1997/10/06 by Dimitry Leites, Leites, Dimitry, Xuan Peiqi +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Nonlinear Waves and Solitons #hep-th
paper · pdf · doi:10.48550/arxiv.hep-th/9710045
LaTeX, 15 pages
arxiv created 1997/10/06 · openalex publication_date 1997/10/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 70's A.A. Kirillov interpreted the stationary Schroedinger (Sturm-Liouville) operator as an element of the dual space to the Virasoro algebra, i.e., the nontrivial central extension of the Witt algebra. He interpreted the KdV operator in terms of the stabilizer of the Schroedinger operator. By studying the coadjoint representation of the simplest nontrivial central extension of a simplest stringy superalgebra, the Neveu--Schwarz superalgebra, Kirillov connected solutions of the KdV and Schroedinger equations. We extend Kirillov's results and find all supersymmetric extension of the Schroedinger and Korteweg-de Vries operators associated with the 12 distinguished stringy superalgebras. We also take into account the odd parameters and the possibility for Time to be a (1|1)-dimensional supermanifold. The superization of construction due to Khesin e.a. (Drinfeld--Sokolov's reduction for the pseudodifferential operators) relates the complex powers of the Schroedinger operators we describe to the superized KdV-type hierarchies labeled by complex parameter. Our construction brings the KdV-type equations directly in the Lax form guaranteeing their complete integrability.