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A first approach to the Burchnall-Chaundy theory for quadratic algebras having PBW bases

Arturo Niño, María Camila Ramírez, Armando Reyes

Abstract

In this paper, we present a first approach toward a Burchnall-Chaundy theory for the skew Ore polynomials of higher order generated by quadratic relations defined by Golovashkin and Maksimov \cite{GolovashkinMaksimov1998}.

A first approach to the Burchnall-Chaundy theory for quadratic algebras having PBW bases

Abstract

In this paper, we present a first approach toward a Burchnall-Chaundy theory for the skew Ore polynomials of higher order generated by quadratic relations defined by Golovashkin and Maksimov \cite{GolovashkinMaksimov1998}.
Paper Structure (15 sections, 15 theorems, 50 equations)

This paper contains 15 sections, 15 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Products of homogeneous elements
  4. Case $Q(a,-1,c)$
  5. Case $Q(a,0,0)$
  6. Case $Q(0,b,0)$
  7. Sylvester matrix and Resultants
  8. Sylvester matrix for $Q(a,b,c)$
  9. Case $Q(a,-1,c)$
  10. Case $Q(a,0,0)$
  11. Case $Q(0,b,0)$
  12. Resultants and right common factors
  13. Burchnall-Chaundy theory for quadratic algebras
  14. Examples
  15. Future work

Key Result

Proposition 2.1

In $Q(a,-1,c)$, the following formulas hold:

Theorems & Definitions (42)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • proof
  • ...and 32 more