On tensor products of representations of Lie superalgebras

We consider typical finite dimensional complex irreducible representations of a basic classical simple Lie superalgebra, and give a sufficient condition on when unique factorization of finite tensor products of such representations hold. We also prove unique factorization of tensor products of singly atypical finite dimensional irreducible modules for $\mathfrak{s}\mathfrak{l}(m+1,n+1)$, $\mathfrak{o}\mathfrak{s}\mathfrak{p}(2,2n)$, $G(3)$ and $F(4)$ under an additional assumption. This result is a Lie superalgebra analogue of Rajan’s fundamental result \cite{MR2123935} on unique factorization of tensor products for finite dimensional complex simple Lie algebras.