Partial Representations and Partial Group Algebras

ISOMORPHISMS OF PARTIAL GROUP RINGS Research partially supported by CNPq Procs. 30111595-8 and 30024379-0 and FAPESP Proc. 00/07291-0.

“…In this paper we fill a gap in formula (14) of [3] and prove a similar result on isomorphisms in the modular case; i.e., when the characteristic of R divides the order of G. We also show that if R is of characteristic p > 0 and G and H are arbitrary finite groups such that their partial group rings over R are isomorphic, then the group rings of the corresponding Sylow p-subgroups are also isomorphic. Furthermore, we prove that if ‫ޚ‬ par G ∼ = ‫ޚ‬ par H for finite groups G and H, then also ‫ޚ‬G ∼ = ‫ޚ‬H showing that the hypothesis of having isomorphic partial group rings over the integers is even stronger than having isomorphic integral group rings.…”

“…The partial group ring of a group G over a ring R was defined in [3] and plays a role in the theory of partial representations similar to that of the group ring in representation theory. DEFINITION 1.1.…”

“…The structure of partial group rings was described in [3] where it was also shown that if G and H are finite abelian groups and R is an integral domain whose characteristic does not divide the order of G then R par G ∼ = R par H if and only if G ∼ = H. An example was given to show that there do exist non-isomorphic finite groups such that their partial group rings over an algebraically closed field of characteristic 0 are actually isomorphic.…”

Twisted partial actions of Hopf algebras

“…Then by (2) of [13] the following equality holds γ(g)e r = e gr γ(g), where e g = γ(g)γ(g −1 ). This corresponds to item (v) if we take H = κG.…”

“…The algebraic study of partial actions and partial representations was initiated in [23], [13] and [12], motivating investigations in diverse directions. In particular, the Galois theory of partial group actions developed in [16] inspired further Galois theoretic results in [9], as well as the introduction and study of partial Hopf actions and coactions in [10].…”