Classification of irreducible A(L_k(sl3))-modules at level k = -3 + 2/(2m+1)

Classify the irreducible modules in the Bernstein–Gelfand–Gelfand category O for the Zhu’s algebra A(L_k(sl_3)) of the simple affine vertex operator algebra L_k(sl_3) at the non-admissible levels k = -3 + 2/(2m+1) with m > 0, by proving that the following set is complete: {L(tΛ_1 − (2i/(2m+1))Λ_3), L(tΛ_2 − (2i/(2m+1))Λ_3), L(Λ_1 − ((t + 2i + 1)/(2m+1))Λ_3) | t ∈ C, i = 0,1,...,2m}, where L(·) denotes the corresponding highest-weight sl_3-module regarded as an A(L_k(sl_3))-module.

Background

The paper determines associated varieties of L_k(sl_3) for a family of non-admissible levels and studies maximal ideals and singular vectors needed for these determinations. Motivated by representation-theoretic consequences, the authors propose a precise classification of irreducible modules of the Zhu’s algebra A(L_k(sl_3)) in the category O at levels k = -3 + 2/(2m+1).

They note that when m = 0 (i.e., k = -1), the conjecture holds by prior work, and when m = 1, it can be verified computationally, indicating evidence for the full classification at all m > 0.