Fundamental Theorems of Doi–Hopf Modules in a Nonassociative Setting

Abstract

In this paper we introduce the notion of weak non-asssociative Doi-Hopf module and give the Fundamental Theorem of Hopf modules in this setting. Also we prove that there exists a categorical equivalence that admits as particular instances the ones constructed in the literature for Hopf algebras, weak Hopf algebras, Hopf quasigroups, and weak Hopf quasigroups.

Weak quasi-entwining structures

“…Similarly, we obtain (d3) by (24) and by h • Π L H = h * h −1 (this last equality follows by the multiplicative condition for h). Finally, by Proposition 2.6 of [8] we know that…”

“…(by (8) and 18of [5]) Therefore, (a2) of Definition 2.1 holds. Also, by the naturality of c, the comodule condition for A and (b1) we obtain (a3) of Definition 2.1.…”

“…Taking into account the level of generality of weak Hopf quasigroups, as a consequence of the above identities, if H is a Hopf quasigroup (Hopf algebra) and (A, ρ A ) is a right H-comodule magma (monoid) the identity ρ A • η A = η A ⊗ η H is a consequence of (8). Also, if H is a weak Hopf quasigroup (weak Hopf algebra) and (A, ρ A ) is a right H-comodule magma (monoid), the equality (8). Let (A, ρ A ) be a right H-comodule magma.…”

“…Let H be a weak Hopf quasigroup and let (A, ρ A ) be a right H-comodule magma. Following Definition 2.7 of[8]…”