Finite dimensional irreducible representations of Lie superalgebra D (2, 1; α)

This paper focuses on the finite dimensional irreducible representations of Lie superalgebra D(2, 1; alpha). We explicitly construct the finite dimensional representations of the superalgebra D(2, 1; alpha) by using the shift operator and differential operator representations. Unlike ordinary Lie algebra representation, there are typical and atypical representations for most superalgebras. Therefore, its typical and atypical representation conditions are also given. Our results are expected to be useful for the construction of primary fields of the corresponding current superalgebra of D(2, 1; alpha).