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Abstract. We find all values of k ∈ C, for which the embedding of the maximal affine vertex algebra in a simple minimal W-algebra W k (g, θ) is conformal, where g is a basic simple Lie superalgebra and −θ its minimal root. In particular, it turns out that if W k (g, θ) does not collapse to its affine part, then the possible values of these k are either − 2 3, where h ∨ is the dual Coxeter number of g for the normalization (θ, θ) = 2. As an application of our results, we present a realization of simple affine vertex algebra V − n+1 2

We provide an explicit bijection between the ad-nilpotent ideals of a Borel subalgebra of a simple Lie algebra g and the orbits ofQ/(h + 1)Q under the Weyl group (Q being the coroot lattice and h the Coxeter number of g). From this result we deduce in a uniform way a counting formula for the ad-nilpotent ideals.1991 Mathematics Subject Classification. Primary: 17B20; Secondary: 20F55.

Abstract. We study the combinatorics of ad-nilpotent ideals of a Borel subalgebra of sl(n + 1, C). We provide an inductive method for calculating the class of nilpotence of these ideals and formulas for the number of ideals having a given class of nilpotence. We study the relationships between these results and the combinatorics of Dyck paths, based upon a remarkable bijection between ad-nilpotent ideals and Dyck paths. Finally, we propose a (q, t)-analogue of the Catalan number Cn. These (q, t)-Catalan numbers count, on the one hand, ad-nilpotent ideals with respect to dimension and class of nilpotence and, on the other hand, admit interpretations in terms of natural statistics on Dyck paths.

We discover a large class of simple affine vertex algebras V k (g), associated to basic Lie superalgebras g at non-admissible collapsing levels k, having exactly one irreducible g-locally finite module in the category O. In the case when g is a Lie algebra, we prove a complete reducibility result for V k (g)-modules at an arbitrary collapsing level. We also determine the generators of the maximal ideal in the universal affine vertex algebra V k (g) at certain negative integer levels. Considering some conformal embeddings in the simple affine vertex algebras V −1/2 (Cn) and V −4 (E 7 ), we surprisingly obtain the realization of non-simple affine vertex algebras of types B and D having exactly one non-trivial ideal., are non-simple, with a unique non-trivial ideal.The decompositions of the embeddings above is still an open problem, and will be a subject of our forthcoming papers.