On algebraic curves for commuting elements in $q$-Heisenberg algebras
(2009) In Journal of Generalized Lie Theory and Applications 3(4). p.321-328- Abstract
- In the present article we continue investigating the algebraic dependence of commuting
elements in q-deformed Heisenberg algebras. We provide a simple proof that the
0-chain subalgebra is a maximal commutative subalgebra when q is of free type and that
it coincides with the centralizer (commutant) of any one of its elements dierent from
the scalar multiples of the unity. We review the Burchnall-Chaundy-type construction for
proving algebraic dependence and obtaining corresponding algebraic curves for commuting
elements in the q-deformed Heisenberg algebra by computing a certain determinant
with entries depending on two commuting variables and one of the generators. The... (More) - In the present article we continue investigating the algebraic dependence of commuting
elements in q-deformed Heisenberg algebras. We provide a simple proof that the
0-chain subalgebra is a maximal commutative subalgebra when q is of free type and that
it coincides with the centralizer (commutant) of any one of its elements dierent from
the scalar multiples of the unity. We review the Burchnall-Chaundy-type construction for
proving algebraic dependence and obtaining corresponding algebraic curves for commuting
elements in the q-deformed Heisenberg algebra by computing a certain determinant
with entries depending on two commuting variables and one of the generators. The coe
cients in front of the powers of the generator in the expansion of the determinant are
polynomials in the two variables dening some algebraic curves and annihilating the two
commuting elements. We show that for the elements from the 0-chain subalgebra exactly
one algebraic curve arises in the expansion of the determinant. Finally, we present several
examples of computation of such algebraic curves and also make some observations on
the properties of these curves. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/2440747
- author
- Richter, Johan ^{LU} and Silvestrov, Sergei ^{LU}
- organization
- publishing date
- 2009
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Burchnall-Chaundy theory, Heisenberg algebra
- in
- Journal of Generalized Lie Theory and Applications
- volume
- 3
- issue
- 4
- pages
- 321 - 328
- publisher
- Ashdin Publishing
- ISSN
- 1736-5279
- language
- English
- LU publication?
- yes
- id
- a6b5aeb4-3b8d-4880-b845-daabfe365f85 (old id 2440747)
- alternative location
- http://www.ashdin.com/journals/jglta/2009/4/v3_n4_5.pdf
- date added to LUP
- 2012-08-31 15:57:29
- date last changed
- 2016-04-15 20:00:57
@article{a6b5aeb4-3b8d-4880-b845-daabfe365f85, abstract = {In the present article we continue investigating the algebraic dependence of commuting<br/><br> elements in q-deformed Heisenberg algebras. We provide a simple proof that the<br/><br> 0-chain subalgebra is a maximal commutative subalgebra when q is of free type and that<br/><br> it coincides with the centralizer (commutant) of any one of its elements dierent from<br/><br> the scalar multiples of the unity. We review the Burchnall-Chaundy-type construction for<br/><br> proving algebraic dependence and obtaining corresponding algebraic curves for commuting<br/><br> elements in the q-deformed Heisenberg algebra by computing a certain determinant<br/><br> with entries depending on two commuting variables and one of the generators. The coe<br/><br> cients in front of the powers of the generator in the expansion of the determinant are<br/><br> polynomials in the two variables dening some algebraic curves and annihilating the two<br/><br> commuting elements. We show that for the elements from the 0-chain subalgebra exactly<br/><br> one algebraic curve arises in the expansion of the determinant. Finally, we present several<br/><br> examples of computation of such algebraic curves and also make some observations on<br/><br> the properties of these curves.}, author = {Richter, Johan and Silvestrov, Sergei}, issn = {1736-5279}, keyword = {Burchnall-Chaundy theory,Heisenberg algebra}, language = {eng}, number = {4}, pages = {321--328}, publisher = {Ashdin Publishing}, series = {Journal of Generalized Lie Theory and Applications}, title = {On algebraic curves for commuting elements in $q$-Heisenberg algebras}, volume = {3}, year = {2009}, }