Tensor structure on the category of affine Lie algebra modules plays an important role in representation theory of affine Lie algebras. In a series of celebrated work, D. Kazhdan and G. Lusztig constructed rigid braided tensor categories of affine Lie algebras at non-positive rational levels, Y.-Z. Huang and J. Lepowsky gave modular tensor category structures at positive integral levels using tensor category theory of vertex operator algebras.

In this talk, we will discuss the existence of tensor categories of affine Lie algebras at positive rational levels as well as several relevant problems. This is joint work with T. Creutzig and Y.-Z. Huang.