On algebraic curves for commuting elements in $q$-Heisenberg algebras
Author
Summary, in English
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.
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.
Department/s
Publishing year
2009
Language
English
Pages
321-328
Publication/Series
Journal of Generalized Lie Theory and Applications
Volume
3
Issue
4
Links
Document type
Journal article
Publisher
Ashdin Publishing
Topic
- Mathematics
Keywords
- Burchnall-Chaundy theory
- Heisenberg algebra
Status
Published
Research group
- Algebra
- Non-commutative Geometry
ISBN/ISSN/Other
- ISSN: 1736-5279