A direct and simple proof of Jacobi identities for determinants

TL;DR

This paper provides a straightforward proof of the Jacobi identities for determinants using Plücker relations, which are crucial in solving integrable systems in soliton theory.

Contribution

It introduces a simple, direct proof of Jacobi identities for determinants, enhancing understanding and application in integrable systems.

Findings

Proof simplifies the understanding of Jacobi identities

Uses Plücker relations for a direct proof

Facilitates explicit solutions in soliton theory

Abstract

The Jacobi identities play an important role in constructing the explicit exact solutions of a broad class of integrable systems in soliton theory. In the paper, a direct and simple proof of the Jacobi identities for determinants is presented by employing the Pl$\ddot{u}$cker relations.