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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Amitsur, S.A.: Commutative linear differential operators. Pacific J. Math. 8, 1–10 (1958)
Burchnall, J.L., Chaundy, T.W.: Commutative ordinary differential operators. Proc. London Math. Soc. (Ser. 2) 21, 420–440 (1922)
Burchnall, J.L., Chaundy, T.W.: Commutative ordinary differential operators. Proc. Roy. Soc.London A 118, 557–583 (1928)
Burchnall, J.L., Chaundy, T.W.: Commutative ordinary differential operators. II. — The Identity P n = Q m. Proc. Roy. Soc. London A 134, 471–485 (1932)
Hellström, L.: Algebraic dependence of commuting differential operators. Disc. Math. 231, no. 1–3, 246–252 (2001)
Hellström, L., Silvestrov, S.D.: Commuting elements in q-deformed Heisenberg algebras.World Scientific, New Jersey (2000)
Hellström, L., Silvestrov, S.: Ergodipotent maps and commutativity of elements in non-commutative rings and algebras with twisted intertwining. J. Algebra 314, 17–41 (2007)
de Jeu, M., Svensson, P.C., Silvestrov, S.: Algebraic curves for commuting elements in the q-deformed Heisenberg algebra, arXiv:0710.2748v1 [math.RA], 17pp., to appear (2007)
Krichever, I.M.: Integration of non-linear equations by the methods of algebraic geometry.Funktz. Anal. Priloz. 11, no. 1, 15–31 (1977)
Krichever, I.M.: Methods of algebraic geometry in the theory of nonlinear equations. Uspekhi Mat. Nauk, 32, no. 6, 183–208 (1977)
Larsson, D., Silvestrov, S.D.: Burchnall—Chaundy theory for q-difference operators and q-deformed Heisenberg algebras. J. Nonlin. Math. Phys. 10, suppl. 2, 95–106 (2003)
Mumford, D.: An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg—de Vries equation and related non-linear equations. Proc.Int. Symp. on Algebraic Geometry, Kyoto, 115–153 (1978)
Nesterenko, Yu.: Modular functions and transcendence problems. C.R. Acad. Sci. Paris Ser. I Math. 322, no. 10, 909–914 (1996)
Nesterenko, Yu., Philippon, P.: Introduction to algebraic independence theory, Lecture Notes in Mathematics 1752. Springer, Berlin (2001)
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Silvestrov, S., Svensson, C., Jeu, M.d. (2009). Algebraic Dependence of Commuting Elements in Algebras. In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A. (eds) Generalized Lie Theory in Mathematics, Physics and Beyond. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85332-9_23
Download citation
DOI: https://doi.org/10.1007/978-3-540-85332-9_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-85331-2
Online ISBN: 978-3-540-85332-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)