Linear Representations and Frobenius Morphisms of Groupoids

Abstract

Given a morphism of (small) groupoids with injective object map, we provide sufficient and necessary conditions under which the induction and co-induction functors between the categories of linear representations are naturally isomorphic. A morphism with this property is termed a Frobenius morphism of groupoids. As a consequence, an extension by a subgroupoid is Frobenius if and only if each fibre of the (left or right) pull-back biset has finitely many orbits. Our results extend and clarify the classical Frobenius reciprocity formulae in the theory of finite groups, and characterize Frobenius extension of algebras with enough orthogonal idempotents.

Correspondence theorems for Hopf algebroids with applications to affine groupoids

“…Recall that an o -subring of H via is a right 2 It is a well-known fact that if ( , H ) is a left bialgebroid, then the category H Comod of left H -comodules is a monoidal category with tensor product ⊗ and unit object , where the H -coaction on is provided by the source map (see, e.g., [3,Theorem 3.18], where the property is stated for the right-hand scenario) and where every left H -comodule is a right -module with action…”

“…To prove that (4) implies (5), consider the morphisms 0 → M g → N and apply the functor B ⊗ A −. The implication from ( 5) to (2) follows by considering the morphism…”

“…for all a ∈ A, b ∈ B (see (2.10)). Moreover, (3.1) is a split short exact sequence of right A-modules (in fact, left A o -modules via t ′ ), where t ′ provides an A-linear section for ε ′ in view of (B6): 2 It is a well-known fact that if (A, H) is a left bialgebroid, then the category H Comod of left H-comodules is a monoidal category with tensor product ⊗ A and unit object A, where the H-coaction on A is provided by the source map (see, e.g., [3,Theorem 3.18], where the property is stated for the right-hand scenario) and where every left H-comodule M is a right A-module with action…”