I am interested in studying the category of [typical representations][1] over basic classical simple Lie superalgebras. In particular, I want to know 

1) is this category semisimple (character determines a typical representation)?

2)is this category closed under tensor product? 

3) are there non-trivial one-dimensional representations in this category? and

4) What is the connection between this category to [the category $\mathcal O$][2] for Lie superalgebras?

In Theorem 1, [Kac][1] has given that, 
typical representation splits in any finite-dimensional representation (i.e. if it is a submodule or a factor-module of a finite-dimensional G-module, then it is a direct summand).

Is this means that 1) is true?


Basically, I want to know how much bad (or not nice) is this category compared to the BCG category $\mathcal{O}$ or $\mathcal{O}^{int}$ of Kac-Moody Lie algebras.  

Kindly share your thoughts and some references to learn about this category.

Thank you :) .


  [1]: https://link.springer.com/chapter/10.1007/BFb0063691
  [2]: https://pages.uoregon.edu/brundan/papers/superofont.pdf