Algebraic dependence of commuting elements in algebras
(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:
http://lup.lub.lu.se/record/959418
- author
- Silvestrov, Sergei ^{LU} ; Svensson, Christian and De Jeu, Marcel
- organization
- publishing date
- 2007
- 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- unpublished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}, }