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Algebraic dependence of commuting elements in algebras

Silvestrov, Sergei LU ; Svensson, Christian and De Jeu, Marcel (2007) In LUTFMA-5090-2007/1-14/(2007)
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... (More)
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. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
unpublished
subject
keywords
eliminant, commuting elements, Burchnall-Chaundy construction, algebraic dependence, q-difference operators, q-deformed Heisenberg algebras
in
LUTFMA-5090-2007/1-14/(2007)
issue
24
pages
14 pages
ISSN
1403-6207
project
Non-commutative Analysis of Dynamics, Fractals and Wavelets
Non-commutative Geometry in Mathematics and Physics
language
English
LU publication?
yes
id
8852b837-07e8-47ac-84a6-e0c82d639ba7 (old id 959418)
alternative location
http://arxiv.org/abs/0710.3250
date added to LUP
2008-01-28 14:45:12
date last changed
2016-04-16 07:00:30
@misc{8852b837-07e8-47ac-84a6-e0c82d639ba7,
  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 <br/><br>
Burchnall-Chaundy eliminant construction indeed provides annihilating curves for commuting elements in the <br/><br>
q-deformed Heisenberg algebras for q not a root of unity.},
  author       = {Silvestrov, Sergei and Svensson, Christian and De Jeu, Marcel},
  issn         = {1403-6207},
  keyword      = {eliminant,commuting elements,Burchnall-Chaundy construction,algebraic dependence,q-difference operators,q-deformed Heisenberg algebras},
  language     = {eng},
  number       = {24},
  pages        = {14},
  series       = {LUTFMA-5090-2007/1-14/(2007)},
  title        = {Algebraic dependence of commuting elements in algebras},
  year         = {2007},
}