Classification of finite irreducible conformal modules for K4′

Abstract

We classify finite irreducible modules over the conformal superalgebra [Formula: see text] by their correspondence with finite conformal modules over the associated annihilation superalgebra [Formula: see text]. This is achieved by a complete classification of singular vectors in generalized Verma modules for [Formula: see text]. We also show that morphisms between generalized Verma modules can be arranged in infinitely many bilateral complexes.

Computation of the Homology of the Complexes of Finite Verma Modules for ${K}_{4}^{\prime }$

“…A classification of all finite irreducible modules over the conformal superalgebra CK 6 was obtained in [4] and [23] with different approaches. Finally the classification of all finite irreducible modules over the conformal superalgebra K 4 was obtained in [2].…”

“…In [2] the classification of all finite irreducible modules over the conformal superalgebra K 4 was obtained by their correspondence with irreducible finite conformal modules over the annihilation superalgebra g := A(K 4 ) associated with K 4 . In order to obtain this classification, the authors classified all highest weight singular vectors, i.e.…”

“…highest weight vectors which are annihilated by A(K 4 ) >0 , of finite Verma modules, that are the induced modules Ind(F ) = U(g) ⊗ U(g ≥0 ) F such that F is a finite−dimensional irreducible g ≥0 -module [13,19]. In [2] the authors showed that for A(K 4 ) there are four families of singular vectors of degree 1, four families of singular vectors of degree 2 and two singular vectors of degree 3. Since the classification of singular vectors of finite Verma modules is equivalent to the classification of morphisms between such modules, in [2] it was shown that these morphisms can be arranged in an infinite number of bilateral complexes as in Fig.…”

“…In [2] the authors showed that for A(K 4 ) there are four families of singular vectors of degree 1, four families of singular vectors of degree 2 and two singular vectors of degree 3. Since the classification of singular vectors of finite Verma modules is equivalent to the classification of morphisms between such modules, in [2] it was shown that these morphisms can be arranged in an infinite number of bilateral complexes as in Fig. 1, which is similar to those obtained for the exceptional Lie superalgebras E (1,6), E (3,6), E (3,8) and E (5,10) (see [6,8,[18][19][20][21]24]).…”

“…In Section 2 we recall some notions on conformal superalgebras. In Section 3 we recall the definition of the conformal superalgebra K 4 and the classification of singular vectors obtained in [2]. In Section 4 we find an explicit expression for the morphisms represented in Fig.…”