Highest weight categories appear frequently in algebraic Lie theory, for example as blocks of the BGG category \(\mathcal{O}\) of a complex semi-simple Lie algebra. Since its inception, many different generalisations of highest weight categories have appeared.
Going from highest weight categories to module categories of standardly stratified algebras two types of standard modules arise: the proper standard modules and the standard modules. The multiplicity of the top of the standard module in the corresponding proper standard module is exactly one, whereas this condition is relaxed for the corresponding standard module. In particular, instead of just modules admitting a filtration by standard modules, one can study modules admitting a filtration by a mix of standard and proper standard modules. In fact, various generalisations of quasi-hereditary algebras are obtained by imposing the regular module to admit such mix filtrations: for instance, standardly stratified algebras, properly stratified algebras and CPS-algebras, among many others.
However, the formalism of the previous notions mentioned is not yet enough to cover further additional cases of interest like the category of finite-dimensional rational representations of a reductive algebraic group. Also, the terminology used for generalisations of quasi-hereditary algebras is sometimes not consistent in the literature, with the same term being used for different concepts.
In this paper, the authors unify many generalisations of highest weight categories (over algebraically closed fields) and hugely develop the theory to the infinite-dimensional world by dealing with generalisations of highest weight categories in four different settings: finite abelian, essentially finite abelian, Schurian and locally finite abelian categories leading to the concepts of lower finite and upper finite highest weight categories. For each of these, they study their strata and properties, including their fundamental homological properties. Although with a different name, the concept of a lower finite highest weight category can be traced back to the classic work of E. Cline et al. [J. Reine Angew. Math. 391, 85–99 (1988; Zbl 0657.18005)].
The authors show that these categories admit objects called \(\varepsilon\)-tilting generators, capturing the idea of characteristic tilting and cotilting modules from the classical case. Using these, the authors extend Ringel duality to these categories. In particular, the authors extend Ringel duality to a correspondence between lower finite highest weight categories and upper finite highest weight categories.
This paper is divided in 6 chapters and all categories in this paper are linear over an algebraically closed field. Chapter 1 is devoted to the introduction. In Chapter 2, the four settings of study are introduced and elementary properties about these are given. The reader is warned that the terminology used in this paper might differ sometimes with other sources in the literature. Here, by a Schurian category the authors mean any category which is equivalent to the category of locally finite-dimensional modules over a locally finite-dimensional locally unital algebra. Here a finite abelian (resp. locally finite abelian, essentially finite abelian) category means any category equivalent to the category of all finite-dimensional modules over a finite dimensional algebra (resp. unital algebra, essentially finite-dimensional locally unital algebra) over an algebraically closed field.
In Chapter 3, the authors present the definitions and general properties of the strata of finite (resp. essentially finite, upper finite, lower finite) \(\varepsilon\)-stratified categories. The concept of finite (resp. essentially finite, upper finite, lower finite) \(\varepsilon\)-stratified category is the generalisation of highest weight category in the setup of finite abelian (resp. essentially finite abelian, Schurian, locally finite abelian) categories. Since essentially finite abelian categories are locally finite abelian categories with enough projectives and injectives, the two extreme cases are the upper finite \(\varepsilon\)-stratified categories and the lower finite \(\varepsilon\)-stratified categories.
In Chapter 4, the concept of characteristic tilting module is generalised to the semi-infinite setup. These new objects are called \(\varepsilon\)-tilting generators. Depending on the choice of sign function \(\varepsilon\), this object might be a full tilting object, a full cotilting object or neither. This object is used to define the Ringel dual, and the phenomenon of Ringel duality is then extended to these general setups. The authors prove that the \(\varepsilon\)-tilting generators are independent of the sign function \(\varepsilon\) whenever the underlying category possesses Gorensteiness properties and a quasi-Frobenius strata.
In the classical case, highest weight categories are realised as module categories of quasi-hereditary algebras. In Chapter 5, the authors use Ringel duality to obtain this analogue for upper finite highest weight categories. In Chapter 6, the authors exhibit several examples that fit in their setup. For instance, the category of finite-dimensional rational representations of a reductive algebraic group is an example of a lower finite highest weight category.