DUALITY FOR PARTIAL GROUP ACTIONS

Authors : Christian LOMP

Pages : 53-62

View : 38 | Download : 11

Publication Date : 2008-12-01

Article Type : Research Paper

Abstract :Given a finite group G acting as automorphisms on a ring A, the skew group ring A ∗ G is an important tool for studying the structure of G-stable ideals of A. The ring A ∗ G is G-graded, i.e. G coacts on A ∗ G. The Cohen-Montgomery duality says that the smash product A ∗G#k[G]∗ of A ∗G with the dual group ring k[G]∗ is isomorphic to the full matrix ring Mninsert ignore into journalissuearticles values(A); over A, where n is the order of G. In this note we show how much of the Cohen-Montgomery duality carries over to partial group actions in the sense of R.Exel. In particular we show that the smash product insert ignore into journalissuearticles values(A ∗α G); #k[G]∗ of the partial skew group ring A ∗α G and k[G]∗ is isomorphic to a direct product of the form K × eMninsert ignore into journalissuearticles values(A);e where e is a certain idempotent of Mninsert ignore into journalissuearticles values(A); and K is a subalgebra of insert ignore into journalissuearticles values(A ∗α G); #k[G]∗. Moreover A ∗α G is shown to be isomorphic to a separable subalgebra of eMninsert ignore into journalissuearticles values(A);e. We also look at duality for infinite partial group actions.
Keywords : Partial Group actions, Cohen Montgomery duality

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