Algebraic dependence of commuting elements in algebras

TL;DR

This paper explores algebraic dependence among commuting elements in various algebras, reviewing classical and recent results, including the Burchnall-Chaundy construction, and extends these ideas to q-deformed Heisenberg algebras.

Contribution

It extends the Burchnall-Chaundy approach to q-deformed Heisenberg algebras, providing new insights into algebraic dependence of commuting elements.

Findings

Burchnall-Chaundy construction applies to q-deformed Heisenberg algebras

Existence of annihilating algebraic curves for commuting elements

Comparison of algorithmic and Burchnall-Chaundy proofs

Abstract

The aim of this paper to draw attention to several aspects of the algebraic dependence in algebras. The article starts with discussions of the algebraic dependence problem in commutative algebras. Then the Burchnall-Chaundy construction for proving algebraic dependence and obtaining the corresponding algebraic curves for commuting differential operators in the Heisenberg algebra is reviewed. Next some old and new results on algebraic dependence of commuting q-difference operators and elements in q-deformed Heisenberg algebras are reviewed. The main ideas and essence of two proofs of this are reviewed and compared. One is the algorithmic dimension growth existence proof. The other is the recent proof extending the Burchnall-Chaundy approach from differential operators and the Heisenberg algebra to the q-deformed Heisenberg algebra, showing that the Burchnall-Chaundy eliminant construction indeed provides annihilating curves for commuting elements in the q-deformed Heisenberg algebras for q not a root of unity.