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

TL;DR

This paper classifies non-irreducible finite Verma modules over the exceptional Lie superalgebra E(5,10), identifying new singular vectors and constructing infinite complexes analogous to de Rham complexes.

Contribution

It provides a complete classification of finite Verma modules over E(5,10) and discovers new singular vectors, extending the understanding of their structure and morphisms.

Findings

Identified singular vectors of degrees 7 and 11.

Constructed infinite complexes of Verma modules.

Extended the theory of de Rham complexes to E(5,10).

Abstract

Given a Lie superalgebra $\frak g$ with a subalgebra $\frak g_{\geq 0}$, and a finite-dimensional irreducible $\frak g_{\geq 0}$-module $F$, the induced $\frak g$-module $M(F)=U({\frak g}) \otimes_{U(\frak g_{\geq 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 ${\frak g}=E(5,10)$ with the subalgebra $\frak g_{\geq 0}$ 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 $E(5,10)$.