Strong Hopf modules for weak Hopf quasigroups

Abstract

In this paper we introduce the category of strong Hopf modules for a weak Hopf quasigroup H in a braided monoidal category. We also prove that this category is equivalent to the category of right modules over the image of the target morphism of H.Let F be a field and C = F − V ect. Let M be a right H-module and a right H-comodule. If, for all m ∈ M and h ∈ H, we write m.h for the action and we use the Sweedler notation ρ M (m) = m [0] ⊗ m [1] for the coaction, we will say that M is a Hopf module if the equalitythe coproduct of H and m [1] h (2) the product in H of m [1] and h (2) . A morphism between two Hopf modules is a F-linear map that is H-linear and H-colinear. Hopf modules and morphisms of Hopf modules constitute the category of Hopf modules denoted by M H H . In 1969 Larsson and Sweedler proved a result, called Fundamental Theorem of Hopf Modules, that asserts the following: If M ∈ M H H and M coH = {m ∈ M | ρ M (m) = m ⊗ 1 H } are the coinvariants of H in M , M is isomorphic to M coH ⊗ H as Hopf modules (see [11] and [16]). On the other hand, if N is a F-vector space, the tensor product N ⊗ H, with the action and coaction induced by the product and the coproduct of H, is a Hopf module. This construction is functorial, so we have a functor F = − ⊗ H : C → M H H . Also, for all M ∈ M H H , the construction of M coH is functorial and we have a functor G = ( ) coH : M H H → C and F ⊣ G. Moreover, F and G is a pair of inverse equivalences, and therefore, M H H is equivalent to the category of F-vector spaces. The Fundamental Theorem of Hopf Modules also holds for weak Hopf algebras as was proved by Böhm, Nill and Szlachányi in [5]. In this case, if H is a weak Hopf algebra, the category of Hopf modules is defined in the same way as in the Hopf algebra setting. For M ∈ M H H , the coinvariants of H in M are defined by M coH = {m ∈ M | ρ M (m) = m [0] ⊗ Π L H (m [1] )}, where Π L H is the target morphism associated to H. Then, Böhm, Nill and Szlachányi proved that M is isomorphic to M coH ⊗ HL H as Hopf modules, where H L is the image of Π L H . Moreover, if C HL is the category of right H L -modules, there exist two functors F = − ⊗ HL H : C HL → M H H and G = ( ) coH : M H H → C HL such that F is left adjoint of G and they induce a pair of inverse equivalences (see [7]). Therefore, in the weak setting, M H H is equivalent to C HL . In this case, is a relevant fact the following property: the tensor product M coH ⊗ HL H is isomorphic as Hopf modules to M coH × H where M coH × H is the image of a suitable idempotent ∇ M : M coH ⊗ H → M coH ⊗ H. Note that, as a consequence, in the weak framework the Fundamental Theorem of Hopf Modules can be written using M coH × H instead of M coH ⊗ HL H.