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A new class of locally unital and locally finite dimensional algebras A over an arbitrary algebraically closed field is discovered. Each of them admits an upper finite weakly triangular decomposition, a generalization of an upper finite triangular decomposition. Any locally unital algebra which admits an upper finite Cartan decomposition is Morita equivalent to some special locally unital algebra A which admits an upper finite weakly triangular decomposition. It is established that the category A-lfdmod of locally finite dimensional left A-modules is an upper finite fully stratified 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 and give categorical actions of some Lie algebras on the subcategory of A-lfdmod consisting of all objects which have a finite standard filtration. In the case A is the locally unital algebra associated to one of cyclotomic oriented Brauer categories, cyclotomic Brauer categories and cyclotomic Kauffman categories, A admits an upper finite weakly triangular decomposition. This leads to categorifications of representations of the classical limits of coideal algebras, which come from all integrable highest weight modules of sl∞ or ŝle. Finally, we study representations of A associated to either cyclotomic Brauer categories or cyclotomic Kauffman categories in details, including explicit criteria on the semisimplicity 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 algebras.

In this paper, a notion of affine walled Brauer-Clifford superalgebras BC aff r,t is introduced over an arbitrary integral domain R containing 2 −1 . These superalgebras can be considered as affinization of walled Brauer superalgebras in [9]. By constructing infinite many homomorphisms from BC aff r,t to a class of level two walled Brauer-Clifford superagebras over C, we prove that BC aff r,t is free over R with infinite rank. We explain that any finite dimensional irreducible BC aff r,t -module over an algebraically closed field F of characteristic not 2 factors through a cyclotomic quotient of BC aff r,t , called a cyclotomic (or level k) walled Brauer-Clifford superalgebra BC k,r,t . Using a previous method on cyclotomic walled Brauer algebras in [15], we prove that BC k,r,t is free over R with super rank (k r+t 2 r+t−1 (r + t)!, k r+t 2 r+t−1 (r + t)!) if and only if it is admissible in the sense of Definition 6.4. Finally, we prove that the degenerate affine (resp., cyclotomic) walled Brauer-Clifford superalgebras defined by are isomorphic to our affine (resp., cyclotomic) walled Brauer-Clifford superalgebras. Contents 1. Introduction 1 2. Walled Brauer-Clifford superalgebras 3 3. Affine walled Brauer-Clifford superalgebras 10 4. A basis of BC 2,r,t with special parameters 13 5. Homomorphisms between BC r,t and BC 2,r+k,t+k 18 6. A basis of the cyclotomic Brauer-Clifford superalgebra 25 7. Isomorphisms between affine (resp., cyclotomic) Brauer-Clifford algebras and Comes-Kujawa's affine (resp., cyclotomic) algebras 30 References 33

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Let A be the locally unital algebra associated to a cyclotomic oriented Brauer category over an arbitrary algebraically closed field k of characteristic p ≥ 0. The category of locally finite dimensional representations of A is used to give the tensor product categorification (in the general sense of Losev and Webster) for an integrable lowest weight with an integrable highest weight representation of the same level for the Lie algebra g, where g is a direct sum of sl∞ (resp., ŝlp ) if p = 0 (resp., p > 0). Such a result was expected in [3] when k = C and proved previously in [2] when the level is 1.