Quasi-Pfaffians and applications

TL;DR

This paper introduces quasi-Pfaffians, a non-commutative extension of Pfaffians, and explores their properties and applications in solving non-commutative linear systems and integrable systems.

Contribution

It proposes the concept of quasi-Pfaffians, providing foundational identities and demonstrating their use in non-commutative integrable systems.

Findings

Defined quasi-Pfaffians and derived their key identities

Applied quasi-Pfaffians to solve non-commutative linear systems

Presented a new non-commutative integrable system

Abstract

This paper presents a non-commutative generalization of the Pfaffian which we call a quasi-Pfaffian. This novel concept arises from solving linear systems with non-commutative skew-symmetric coefficients. A new non-commutative integrable system whose solutions are expressed in terms of these quasi-Pfaffians is presented. Derivative formulae and identities satisfied by these quasi-Pfaffians are presented.