From Polynomial Lie Algebras gl∞(n) and so∞(n) to Matrix Lax Structures

ABSTRACT

This paper presents a unified and systematic approach to constructing the modified Kadomtsev–Petviashvili (mKP), BKP, and modified BKP (mBKP) hierarchies through the representation theory of polynomial Lie algebras. Based on the polynomial Lie algebras and , we naturally derive the complete structure of these integrable systems, including bilinear identities, tau functions, wave matrices, dressing operators, and Lax pairs, emerge naturally without external constraints. A key result is the automatic emergence of the BKP constraint from symmetry, contrasting with traditional approaches where such constraints are artificially imposed. The polynomial algebra framework provides a unifying principle where matrix structures arise intrinsically through fermionic Fock space decompositions, offering a powerful methodology for constructing and analyzing integrable hierarchies in higher dimensions.