Ichi Fujii

Construction of Z₂² Integrable Hierarchy Using Z₂²-loop osp(1|2)

Integrable nonlinear partial differential equations defined in two-dimensional spacetime, such as the Liouville, sinh-Gordon, and mKdV equations, play crucial roles in the theories of solitons and classical/quantum fields. It is well known that a hierarchy of integrable equations, including these examples, can be constructed from
the loop extension of the Lie algebra sl(2).  In this talk, we present an extension of this construction to Z₂²-graded Lie superalgebras and introduce a new class of integrable systems characterized by such algebras. This extension enables the derivation of soliton equations involving exotic fields beyond the conventional bosonic and fermionic ones. In particular, we present the case of the Z₂²-graded extension of the loop algebra osp(1|2),  which contains sl(2) as a subalgebra, and demonstrate that this construction yields an integrable Z₂²-graded extension of the aforementioned equations.