I am interested in studying the category of typical representations 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? and

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

In Theorem 1, Kac 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 :) .

I am interested in studying the category of typical representations 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 $\mathcal O$ for Lie superalgebras?

In Theorem 1, Kac 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 :) .

I am interested in stying the category of typical representations 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? and

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

In Theorem 1, Kac 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 :) .

I am interested in studying the category of typical representations 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? and

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

In Theorem 1, Kac 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 :) .