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Chen, Jian Min; Dong, Qiang; Lin, Ya Nan

On connected components of skew group algebras. (English) Zbl 07710151

Acta Math. Sin., Engl. Ser. 39, No. 5, 799-813 (2023).
Summary: Let \(A\) be a basic connected finite dimensional associative algebra over an algebraically closed field \(k\) and \(G\) be a cyclic group. There is a quiver \(Q_G\) with relations \(\rho_G\) such that the skew group algebras \(A[G]\) is Morita equivalent to the quotient algebra of path algebra \(kQ_G\) modulo ideal \(( \rho_G)\). Generally, the quiver \(Q_G\) is not connected. In this paper we develop a method to determine the number of connect components of \(Q_G\). Meanwhile, we introduce the notion of weight for underlying quiver of \(A\) such that \(A\) is \(G\)-graded and determine the connect components of smash product \(A\# kG^*\).

MSC:

16G20 Representations of quivers and partially ordered sets
16S35 Twisted and skew group rings, crossed products
16W50 Graded rings and modules (associative rings and algebras)
16D90 Module categories in associative algebras

Keywords:

smash product; skew group algebra; \(n\)-translation algebra; path algebra; connected component
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References:

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