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Quasi-Pfaffians and applications

Claire Gilson, Shi-Hao Li, Guo-Fu Yu

TL;DR

The paper extends Pfaffian techniques to a non-commutative setting by introducing quasi-Pfaffians, defined via quasi-determinants on division rings with an anti-involution. It establishes derivative formulae for both Grammian- and Wronskian-type quasi-Pfaffians and proves a heredity principle, enabling a condensation-based computation. These tools yield a non-commutative B-Toda lattice whose solutions are expressed in terms of quasi-Pfaffians and are governed by a Lax pair built from $ ext{R}$-valued skew-orthogonal polynomials. The work thus provides new exact, non-commutative integrable structures and suggests rich connections to matrix-valued orthogonal polynomials and non-commutative geometry, with open questions about determinant–Pfaffian analogues and broader applications.

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.

Quasi-Pfaffians and applications

TL;DR

The paper extends Pfaffian techniques to a non-commutative setting by introducing quasi-Pfaffians, defined via quasi-determinants on division rings with an anti-involution. It establishes derivative formulae for both Grammian- and Wronskian-type quasi-Pfaffians and proves a heredity principle, enabling a condensation-based computation. These tools yield a non-commutative B-Toda lattice whose solutions are expressed in terms of quasi-Pfaffians and are governed by a Lax pair built from -valued skew-orthogonal polynomials. The work thus provides new exact, non-commutative integrable structures and suggests rich connections to matrix-valued orthogonal polynomials and non-commutative geometry, with open questions about determinant–Pfaffian analogues and broader applications.

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.

Paper Structure

This paper contains 16 sections, 29 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Quasi-determinants and quasi-Pfaffians
  4. Quasi-determinants
  5. Definition of quasi-Pfaffians
  6. A quasi-Pfaffian identity
  7. Derivative formulae for quasi-Pfaffians
  8. Derivative formulae for Wronskian type quasi-Pfaffians
  9. Derivatives of Grammian type quasi-Pfaffians
  10. On a non-commutative B-Toda lattice equation
  11. B-Toda lattice and its Pfaffian solution
  12. Non-commutative B-Toda equation and its quasi-Pfaffian solution
  13. Lax pair of the system
  14. Concluding remarks
  15. Linear system and definition of quasi-Pfaffian
  16. ...and 1 more sections

Key Result

Proposition 2.3

In the commutative setting, quasi-Pfaffian defined by def is a ratio of Pfaffians. In other words, we have

Theorems & Definitions (46)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • Proposition 2.6
  • Theorem 2.7
  • Lemma 2.8
  • ...and 36 more