Large ordered tensor categories: generalities and new examples

Ordered tensor categories are nonrigid tensor categories, which are at the same time highest
weight categories.  The best known examples of ordered tensor categories are the categories
of tensor modules over the finitary Lie algebras sl(\infty), so(\infty), and sp(\infty).

After reviewing the generalities about ordered tensor categories, in this talk I will introduce
new examples.  These are categories of tensor representations of the Mackey Lie algebras
of endomorphisms of split nondegenerate pairings V\otimes V_*\to C where V is a vector
space of dimension \aleph_k and k=0,1,2,... .