Algebraic Dependence of Commuting Elements in Algebras

Sergei Silvestrov (), Christian Svensson () and Marcel de Jeu ()
Additional contact information
Sergei Silvestrov: Lund Institute of Technology, Lund University, Centre for Mathematical Sciences, Division of Mathematics
Christian Svensson: Lund Institute of Technology, Lund University, Centre for Mathematical Sciences, Division of Mathematics
Marcel de Jeu: Leiden University, Mathematical Institute

Chapter 23 in Generalized Lie Theory in Mathematics, Physics and Beyond, 2009, pp 265-280 from Springer

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.

Date: 2009
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-540-85332-9_23

Ordering information: This item can be ordered from
http://www.springer.com/9783540853329

DOI: 10.1007/978-3-540-85332-9_23

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().