On observable module categories

Abstract

Module categories should play the same fundamental role in the theory of tensor categories, as representations do in the theory of groups. They were introduced in [Be] and studied in several papers in category theory; they appeared also in mathematical physics, see for example [FK]. Recently a systematic study of module categories over fusion, or more generally finite, tensor categories was undertaken in [O1, O2, ENO, EO]. In particular, indecomposable module categories over the category of representations of a finite group were classified in [O1] (characteristic 0) and [EO] (arbitrary characteristic). In the papers [ENO, EO], the authors consider rigid tensor categories with appropriate finiteness conditions. One of the motivations for the discussions in the present paper is the study of module categories over the tensor category GM of rational modules over an algebraic group G. Furthermore, we are interested in the induction functor from the category HM of representations of a closed subgroup H of G to GM. For this, one needs to consider any rational module, not only the rigid (= finite-dimensional) ones. We are naturally led to the notions of ind-rigid and geometric tensor categories, see Definition 1.2. The purpose of the present paper is to begin the study of a class of module categories over a tensor category C that we call observable module categories. These module categories are simple in a suitable sense (that we introduce in this paper). If C = GM then the archetypical example is the module category HM where H is an observable subgroup of G. We extend some well-known results on observable subgroups to the setting of quotients of Hopf algebras. The notion of observable subgroup has the following geometric characterization: a closed subgroup H of an algebraic group G is observable iff the homogeneous space G/H is quasiaffine. This suggests that the study of observable module categories could have a “noncommutative algebra” flavor: they should correspond to “non-commutative quasi-affine varieties”. Throughout, k denotes an arbitrary algebraically closed field, and all vector spaces, algebras, varieties, etc. are considered over k.