Large annihilator category O for sl(∞),o(∞),sp(∞)

Abstract

Abstract We construct a new analogue of the BGG category O for the infinite-dimensional Lie algebras g = sl ( ∞ ) , o ( ∞ ) , sp ( ∞ ) . A main difference with the categories studied in [9] and [2] is that all objects of our category satisfy the large annihilator condition introduced in [5] . Despite the fact that the splitting Borel subalgebras b of g are not conjugate, one can eliminate the dependency on the choice of b and introduce a universal highest weight category OLA of g -modules, the letters LA coming from “large annihilator”. The subcategory of integrable objects in OLA is precisely the category T g studied in [5] . We investigate the structure of OLA , and in particular compute the multiplicities of simple objects in standard objects and the multiplicities of standard objects in indecomposable injectives. We also complete the annihilators in U ( g ) of simple objects of OLA .