On Connected Components of Skew Group Algebras

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*.