Verma Modules over a ${\mathbb Z}_2 \otimes {\mathbb Z}_2$ Graded Superalgebra and Invariant Differential Equations

TL;DR

This paper constructs Verma modules over a specific ${ m Z}_2 imes { m Z}_2$ graded superalgebra, analyzes their reducibility via singular vectors, and derives invariant differential equations under the associated supergroup.

Contribution

It provides an explicit construction of Verma modules and singular vectors for the ${ m Z}_2 imes { m Z}_2$ graded superalgebra, and derives invariant differential equations.

Findings

Explicit formulas for singular vectors

Demonstration of module reducibility

Derivation of invariant differential equations

Abstract

Lowest weight representations of the ${\mathbb Z}_2 \otimes {\mathbb Z}_2$ graded superalgebra introduced by Rittenberg and Wyler are investigated. We give a explicit construction of Verma modules over the ${\mathbb Z}_2 \otimes {\mathbb Z}_2$ graded superalgebra and show their reducibility by using singular vectors. The explicit formula of singular vectors are given and are used to derive partial differential equations invariant under the color supergroup generated by the ${\mathbb Z}_2 \otimes {\mathbb Z}_2$ graded superalgebra.