I have the following question concerning the topic of highest weight representations of a Lie superalgebra $\mathfrak g$.
In the case of standard Lie algebras, there exists the concept of fundamental representations $\Lambda_i$ with $i=1,\cdots ,\textrm{rk}(\mathfrak g)$. These serve as "building blocks" from which all the other representations can be obtained taking tensor products. Each of these representations is associated with a node of the Dynkin diagram of $\mathfrak g$ and they enjoy the following decomposition property: $$\Lambda_i \wedge \Lambda_i = \left(\bigotimes_{j\sim i} \Lambda_j \right) \oplus \cdots \; ,$$ where the symbol $\wedge$ stands for the antisymmetrised tensor product and $j \sim i$ denotes the integers $j$ such that the $j$-th node of the Dynkin diagram is a neighbor of the $i$-th.
I would like to know whether a similar construction can be performed with a Lie superalgebra.
Thanks a lot!