Algebraic structures of higher-rank Z-graded Lie superalgebras

Develop a precise understanding of the algebraic structures of higher-rank Z2×Z2-graded Lie superalgebras, including their root systems and irreducible representations, to enable the extension of the integrable hierarchy constructed from the loop extension of Z2×Z2-graded osp(1|2).

Background

The paper constructs an integrable hierarchy using the loop extension of the Z2×Z2-graded Lie superalgebra osp(1|2), yielding Z-graded extensions of classical integrable equations (Liouville, sinh-Gordon, cosh-Gordon, mKdV) and deriving a corresponding Z-graded KdV equation via a Miura transformation.

The authors note that while their framework is applicable to other Z-graded Lie (super)algebras, extending the construction to higher-rank cases requires foundational algebraic structures—such as root systems and irreducible representations—that are currently not fully understood. Clarifying these structures is necessary for systematic development of broader classes of Z-graded integrable systems.