SL(2 vertical stroke 1) and D(2 vertical stroke 1;α) as vertex operator extensions of dual affine SL(2) algebras

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