Title: Integrable $sl(\infty)$-modules and Category $\mathcal O$ for $\mathfrak{gl}(m|n)$,
Abstract: We introduce and study new categories T(g,k)of integrable sl(\infty)-modules
which depend on the choice of a certain reductive subalgebra k in g=sl(\infty).
The simple objects of these categories are tensor modules as in the previously
studied category, however, the choice of k provides more flexibility of
nonsimple modules. We then choose k to have two infinite-dimensional diagonal
blocks, and show that a certain injective object K(m|n) in T(g,k) realizes a
categorical sl(\infty)-action on the integral category O(m|n) of the Lie
superalgebra gl(m|n). We show that the socle of K(m|n) is generated by the
projective modules in O(m|n), and compute the socle filtration of K(m|n)
explicitly. We conjecture that the socle filtration of K(m|n) reflects a
"degree of atypicality filtration" on the category O(m|n). We also conjecture
that a natural tensor filtration on K(m|n) arises via the Duflo--Serganova
functor sending the category O(m|n) to O(m-1|n-1). We prove this latter
conjecture for a direct summand of K(m|n) corresponding to the
finite-dimensional gl(m|n)-modules.