Determinants and symmetries in 'Yetter-Drinfeld' categories

A Yetter-Drinfeld category over a Hopf algebra H with a bijective antipode, is equipped with a 'braiding' which may be symmetric for some of its subcategories (e.g, when H is a triangular Hopf algebra). We prove that under an additional condition (which we term the u-condition) such symmetric subcategories completely resemble the category of vector spaces over a field k, with the ordinary 'flip' map. Consequently, when Char k = 0, one can define well behaving exterior algebras and non-commutative determinant functions.