📝 SummarXiv

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 the finite dimensional algebras $A$ (with unit) admitting a $G$-grading such that the corresponding regular decomposition is minimal. Moreover we compute the graded codimension sequence of these algebras assuming complete support. It turns out that the graded PI exponent of every finite dimensional $G$-graded algebra with regular grading such that its regular decomposition is minimal and it has complete support, coincides with its ordinary (ungraded) PI exponent. Finally we have shown that the regular decomposition of a finite dimensional regular $G$-grading $A$ is minimal if and only if $\exp(A)=|G|$.