Representations of weakly triangular categories

A new class of locally unital and locally finite dimensional al-gebras A over an arbitrary algebraically closed field is discov-ered. Each of them admits an upper finite weakly triangular decomposition, a generalization of a split triangular decompo-sition. It is established that the category A-lfdmod of locally finite dimensional left A-modules is an upper finite fully strati-fied category in the sense of Brundan-Stroppel. Moreover, A is semisimple if and only if its centralizer subalgebras associated to certain idempotent elements are semisimple. Furthermore, certain endofunctors are defined so as to give categorical ac-tions of some Lie algebras on the subcategory of A-lfdmod consisting of all objects which have a finite standard filtra-tion.As an application, we study representations of A associated to either cyclotomic Brauer categories or cyclotomic Kauffman categories in details, including explicit criteria on the semisim-plicity of A over an arbitrary field, and on A-lfdmod being upper finite highest weight category in the sense of Brundan-Stroppel, and on Morita equivalence between A and direct sum of infinitely many (degenerate) cyclotomic Hecke alge-bras. Finally, we obtain categorifications of representations of the classical limits of coideal algebras, which come from all integrable highest weight modules of sl infinity or s circumflex accent le.(c) 2022 Elsevier Inc. All rights reserved.