Sheets and associated varieties of affine vertex algebras

Abstract

We show that sheet closures appear as associated varieties of affine vertex algebras. Further, we give new examples of non-admissible affine vertex algebras whose associated variety is contained in the nilpotent cone. We also prove some conjectures from our previous paper and give new examples of lisse affine W -algebras.2010 Mathematics Subject Classification. 17B67, 17B69, 81R10.

On the irreducibility of associated varieties of W-algebras

“…In cases (3)-(5) of Table 1, O min is the minimal nilpotent orbit of g. The statement of the case (6) of Table 1 was conjectured in [AM16] and will be proven in Section 6. Here O (2 r−2 ,1 4 ) denotes the nilpotent orbit of so 2r corresponding to the partition (2 r−2 , 1 4 ) of 2r (see Section 4).…”

“…Let ∆ = {±ε i ± ε j | 1 i, j r, i = j} be the root system of g and take ∆ + = {ε i ± ε j | 1 i < j r} for the set of positive roots. By Theorem 1.3 of [AM16] we know that X V k (g) ⊂ O (2 r−2 ,1 4 ) . Hence it is sufficient to show that X V k (g) ⊃ O (2 r−2 ,1 4 ) .…”

“…Therefore, it is enough to show that V (W 2 ) do not contain O (3,2 2 ,1 2r−6 ) . To proceed, we apply [AM16, Lemma 3.3] and we argue as in the proofs of Lemma 7.6 and Lemma 9.2 of [AM16]. Since the verification are similar we omit the details here.…”

“…In our previous paper [AM16] we conjectured that, under reasonable assumptions on the vertex algebra V , the associated variety X V is irreducible if it has only finitely many symplectic leaves (see Conjecture 1).…”

“…Then X H 0 DS,f θs (Ṽ k (g)) = S short ∩ S f θs by Theorem 2.1. Since f θs + e θs is a semisimple element that is conjugate to h θs , we have (see [AM16, (Ṽ k (g)) is strongly generated by L. As V k (g) is a quotient ofṼ k (g), the exactness of H 0 DS,f θs (?) (Theorem 2.1 (1)) implies that H 0 DS,f θs (V k (g)) is also strongly generated by its conformal vector.…”