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 QG with relations ρG such that the skew group algebras A[G] is Morita equivalent to the quotient algebra of path algebra kQG modulo ideal (ρG). Generally, the quiver QG is not connected. In this paper we develop a method to determine the number of connect components of QG. 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*.