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Abstract. This is a generalization of the classic work of Beilinson, Lusztig and MacPherson. In this paper (and an Appendix) we show that the quantum algebras obtained via a BLM-type stabilization procedure in the setting of partial flag varieties of type B/C are two (modified) coideal subalgebras of the quantum general linear Lie algebra,U  andU ı . We provide a geometric realization of the Schur-type duality of Bao-Wang between such a coideal algebra and Iwahori-Hecke algebra of type B. The monomial bases and canonical bases of the Schur algebras and the modified coideal algebraU  are constructed. In an Appendix by three authors, a more subtle 2-step stabilization procedure leading tȯ U ı is developed, and then monomial and canonical bases ofU ı are constructed. It is shown thatU ı is a subquotient ofU  with compatible canonical bases. Moreover, a compatibility between canonical bases for modified coideal algebras and Schur algebras is established.

Abstract. In this paper we use topological techniques to construct generalized trace and modified dimension functions on ideals in certain ribbon categories. Examples of such ribbon categories naturally arise in representation theory where the usual trace and dimension functions are zero, but these generalized trace and modified dimension functions are nonzero. Such examples include categories of finite dimensional modules of certain Lie algebras and finite groups over a field of positive characteristic and categories of finite dimensional modules of basic Lie superalgebras over the complex numbers. These modified dimensions can be interpreted categorically and are closely related to some basic notions from representation theory.

15 pagesWe provide a necessary and sufficient condition for a simple object in a pivotal k-category to be ambidextrous. As a consequence we prove that they exist for factorizable ribbon Hopf algebras, modular representations of finite groups and their quantum doubles, complex and modular Lie (super)algebras, the (1,p) minimal model in conformal field theory, and quantum groups at a root of unity

Abstract. We introduce the marked Brauer algebra and the marked Brauer category. These generalize the analogous constructions for the ordinary Brauer algebra to the setting of a homogeneous bilinear form on a Z2-graded vector space. We classify the simple modules of the marked Brauer algebra over any field of characteristic not two. Under suitable assumptions we show that the marked Brauer algebra is in Schur-Weyl duality with the Lie superalgebra, g, of linear maps which leave the bilinear form invariant. We also provide a classification of the indecomposable summands of the tensor powers of the natural representation for g under those same assumptions. In particular, our results generalize Moon's work on the Lie superalgebra of type p(n) and provide a unifying conceptual explanation for his results.

We construct the finite dimensional simple integral modules for the (degenerate) affine Hecke-Clifford algebra (AHCA),

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Abstract. We provide a presentation of the Schur superalgebra and its quantum analogue which generalizes the work of Doty and Giaquinto for Schur algebras. Our results include a basis for these algebras and a presentation using weight idempotents in the spirit of Lusztig's modified quantum groups.