Classification of Degenerate Verma Modules for E(5, 10)

Abstract

Given a Lie superalgebra $${\mathfrak {g}}$$ g with a subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 , and a finite-dimensional irreducible $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 -module F, the induced $${\mathfrak {g}}$$ g -module $$M(F)={\mathcal {U}}({\mathfrak {g}})\otimes _{{\mathcal {U}}({\mathfrak {g}}_{\ge 0})}F$$ M ( F ) = U ( g ) ⊗ U ( g ≥ 0 ) F is called a finite Verma module. In the present paper we classify the non-irreducible finite Verma modules over the largest exceptional linearly compact Lie superalgebra $${\mathfrak {g}}=E(5,10)$$ g = E ( 5 , 10 ) with the subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 of minimal codimension. This is done via classification of all singular vectors in the modules M(F). Besides known singular vectors of degree 1,2,3,4 and 5, we discover two new singular vectors, of degrees 7 and 11. We show that the corresponding morphisms of finite Verma modules of degree 1,4,7, and 11 can be arranged in an infinite number of bilateral infinite complexes, which may be viewed as “exceptional” de Rham complexes for E(5, 10).

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

“…The final result is much richer than in the "standard" conformal contact superalgebras K n where, up to duality, there is only one family of singular vectors, all of degree 1: we show 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 "exceptional" singular vectors of degree 3. Since the classification of singular vectors in finite Verma modules is equivalent to the classification of morphisms between such modules, we show that these morphisms can be arranged in an infinite number of bilateral complexes in a picture (see Figure 1) which is similar to those obtained for the exceptional linearly compact Lie superalgebras E(1, 6), E (3,6), E (3,8) and E(5, 10) (see [17,18,19,20,4,6]). In a subsequent publication we will compute the homology of these complexes and provide an explicit construction of all irreducible quotients.…”

Computation of the homology of the complexes of finite Verma modules for $K'_4$

“…In [1] the authors show 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 [1] it is shown that these morphisms can be arranged in an infinite number of bilateral complexes as in Figure 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 [17,18,19,20,5,7]). The aim of this work is to compute the homology of the complexes in Figure 1 and provide an explicit construction of all the irreducible quotients of finite Verma modules over A(K ′ 4 ).…”

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

“…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]).…”

“…Remark 6. 8 The following relations are a direct consequence of the definition of the G X • 's and Remark 6.7:…”

“…Definition 2. 8 The annihilation superalgebra A(R) of a conformal superalgebra R is the subalgebra of LieR spanned by all elements ay n with n ≥ 0 and a ∈ R.…”