sℓ̂(2|1) and D̂(2|1; α) as vertex operator extensions of dual affine sℓ(2) algebras

We discover a realisation of the affine Lie superalgebra sℓ̂(2|1) and of the exceptional affine superalgebra D̂(2|1; α) as vertex operator extensions of two sℓ̂(2) algebras with "dual" levels (and an auxiliary level-1 sℓ̂(2) algebra). The duality relation between the levels is (k₁ + 1)(k₂+ 1) = 1. We construct the representation of sℓ̂(2|1)_k1 on a sum of tensor products of sℓ̂(2)_k1, sℓ̂(2)_k2, and sℓ̂(2)₁ modules and decompose it into a direct sum over the sℓ̂(2|1)_k1 spectral flow orbit. This decomposition gives rise to character identities, which we also derive. The extension of the construction to D̂(2|1; k₂)_k1 is traced to the properties of sℓ̂(2) ⊕ sℓ̂(2) ⊕ sℓ̂(2) embeddings into D̂(2|1; α) and their relation with the dual sℓ̂(2) pairs. Conversely, we show how the sℓ̂(2)_k2, representations are constructed from sℓ̂(2|1)_k1 representations.