the group order.

The structure of algebras with regular gradings is fully characterized.

Abstract

Let $A$ be an associative algebra over an algebraically closed field $K$ of characteristic 0. A decomposition $A=A_1\oplus\cdots \oplus A_r$ of $A$ into a direct sum of $r$ vector subspaces is called a \textsl{regular decomposition} if, for every $n$ and every $1\le i_j\le r$, there exist $a_{i_j}\in A_{i_j}$ such that $a_{i_1}\cdots a_{i_n}\ne 0$, and moreover, for every $1\le i,j\le r$ there exists a constant $\beta(i,j)\in K^*$ such that $a_ia_j=\beta(i,j)a_ja_i$ for every $a_i\in A_i$, $a_j\in A_j$. We work with decompositions determined by gradings on $A$ by a finite abelian group $G$. In this case, the function $\beta\colon G\times G\to K^*$ ought to be a bicharacter. A regular decomposition is {minimal} whenever for every $g$, $h\in G$, the equalities $\beta(x,g)=\beta(x,h)$ for every $x\in G$ imply $g=h$. In this paper we describe completely the structure of the finite dimensional algebras $A$ (with unit) admitting a $G$-regular grading. Moreover, we compute the graded codimension sequence for a class of such algebras assuming complete support and minimal regular decomposition. It turns out that, for these algebras, the graded PI-exponent coincides with the ordinary (ungraded) PI-exponent. Finally, we show that the regular decomposition of a finite-dimensional algebra $A$ with a regular $G$-grading is minimal if and only if $\exp(A)=|G|$.