New tensor categories related to orthogonal and symplectic groups and the strange supergroup P(n)

We start with discussing the category of tensor representations of O(∞). We show that this category is Koszul and that its Koszul dual is the category of tensor representations of the strange supergroup P(∞). One can use this relation to calculate Ext groups between simple objects in both categories. The above categories are monoidal symmetric categories but they are missing the duality functor. It is possible to extend these categories to certain rigid tensor categories satisfying a nice universality property. In the case of O(∞) such extension depends on a parameter t and is closely related to the Deligne's category Rep O(t). When t is integer, this new category is a highest weight category. We finish by formulating several open questions concerning these new categories.