Tensor category KLk(sl2n) via minimal affine W-algebras at the non-admissible level k =-2n+1/2

We prove that the Kazhdan-Lusztig category of slm at level k, KLk(slm), is a semi-simple, rigid braided tensor category for all even m >= 4, and k = - m+1 2 . Moreover, all modules in KLk(slm) are simple-currents and they appear in the decomposition of conformal embeddings glm -> slm+1 at level k = - m+1 2 . For this we inductively identify minimal affine W-algebra Wk-1(slm+2, theta) as simple current extension of Lk(slm) circle times H circle times M, where H is the rank one Heisenberg vertex algebra, and M the singlet vertex algebra for c = -2. The proof uses previously obtained results for the tensor categories of singlet algebra. We also classify all irreducible ordinary modules for Wk-1(slm+2, theta). The semi-simple part of the category of Wk-1(slm+2, theta)-modules comes from KLk-1(slm+2), using quantum Hamiltonian reduction, but this W-algebra also contains indecomposable ordinary modules. (c) 2023 Elsevier B.V. All rights reserved.