On finite dimensional regular gradings

TL;DR

The paper classifies finite-dimensional associative algebras with a regular grading by a finite abelian group $G$ over an algebraically closed field $K$ of characteristic $0$, describing their structure via a graded Wedderburn–Malcev decomposition into twisted group algebras and radicals. It introduces and exploits the notion of complete support and minimal regular decomposition, proving that in this regime the graded PI-exponent equals the ordinary PI-exponent, and equals $|G|$ when the decomposition is minimal. A detailed description shows that under complete support, the semisimple part is a direct sum of twisted group algebras $K^{\alpha}G$ with a single cocycle $\alpha$ inducing the grading bicharacter, while the radical interacts predictably with these pieces. The study also provides explicit graded codimension computations, establishing $c_n^{G}(A)=|G|^{n}$ in key cases and solidifying the relation between minimality and PI-exponent, together with a general decomposition framework for the non-complete-support setting.

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 $β(i,j)\in K^*$ such that $a_ia_j=β(i,j)a_ja_i$ 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 $β\colon G\times G\to K^*$ ought to be a bicharacter. A regular decomposition is {minimal} whenever for every $g$, $h\in G$, the equalities $β(x,g)=β(x,h)$ 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 $\exp(A)=|G|$.

Theorems & Definitions (69)

• Definition 1
• Definition 2
• Example 3
• Example 4
• Example 5
• Example 6
• Example 7
• Theorem 8
• Definition 9
• Theorem 10