2-categories of symmetric bimodules and their 2-representations

Abstract

In this article we analyze the structure of 2-categories of symmetric projective bimodules over a finite dimensional algebra with respect to the action of a finite abelian group. We determine under which condition the resulting 2-category is fiat (in the sense of [MM1]) and classify simple transitive 2-representations of this 2-category (under some mild technical assumption). We also study several classes of examples in detail.

Basic Hopf algebras and symmetric bimodules

“…We thus obtain the analogous classification of simple transitive birepresentations to the one given in [MMZ2] in the case where G is abelian.…”

“…We next obtain results akin to [MMZ2,Propositions 6 and 7]. We only consider (A, π 1 ) on the one hand, but generalise to any bimodule M with any idempotent on the other hand.…”

“…The first such class is given by a certain subbicategory H A of modules over a finite-dimensional radically graded Hopf algebra (under tensor product over k), whose 1-morphisms are given by the additive closure of projective modules and the trivial module. The second example is the bicategory G A of projective symmetric bimodules over a finite-dimensional self-injective basic algebra A with an action of a finite group G, generalising the 2-categories studied in [MMZ2]. We embed both bicategories into the bicategory C A of projective A-A-bimodules (with horizontal composition given by tensor product over A) in Corollary 4.15 resp.…”

“…Moreover, both bicategories are H 0 -simple (Proposition 5.13). Finally, we give a classification of simple transitive birepresentations (extending the case considered in [MMZ2]) of G A in Theorem 5.19, of which there are only finitely many equivalence classes. By contrast, H A has infinitely many equivalence classes of simple transitive birepresentations in general.…”

“…In Section 4, we introduce the relevant bicategories of representations of a Hopf algebra A, and construct a pseudofunctor Γ embedding these into the appropriate bicategories of A-A-bimodules. In Section 5, we introduce the relevant bicategories G A of symmetric A-Abimodules (generalising the setup from [MMZ2]) and likewise embed these into the appropriate bicategories of A-A-bimodules via a pseudofunctor Θ. Moreover, we verify that the essential image of H A under Γ is a 1-full subbicategory or the essential image of G A under Θ.…”