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We study the representation theory of non-admissible simple affine vertexalgebra $L_{-5/2} (sl(4))$. We determine an explicit formula for the singularvector of conformal weight four in the universal affine vertex algebra$V^{-5/2} (sl(4))$, and show that it generates the maximal ideal in $V^{-5/2}(sl(4))$. We classify irreducible $L_{-5/2} (sl(4))$--modules in the category${\mathcal O}$, and determine the fusion rules between irreducible modules inthe category of ordinary modules $KL_{-5/2}$. It turns out that this fusionalgebra is isomorphic to the fusion algebra of $KL_{-1}$. We also prove that$KL_{-5/2}$ is a semi-simple, rigid braided tensor category.  In our proofs we use the notion of collapsing level for the affine$\mathcal{W}$--algebra, and the properties of conformal embedding $gl(4)\hookrightarrow sl(5)$ at level $k=-5/2$ from arXiv:1509.06512. We show that$k=-5/2$ is a collapsing level with respect to the subregular nilpotent element$f_{subreg}$, meaning that the simple quotient of the affine$\mathcal{W}$--algebra $W^{-5/2}(sl(4), f_{subreg})$ is isomorphic to theHeisenberg vertex algebra $M_J(1)$. We prove certain results on vanishing andnon-vanishing of cohomology for the quantum Hamiltonian reduction functor$H_{f_{subreg}}$. It turns out that the properties of $H_{f_{subreg}}$ are moresubtle than in the case of minimal reducition.

On the representation theory of the vertex algebra $L_{-5/2}(sl(4))$