Classification of degenerate Verma modules for E(5,10)

Abstract

Given a Lie superalgebra $\mathfrak{g}$ with a subalgebra ${\mathfrak{g}}_{\ge 0}$, and a finite-dimensional irreducible ${\mathfrak{g}}_{\ge 0}$-module $F$, the induced $\mathfrak{g}$-module $M(F)=U(\mathfrak{g}){\otimes }_{U({\mathfrak{g}}_{\ge 0})}F$ 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 $\mathfrak{g}=E(5,10)$ with the subalgebra ${\mathfrak{g}}_{\ge 0}$ 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 $E(5,10)$.

arXiv comment43 pages, 5 figures

References in corpus (5)

• Representations of the Exceptional Lie superalgebra $E(3,6): II. Four series of degenerate modules
• Lie conformal superalgebras and duality of modules over linearly compact Lie superalgebras
• Morphisms of Verma modules over exceptional Lie superalgebra E(5,10)
• Infinite dimensional primitive linearly compact Lie superalgebras
• A bound on the degree of singular vectors for the exceptional Lie superalgebra $E(5,10)$

Cited by in corpus (3)

• Classification of finite irreducible conformal modules for $K_4^{\prime }$
• An upper bound on the degree of singular vectors for $E(1,6)$
• Computation of the homology of the complexes of finite Verma modules for $K_4^{\prime }$