Classification of finite irreducible conformal modules for $K'_4$

Abstract

We classify the finite irreducible modules over the conformal superalgebra K ′ 4 by their correspondence with finite conformal modules over the associated annihilation superalgebra A(K ′ 4 ). This is achieved by a complete classification of singular vectors in generalized Verma modules for A(K ′ 4 ). We also show that morphisms between generalized Verma modules can be arranged in infinitely many bilateral complexes.

An upper bound on the degree of singular vectors for $E(1,6)$

“…The aim of this paper is to prove that technical lemma stated in [3]. The proof of this lemma completes the classification of singular vectors for E (1,6) given in [3].…”

“…Remark 2.6. Let (g, F ) be a formal distribution superalgebra, endowed with λ−bracket (1). The elements of F satisfy sesquilinearity, skew-symmetry and Jacobi identity with ∂ = ∂ z ; for a proof see Proposition 2.3 in [13].…”

“…In section 3 we recall the definition of the conformal superalgebra CK 6 and some of its properties. Finally, in section 4 we prove the bound on the degree of singular vectors for E (1,6).…”

“…Boyallian, Kac, Liberati and Rudakov classified all finite irreducible modules over the conformal superalgebras of type W and S in [4]; Boyallian, Kac and Liberati classified all finite irreducible modules over the conformal superalgebras of type K N for N ≥ 4 in [2]. All finite irreducible modules over the conformal superalgebra K ′ 4 were classified in [1]. Finally a classification of all finite irreducible modules over the conformal superalgebra CK 6 was obtained in [3] and [16] with different approaches.…”

“…induced modules Ind(F ) = U(g) ⊗ U (g ≥0 ) F , where F is a finite−dimensional irreducible g ≥0 -module [15,10]. In [3] the classification of highest weight singular vectors is based on a technical lemma, whose proof is missing (Lemma 4.4 in [3]), that provides an upper bound on the degree of singular vectors for E (1,6).…”