Dualities and vertex operator algebras of affine type

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We notice that for any positive integer k, the set of (1,2)-specialized characters of level k standard A₁⁽¹⁾-modules is the same as the set of rescaled graded dimensions of the subspaces of level 2k+1 standard A₂⁽²⁾-modules that are vacuum spaces for the action of the principal Heisenberg subalgebra of A₂⁽²⁾. We conjecture the existence of a semisimple category induced by the "equal level" representations of some algebraic structure which would naturally explain this duality-like property, and we study potential such structures in the context of generalized vertex operator algebras.