On parafermion vertex algebras of sl(2) and sl(3) at level-3/2

We study parafermion vertex algebras N-3/2 (sl(2)) and N-3/2 (sl(3)). Using the isomorphism between N-3/2 (sl(3)) and the logarithmic vertex algebra W-0(2)(A2) from [D. Adamovic, A realization of certain modules for the N = 4 superconformal algebra and the affine Lie algebra A(2)((1)), Transform. Groups 21(2) (2016) 299-327], we show that these parafermion vertex algebras are infinite direct sums of irreducible modules for the Zamolodchikov algebra W(2,3) of central charge c = -10, and that N-3/2(sl(3)) is a direct sum of irreducible N-3/2(sl(2))-modules. As a byproduct, we prove certain conjectures about the vertex algebra W-0(p)(A2). We also obtain a vertex-algebraic proof of the irreducibility of a family of W(2, 3)(c) modules at c = -10.