Journal of Generalized Lie Theory and Applications
- J. Gen. Lie Theory Appl.
- Volume 3, Number 4 (2009), 321-328.
On algebraic curves for commuting elements in q-Heisenberg algebras
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 different 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 coefficients in front of the powers of the generator in the expansion of the determinant are polynomials in the two variables defining 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.
J. Gen. Lie Theory Appl., Volume 3, Number 4 (2009), 321-328.
First available in Project Euclid: 6 August 2010
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Zentralblatt MATH identifier
Primary: 16S99: None of the above, but in this section 81S05: Canonical quantization, commutation relations and statistics 39A13: Difference equations, scaling ($q$-differences) [See also 33Dxx]
Associative Rings Associative Algebras Quantum Theory Canonical Quantization Commutation Relations Commutation Statistics Difference Equations Functional Equations Difference scaling q-differences
Richter, Johan; Silvestrov, Sergei. On algebraic curves for commuting elements in q-Heisenberg algebras. J. Gen. Lie Theory Appl. 3 (2009), no. 4, 321--328. doi:10.4303/jglta/S090405. https://projecteuclid.org/euclid.jglta/1281106599